Proof. Induct on the attachments. When a chart is attached at \(W\) in an old chart, its intersection with any other old chart is exactly that chart’s intersection with \(W\). By the inductive hypothesis this is a flat in the old chart containing \(W\) and in the other old chart, with agreeing affine structures. It is therefore a flat of the new chart as well. This proves the pairwise-intersection assertion. Trunk attachment flats have rank less than \(Dr<m\), and leaf attachment flats have rank at most \(m\), so every intersection of distinct charts has rank at most \(m\). Affine subflats of a shared flat are consequently identified consistently.

It follows that intersections of labels are again flats, allowing the empty set. Moreover, whenever one nonempty label is contained in another, it is an affine subflat in every chart containing the larger label. Thus \(L^+\) has finite meets and a top, which suffice to give all joins. A new proper upper bound of two old labels, not itself already an old flat, can contain those old labels only if both lie in the attachment flat \(W\); it then contains their old affine span in \(W\). Hence no old join changes. Meets are set intersections and do not change either. This also proves the claim about private charts, first at their attachment and then after all subsequent attachments. Within a chart the stated affine span is the least proper upper bound. The horizontal sum with the opposite lattice is a self-dual lattice.

Every proper chain in one orientation can be refined one affine dimension at a time, up to dimension \(n\), before reaching the formal top. Thus \(L\) is graded of length \(n+2\). A flat is generated by its points, so \(L^+\) is atomistic; intersecting its \(n\)-dimensional extensions inside one chart recovers the flat, so it is coatomistic. The same properties hold in the opposite orientation and hence in the horizontal sum.

The \(g\ge2\) initial charts are disjoint. An attachment meets all old points in a flat contained in one old chart; it cannot connect two old initial branches by comparabilities. Each orientation therefore has at least two horizontal interior components, and each component contains long affine chains. In particular \(L\) has at least three such components.

Finally, let a nonidentity lattice congruence identify two different comparable elements \(u<v\); such a pair exists whenever any two distinct elements are identified, by meeting and joining them. By atomisticity there is an atom \(a\le v\) with \(a\not\le u\). Meeting the identification with \(a\) gives \(a\equiv0\). Choose an interior element \(b\) in another component. Joining with \(b\) gives \(1\equiv b\). Meeting this with an interior element \(c\) in a third component gives \(c\equiv0\). Since \(a\vee c=1\), we conclude \(1\equiv0\). The congruence is total, proving simplicity. ◻

Proof. A leaf attached at \(X\) supplies a branch of strict superflats of \(X\) incomparable with the previous strict superflats. Later attachments cannot connect two old branches in this reverse link: any connection through a new chart passes through superflats of \(X\) lying in its attachment flat; if any exist, they are already connected within that flat. Every superflat of \(X\) lies in a chart containing \(X\), and in that chart it has extensions and intermediate flats. Each branch therefore contains comparable distinct elements and chains out to \(n\)-flats. Gradedness follows from these dimension chains. The flats extending \(X\) by one dimension, and the intersections of \(n\)-extensions, establish atomisticity and coatomisticity in the reversed order.

On the primal side of \(L\), a bottom interval of length two is a line, with \(b=q\) points; the choice of \(q\) gives the claimed condition on \(b-1\). On the reverse side, including a prepared link, its top flat has dimension \(n-1\). Such a flat belongs to exactly one chart, since distinct charts meet in rank at most \(m<n-2\). Its \(n\)-dimensional extensions are counted by one-dimensional subspaces of a vector space of dimension \(N-n+1\), so \[b=\frac{q^{N-n+1}-1}{q-1},\qquad b-1=q(1+q+\cdots+q^{N-n-1}).\] The second factor is greater than one because \(N\ge n+2\), and is congruent to \(1\) modulo the characteristic prime. Thus \(b-1\) has at least two prime divisors. This proves [f:link-diamonds].

For [f:link-upper-diamonds], choose a primal \((n-1)\)-flat, or, on the reverse side, a flat whose rank exceeds that of the empty flat or the prepared \(X\) by two. The appropriate upper interval is a diamond, with at least two middle flats. Such choices can be made in every branch.

For [f:link-separation], a primal atom is a point. In one chart through it there is a line escaping the given proper flat \(\omega\). A reverse atom is an \(n\)-flat, belonging to a unique chart; its proper upper bound corresponds to a nonempty subflat. Choose a hyperplane of the \(n\)-flat that does not contain that subflat. It is the required rank-two reverse label. ◻

Proof. Induct on the number of coordinates. Project away the last coordinate. The kernel in the last simple factor is normal in that factor, by surjectivity, so is either the whole factor or trivial. In the first case the last factor splits off. In the second, the subgroup is the graph of a surjective homomorphism from the shorter subdirect product to the last simple group. By induction the shorter group is a product of nonabelian simple diagonal factors. The images of those factors are commuting normal subgroups of a simple group; exactly one has nontrivial image, and on that factor the map is an isomorphism. This ties the last coordinate to just that part. The description of containment follows by projecting to pairs of coordinates: a required tie must hold in the smaller subgroup with the same identifying automorphism. Containing the untwisted full diagonal forces each such automorphism to be the identity. ◻

Proof. We begin with the least-carrier restriction argument of Pálfy and Pudlák (1980, Lemma 1). The final diagonal construction below will turn the resulting permutation action into an ordinary interval. Choose a representing algebra with smallest possible carrier \(A\). Let \(F\) be the monoid of all maps \(A\to A\) preserving its represented congruences. The unary algebra \((A;F)\) has exactly those congruences: each represented congruence is preserved by definition, while \(F\) contains all one-variable translations of the original operations, whose preservation forces an equivalence to be an original congruence.

Let \(e\in F\) be idempotent. On \(eA\) use the unary operations \(ef|_{eA}\), \(f\in F\). Restriction induces a surjective lattice homomorphism \[ \operatorname{Con}(A;F)\longrightarrow \operatorname{Con}(eA;\{ef|_{eA}:f\in F\}). \tag{5}\] Meets are preserved by restriction. To check joins, retract an equivalence path by \(e\): its vertices then lie in \(eA\) and each step remains in the same congruence. For surjectivity, if \(\delta\) is a congruence on the displayed algebra on \(eA\), define \[x\mathrel{\widehat\delta}x' \quad\Longleftrightarrow\quad (efx,efx')\in\delta\ \text{for every }f\in F.\] This is an \(F\)-invariant equivalence. Its restriction to \(eA\) is exactly \(\delta\): one direction uses its compatibility with all \(ef|_{eA}\), and the other uses \(f=1\) and \(e|_{eA}=1\).

If \(|eA|>1\), the target of (5) is nontrivial. Simplicity of \(L\) makes the homomorphism injective, and minimality of \(|A|\) forces equality of carrier sizes. A surjective idempotent is the identity. Thus every nonunit idempotent of \(F\) is constant.

Let \(I\) be the ideal of nonunits of \(F\). Every sufficiently long product of elements of \(I\) is constant. Indeed, otherwise take more than \(|F|\) successive prefixes in order of application of a product whose final image has size greater than one. Every prefix image then has size greater than one, and two prefix maps coincide. The intervening nonempty product fixes their common image pointwise. An idempotent power of that product is a nonunit idempotent fixing at least two points, a contradiction.

For \(\theta\in\operatorname{Con}(A;F)\) let \(\Phi(\theta)\) be the equivalence generated by \[\{(f(x),f(y)):f\in I,\ (x,y)\in\theta\}.\] This is a congruence, by the ideal property of \(I\), and \(\Phi(\theta)\le\theta\). It preserves joins: both equivalence generation and images of paths show that applying it to a join gives the join of the images. Iterating \(\Phi\) generates precisely the pairs obtained from products of the corresponding length in \(I\). The inserted equivalence closures contribute nothing extra, since an image of a path is a path; diagonal pairs accommodate collapsed steps. The product observation makes some iterate of \(\Phi\) identically zero. For an atom \(\alpha\), the deflation \(\Phi(\alpha)\le\alpha\) must therefore be zero, since the only other possibility would fix \(\alpha\) forever. Atomisticity now gives \(\Phi(1)=0\). Hence every map in \(I\) sends every pair to a diagonal pair: all nonunits are constant.

The units consequently form a permutation group \(J\) with exactly \(L\) as its lattice of invariant equivalences; constants impose no conditions. Choose any finite nonabelian simple group \(C\), and form the coordinate permutation semidirect product \(C^A\rtimes J\). Then \[ [\Delta(C)\rtimes J,\ C^A\rtimes J] \tag{6}\] is dual to that invariant-equivalence lattice. Indeed, intersection with \(C^A\) identifies this interval with the \(J\)-invariant subgroups above the full diagonal. By 2 these are the diagonal products determined by partitions of \(A\), and subgroup containment reverses refinement of the corresponding equivalences. Since \(L\) is self-dual, (6) represents \(L\) itself. ◻

Proof. If \(R\trianglelefteq G\), then \(HR\) permutes as a set with every overgroup of \(H\). Dedekind’s law consequently makes \(HR\) left-modular in \([H,G]\). There is no proper interior left-modular label: for a putative such label \(u\), choose comparable distinct interior labels \(v<w\) in another horizontal component. Then \[(v\vee u)\wedge w=w\ne v=v\vee(u\wedge w).\] Thus either \(R\le H\) or \(HR=G\). In the core quotient the first alternative forces \(R=1\).

An abelian minimal normal supplement would identify the interval with an invariant-subgroup interval in that abelian group. Such an interval is modular, contradicting the displayed failure of the modular identity. If two distinct minimal normal subgroups existed, they would commute. Using one to supplement \(H\), the conjugation action of \(H\) on the other would induce all its inner actions. The interval would again identify with a modular interval, now of invariant normal subgroups of that other minimal normal subgroup. Both possibilities are impossible. The unique minimal normal subgroup is therefore a power \(S^s\) of a nonabelian simple group. Its factor orbits under \(H\) are normal in \(HM=G\), so minimality makes the action transitive. Its centralizer is normal; if nontrivial it would contain \(M\), which is impossible because \(M\) is centerless. Finally \(HM=G\), and the elementary intersection–product correspondence gives (7). ◻

Proof. By 2, a subdirect group has tied diagonal parts that form an \(H\)-invariant partition of the coordinates. The part containing \(o\) is a block for the transitive action; it equals \(To\), where \(T\) is its setwise stabilizer and \(A_0\le T\). The action of \(T\) on that diagonal, identified with \(S\) by projection to \(o\), gives \(\beta\) extending \(\alpha\).

Conversely, given (8), define the diagonal on \(To\) using left-action notation. At coordinate \(ho\), for \(h\in T\), send \(u\in S\) to the transport by \(h\) of \(\beta(h)^{-1}(u)\) from coordinate \(o\). Replacing \(h\) by \(ha\), with \(a\in A_0\), gives the same element because \(\beta(a)=\alpha(a)\). This defines a \(T\)-invariant full diagonal. Transport it to the other blocks and multiply the resulting disjoint diagonal factors. The result is \(H\)-invariant and subdirect, and recovers the original data. Refinement of tied parts, with agreement of identifications, is exactly restriction of the extension. The last assertion follows because such restrictions give larger actual subgroups, still containing \(P\). ◻

Proof. We check all the cases, retaining the prepared-interval hypotheses.

Subdirect bottom. If \(c(P)=S^s\), every label is subdirect and refines the fixed diagonal ties of \(P\). Restrict to one tied part of \(P\). Its setwise stabilizer in \(H\) acts transitively on its coordinates. A refinement of its coordinates that is invariant under this stabilizer extends uniquely to the other parts by \(H\), keeping the twists fixed by \(P\). Thus the interval is dual to the invariant-equivalence lattice of that transitive action. By 1, the latter is an ordinary interval above a point stabilizer in the part stabilizer. Its top has order at most \(|H|<|G|\), proving [f:case-subdirect].

A proper nonproduct bottom. Suppose \(P<c(P)<S^s\). Choose a horizontal interior component different from that of \(c(P)\). Every label \(Y\) in it has \(Y\vee c(P)=S^s\) and \(Y\cap c(P)=P\). Its coordinate projections already contain those of \(P\), so adjoining \(c(P)\) does not enlarge them. The join condition therefore makes \(Y\) subdirect. In its diagonal description, intersection with \(c(P)\) restricts each coordinate to the corresponding nontrivial projection of \(P\), with the same ties. These restrictions project onto every such subgroup, because the intersection contains \(P\). There are comparable distinct labels in the chosen component. Their strict containment changes at least one diagonal tie, and releasing a tie strictly enlarges the intersection: its tied coordinate subgroup is nontrivial and may now be chosen independently on the separated parts. This contradicts the fact that both intersections equal \(P\).

A nontrivial product bottom. Suppose \(P=c(P)\ne1\), necessarily a proper product. A subdirect group containing independent nontrivial subgroups in each coordinate cannot have any nontrivial ties. Thus the only subdirect label above \(P\) is \(S^s\). For a coatom \(Y\), the product \(c(Y)\) is a label containing \(Y\); maximality and the preceding observation imply \(Y=c(Y)\). Since the interval is coatomistic, every label is a product. At one coordinate let \(Q\) be the projection of \(P\). Then \(1<Q\le I\): the coordinate elements of \(P\le H\) induce their inner automorphisms. Products identify the interval with \([Q,S]^\Lambda\). For any coatom \(T\) of this invariant interval, \(I\) normalizes \(T\). If \(TI=S\), then \(T\) is normal in \(TI\), hence in \(S\), impossible because \(1<Q\le T<S\). Thus \(TI<S\), and maximality gives \(I\le T\). Intersecting all coatoms yields \(I\le Q\), so \(I=Q\). Intersection with \(S\) and multiplication by \(\Lambda\) now identify \([Q,S]^\Lambda\) with \([\Lambda,S\Lambda]\) inside \(\operatorname{Aut}(S)\), proving [f:case-product].

Trivial bottom and no inner operators. Suppose \(P=1\) and \(I=1\). Schreier’s Theorem makes \(\Lambda\) soluble, since it embeds in \(\operatorname{Out}(S)\). We first show that a soluble operator group on a finite nonabelian simple group has a nontrivial proper invariant subgroup. We may replace it by its effective image. If that image is trivial, choose any subgroup of prime order. Otherwise choose a minimal normal elementary abelian \(p\)-subgroup \(A\) of the operator group. Its fixed group is invariant, and if nontrivial it is proper by effectiveness. If its fixed group is trivial, orbit counting on the simple group gives \(p\nmid|S|\). For each prime divisor \(\ell\) of \(|S|\), the number of Sylow \(\ell\)-subgroups divides \(|S|\) and is prime to \(p\), so the \(p\)-group \(A\) fixes at least one such Sylow subgroup. There is at most one: the set of conjugators between two \(A\)-invariant Sylow subgroups is \(A\)-invariant and has size \(|N_S(P_\ell)|\), prime to \(p\), so has an \(A\)-fixed conjugator. The trivial fixed group forces that conjugator to be \(1\). Uniqueness makes the Sylow subgroup invariant under the whole operator group. This proves the assertion.

Choose a minimal nontrivial \(\Lambda\)-invariant subgroup \(Q\) of \(S\). Its characteristic subgroups are invariant, so \(Q\) is characteristically simple. If it were a power of a nonabelian simple group, apply the preceding assertion to the soluble operator image of a factor stabilizer, and transport the resulting proper invariant subgroup through the transitive factor action. This would give a nontrivial proper invariant subgroup of \(Q\), a contradiction. Therefore \(Q\) is elementary abelian. Its normalizer is proper because \(S\) is simple. It strictly contains \(Q\): if \(Q\) is not Sylow, use the normalizer property in a Sylow subgroup containing it; if it is Sylow and were self-normalizing, Burnside transfer would give a normal prime complement, again contradicting simplicity.

Transport \(Q\) and \(N_S(Q)\) to all coordinates and take their products. They are labels, the latter strictly above the former. If the product of the \(Q\)’s has lattice rank at least two, choose a rank-two label below it and any atom in that interval. Otherwise the product itself is an atom; choose a rank-two label above it and below the normalizer product. In both cases the chosen atom \(A\) is an elementary abelian normal subgroup of the rank-two group \(R\). Every other atom is an \(H\)-invariant complement to \(A\) in \(R\), and conversely every such complement is an atom: intersection with \(A\) is trivial, and the join with \(A\) is \(R\) in the diamond. Fix one invariant complement \(C\). In the semidirect product \(R=A\rtimes C\), other complements are graphs of derivations \(C\to A\). Since \(H\) preserves both factors, the invariant complements correspond precisely to the \(H\)-equivariant derivations. These form a vector space over the prime field of \(A\). Their number is a prime power, allowing the zeroth power. This is the operator-invariant version of the complement-counting argument in Pálfy and Pudlák (1980, Theorem 3). The number equals \(b-1\) in the bottom diamond \(M_b\), contradicting 3[f:link-diamonds].

Trivial bottom and proper nontrivial inner operators. Suppose \(P=1\) and \(1<I<S\). Transporting \(I\) gives a proper interior product label. A label \(Y\) in another horizontal component is subdirect: its nontrivial coordinate projection \(R\) satisfies \(RI=S\), because its join with the product of the \(I\)’s is full. The subgroup \(I\) normalizes \(R\), so \(R\trianglelefteq RI=S\) and hence \(R=S\). Write \((T,\beta)\) for the extension describing \(Y\) in 3. Its image contains all inner automorphisms. Otherwise \(\beta(T)\cap S\) is a nontrivial proper invariant subgroup of the diagonal, containing \(I\). Transporting it over the tied parts and multiplying gives a nonbottom label below \(Y\) with proper coordinate projections. It would lie in the same component, contradicting the preceding conclusion about all its labels.

Choose \(y\) in this component two below the top with a diamond above it. The filter above \(y\) identifies dually, by restriction of \(\beta_y\), with \([A_0,T_y]\). Every domain strictly larger than \(A_0\) has image containing \(S\). Apply this observation to \(A_0\ker\beta_y\): its image is still \(\Lambda\), which does not contain \(S\). Hence \(\ker\beta_y\le A_0\). After quotienting by this kernel, every proper increase over \(\Lambda\) contains \(S\Lambda\), so there is a unique minimal proper increase. This contradicts the diamond’s having at least two middle elements.

Trivial bottom and all inner operators. Finally, if \(P=1\) and \(I=S\), every nonbottom label has nontrivial coordinate projections invariant under all inners, so those projections equal \(S\). The nonbottom labels are exactly the invariant subdirect groups, and 3 identifies their dual with the extension poset. The bottom contributes the artificial top. This proves [f:case-extension] and completes the case analysis. ◻

Proof. Subextensions of pure extensions are pure. If finitely many pure labels have a common extension, their join is its restriction to the subgroup generated by their domains, and is pure as well: its image is generated by copies of \(\psi(H)\).

Suppose an atom \((T,\beta)\) is not pure. The intermediate restriction to \(H\ker\beta\) is pure, so atomhood forces \(H\ker\beta=H\) and \(\ker\beta=D_0\). Since \(H\) already maps onto the inner subgroup \(B\), it also follows that \(\beta^{-1}(B)=P_0\). Thus \(T\) normalizes both \(P_0\) and \(D_0\). The simultaneous normalizer \[N_K(P_0)\cap N_K(D_0)\] acts on \(P_0/D_0\cong B\). Its conjugation homomorphism restricts to \(\psi\) on \(H\), since action on the identified inner automorphisms determines an automorphism of the centerless group \(B\). On each nonpure atom it similarly agrees with \(\beta\). This gives one extension label \(\omega\), strictly below the artificial top, bounding every nonpure atom.

Choose a nonpure atom \(u\). By 3[f:link-separation], there is a rank-two label \(z\ge u\) not below \(\omega\). At most one atom of its diamond lies below \(\omega\), since two distinct atoms join to \(z\). The diamond has at least three atoms, so at least two are pure. Their proper join is \(z\), which is consequently pure; its subextension \(u\) is pure, a contradiction. All atoms, and hence all proper labels by atomisticity and purity of proper joins, are pure.

A pure extension has a kernel satisfying (9) and domain \(HE\). Conversely, such a subgroup \(E\) is normal in \(HE\), and \(HE/E\cong H/D_0\) defines the unique extension killing \(E\) and agreeing with \(\psi\). This proves the parametrization. If \(H\le U\le HE\), then \(U=H(U\cap E)\), so restricting the extension identifies its lower ideal with all of \([H,HE]\). For a proper join, restriction of a common extension to the generated domain shows that the kernel is the subgroup generated by the kernels: their product with \(H\) is the generated domain, and their intersection with \(H\) is \(D_0\). If two labels instead join to the artificial top and their generated kernel \(E\) still met \(H\) in \(D_0\), the same construction would give a common proper extension, impossible. Therefore \(E\cap H\) properly contains \(D_0\). It is normal in \(H\), so its image is a nontrivial normal subgroup of the almost simple group \(\psi(H)\); it contains \(B\). As \(D_0\le E\), this implies \(P_0\le E\). ◻

Proof. We organize the argument into the compatible monolith embeddings, the selection of a plane with small cores, and the final factor action.

Monolith embeddings. For \(i\in X\) put \[ Z_{X,i}=(T_X\cap V_i)/L_i\le W_i. \tag{10}\] The base \(H\) is transitive on the simple factors of \(W_i\). It is therefore transitive on the tied diagonal factors of its invariant subdirect group \(Z_{X,i}\). This makes \(Z_{X,i}\) a nontrivial minimal normal subgroup of \[T_X/L_i=(H/L_i)Z_{X,i}.\] It is not contained in \(H/L_i\), by the strictness in the hypothesis. Since \(T_X\le T_i\), we have \(L_i\le L_X\). Minimal normality and \(L_X\le H\) now give \(L_X\cap V_i=L_i\). Thus the image of \(Z_{X,i}\) in \(T_X/L_X\) is still injective and minimal normal, so equals \(W_X\). We obtain a canonical \(H\)-equivariant subdirect injection \(f_{X,i}:W_X\to W_i\), and \[ V_X=(T_X\cap V_i)L_X. \tag{11}\]

For \(i\in X\subseteq Y\) we have \(H\le T_Y\le T_X\le T_i\) and \(L_i\le L_X\le L_Y\). Intersecting (11) with \(T_Y\) gives \[V_X\cap T_Y=(V_i\cap T_Y)L_X,\qquad V_Y=(V_X\cap T_Y)L_Y,\qquad L_Y\cap V_X=L_X.\] The last equality follows also by using \(L_Y\cap V_i=L_i\). These formulas give injections \(f_{Y,X}:W_Y\to W_X\) composing with the \(f_{X,i}\) as expected: lift representatives in \(V_i\cap T_Y\). They are subdirect. Indeed, every simple factor of \(W_X\) is visible by projection to some simple coordinate of \(W_i\) through \(f_{X,i}\), and the composite image of \(W_Y\) projects onto every such coordinate. The kernel of the action of \(H\) on \(W_X\) is exactly \(L_X\), by the trivial centralizer of the monolith in \(T_X/L_X\).

If \(X\cap Y=\{i\}\), the proper-join condition gives \(T_i=\langle T_X,T_Y\rangle\). Writing each domain as \(H\) times its intersection with \(V_i\) shows that \[ W_i=\langle Z_{X,i},Z_{Y,i}\rangle, \qquad L_i=L_X\cap L_Y. \tag{12}\] For completeness, the subgroup generated by the two \(Z\)’s contains \((H\cap V_i)/L_i\), is normalized by \(H/L_i\), and multiplying it by \(H/L_i\) gives \(T_i/L_i\). Intersecting that product with \(W_i\) therefore recovers all of \(W_i\). An element of \(H\) acts trivially on this generated group exactly when it acts trivially on both embedded monoliths; this proves the asserted equality of cores.

A plane whose cores are killed by \(\psi\). For any affine plane \(U\), write \[L_* =\bigcap_{i\in U}L_i.\] Every line domain \(T_Y\), for a line \(Y\subset U\), normalizes \(L_*\). To see this, fix a target point \(j\notin Y\) and choose two distinct lines through \(j\) meeting \(Y\), at points \(k_1,k_2\). An element of \(T_Y\le T_{k_t}\) normalizes \(L_{k_t}\), so sends \(L_*\le L_{k_t}\) into the core of the line \(jk_t\), because \(L_{k_t}\le L_{jk_t}\). Equation (12) for those two lines puts the resulting image inside \(L_j\). For \(j\in Y\), use \(T_Y\le T_j\) directly. The same argument for inverses proves normalization. Two distinct lines through a point generate its domain, so \(L_*\) is normal in the group generated by all point domains of \(U\).

Since \(L_*\le H\), we have \(L_*\cap E_i=L_*\cap D_0\) for every point \(i\). Both \(E_i\) and \(L_*\) are normal in \(T_i\), so \([E_i,L_*]\le L_*\cap D_0\). Thus every \(E_i\) centralizes \(L_* /(L_*\cap D_0)\). The group generated by kernels of distinct points contains \(P_0\), by hypothesis. Therefore \(\psi(L_*)\) centralizes \(B\), whose centralizer in \(\operatorname{Aut}(B)\) is trivial. Hence \[ L_*\le D_0. \tag{13}\]

There is a point \(i\) with \(L_i\le D_0\). Otherwise every \(\psi(L_i)\) is a nontrivial normal subgroup of \(\psi(H)\) and contains \(B\). Take an iterated commutator of the original normal subgroups \(L_i\) inside \(H\). It lies in \(\bigcap_{i\in U}L_i\) and its \(\psi\)-image still contains \(B\), because \(B\) is perfect. That contradicts (13). Choose two affine planes in the four-space meeting exactly at this point \(i\). Their cores intersect in \(L_i\) by (12), so their two core images cannot both contain \(B\), by the same commutator argument. Hence one of these planes, henceforth denoted \(U\), has \(L_U\le D_0\). All point cores of \(U\) then lie in \(D_0\), since \(L_i\le L_U\) for \(i\in U\).

Quotient the group generated by the point domains of \(U\) by \(\bigcap_{i\in U}L_i\). This subgroup is normal there as shown above, lies in the base, and is contained in all the point cores and all extension kernels in the plane. The core in each quotient is the quotient of the old core. Thus the inclusions, join statements, and monolith identifications survive. We retain the notation and may assume \[ \bigcap_{i\in U}L_i=1,\qquad L_U\le D_0. \tag{14}\]

Compatible perfect subgroups. For a nonempty affine subflat \(X\subseteq U\) set \[ M_X=E_X\cap\bigcap_{i\in X}V_i. \tag{15}\] Every factor \(V_i\cap T_X\) is normal in \(T_X\), as is \(E_X\), so \(M_X\) is normal in \(T_X\). It lies in \(V_X\) by (11). It maps onto \(W_X\): each \(V_i\cap T_X\) maps onto \(W_X\), while the normal image of \(E_X\) modulo \(L_X\) is nontrivial, since \(HE_X=T_X>H\) and \(L_X\le H\), and therefore contains the monolith. Taking nested commutators of these finitely many normal subgroups gives a subgroup in their intersection whose image still contains \(W_X\), by perfection of \(W_X\). As \(M_X\le V_X\), its image is exactly \(W_X\). The supplement condition then yields \[ HM_X=T_X,\qquad D_0M_X=E_X. \tag{16}\] The second equality follows by intersecting the first with \(E_X\).

For \(X=U\), the kernel of \(M_U\to W_U\) is contained in every \(L_i\): it lies in \(L_U\cap V_i=L_i\). Equation (14) makes it trivial. Consequently \(M_U\cong W_U\) is an embedded product of nonabelian simple groups; write its factors as \(C_a\). The normal subgroups \(L_U\) and \(M_U\) of \(T_U\) have trivial intersection, so \(L_U\) centralizes every \(C_a\).

The subdirect embedding \(f_{U,X}\) assigns to each factor \(C_a\) a disjoint set of factors of \(W_X\) on which it has its tied diagonal, and these sets partition all factors of \(W_X\). Let \(M_{X,a}\) be the preimage in \(M_X\) of the entire product of \(W_X\)-factors assigned to \(a\). These groups generate \(M_X\), are permuted by \(H\), and each is normalized by \(L_U\): the latter centralizes \(M_U\), so fixes its factors and their assigned sets. For \(i\in X\), the images of \(M_{X,a}\) in \(W_i\) have support only over the same index \(a\). For \(a\ne b\) it follows that \[ [M_{X,a},C_b]\le L_i\le L_U. \tag{17}\] In fact \(M_{X,a}\) centralizes \(C_b\). Since \(L_U\) centralizes \(C_b\), (17) and the three-subgroups lemma imply \([M_{X,a},[C_b,C_b]]=1\); perfection gives the claim. Likewise, for meeting lines \(X,Y\) (allowing \(X=Y\)) and \(a\ne b\), a common point \(i\) gives \[ [M_{X,a},M_{Y,b}]\le L_i\le L_U. \tag{18}\]

The final action on the simple factors. Choose four pairwise nonparallel lines in \(U\), two through a point \(i\) and two through a distinct point \(j\). There are at least four directions, since the field has at least three elements; the lines can be chosen with the required distinct directions. Set \[R'=\langle M_Y:Y\text{ is one of these four lines}\rangle.\] It is normalized by \(H\). The two proper point joins and then the incompatible join of \(i,j\), together with (16), imply \(P_0\le D_0R'\). Hence \[ \psi(P_0\cap R')=B. \tag{19}\]

It remains to show that this image is soluble. We first show that \(P_0\cap R'\) fixes every factor \(C_a\). Schreier’s Theorem then controls its outer action, and we show that \(\psi\) kills the subgroup inducing inner automorphisms.

Fix an index \(b\). A word in the generating groups \(M_{Y,a}\) can be reordered as \[\ell u v,\] where \(\ell\in L_U\), all letters of \(u\) have indices \(a\ne b\), and all letters of \(v\) have index \(b\). For adjacent letters in the wrong order, put \(e=xyx^{-1}y^{-1}\in L_U\), by (18). If \(p\) is the preceding word, then \[pxy=peyx=e(e^{-1}pe)yx.\] Conjugating the preceding letters by \(e\) preserves their indices because \(L_U\) normalizes every indexed generating subgroup. Thus the swap reduces the inversion count. Induction gives the required reordering.

In right-conjugation notation, \(C_b^{\ell u v}=C_b^v\), since \(\ell\) and \(u\) centralize \(C_b\). The element \(v\) centralizes every other \(C_a\), and \(C_b\) already commutes with them, so \(C_b^v\) commutes with every \(C_a\) for \(a\ne b\). If the word lies in \(P_0\cap R'\le H\), it permutes the factors of \(M_U\); its image of \(C_b\) cannot be another \(C_a\), because that would make a nonabelian simple group commute with itself. It must therefore be \(C_b\), proving the assertion.

The action of \(P_0\cap R'\) on \(W_U\cong M_U\) consequently has soluble quotient modulo its inner action, by Schreier’s Theorem on each fixed simple factor. An element inducing an inner action on \(W_U\) lies in \(V_U\): after multiplying by the corresponding monolith element its image in \(T_U/L_U\) centralizes \(W_U\), and that centralizer is trivial. On the other hand, purity and \(L_U\le D_0\) give \[H\cap V_U\le H\cap(E_UL_U)\le D_0L_U=D_0,\] because the nontrivial normal image of \(E_U\) contains \(W_U\) and hence \(V_U\le E_UL_U\). Thus the inner-action kernel within \(P_0\cap R'\) is killed by \(\psi\). Consequently \(\psi(P_0\cap R')\) is a quotient of a soluble group, contradicting (19) and the nonabelian simplicity of \(B\). ◻

Proof. Choose the least-top-order ordinary representation furnished by 4. The reduction after 6 either already gives the whole \(L\) as an ordinary almost simple interval, or gives an extension poset with artificial top isomorphic to \(L\). In the latter case all proper labels are pure by 4. For each nonempty reverse flat \(X\), write \(E_X,T_X\) for its kernel and domain. Its lower ideal is the ordinary interval \([H,T_X]\).

Take an affine four-subspace of one initial trunk chart and consider all its points, lines, and planes. Their ranks are at most three, hence at most \(m\), so all their reverse links are prepared. The proper and incompatible joins have exactly the domain and kernel properties of 5, by 4. Each ordinary domain interval has the normal-supplement and monolith properties of 5. If at some point \(i\) the nontrivial-product case [f:case-product] of 6 occurs, it gives the required almost simple representation of that point’s link. Otherwise, the remaining possibilities at every point are the subdirect-bottom and the trivial-bottom full-inner cases. In both, every domain label strictly above the base intersects the point monolith subdirectly modulo the core, as required in 5. The strictness in that lemma follows from \(HV_i=T_i\): every incident domain satisfies \(T_X=H(T_X\cap V_i)\), so \(T_X>H\) gives strict containment above the point-base intersection after quotienting by \(L_i\). That lemma gives a contradiction.

For the whole lattice take the empty reverse flat as the starting top. For a point in an initial trunk chart, its rank increment from the empty flat is one, which lies between \(1\) and \(r\); its child path was therefore reserved at the first stage. In either case at least \(D-1\) generations remain by the construction. ◻

Proof. The quotient map is faithful on \(B_W\), since \(B_W\cap C_W=1\). Its inverse on the image of \(B_W\) defines \(B\), which lies in \(K\) and is normal there. The supplement identity lifts because \(C_W\le H\). A lift of an element centralizing \(\overline B\) centralizes \(B\): its commutators with \(B\) lie both in \(B\) and in \(C_W\), so are trivial. Hence the new centralizer is the inverse image of the quotient centralizer and is soluble and contained in \(H\). Quotienting by it gives a faithful automorphism group of \(B\) containing its inner automorphisms.

For an earlier node \(W'\), the inclusion \(B\le B_W\le B_{W'}\) makes its image faithful in \(K/C_{W'}\). The same commutator argument identifies its centralizer there with \(C_K(B)/C_{W'}\), which is soluble. Distinct nonabelian simple normal subgroups would commute, contradicting that solubility. Thus the simple supplied relative to \(W'\) must be this image. All these quotient maps preserve the exact interval, since their kernels lie in \(H\).

For comparable candidates with \(K_U\le K_L\), the simple \(B_U\) acts on \(B_L\). Schreier’s Theorem makes its outer action trivial, so \(B_U\le B_L C_{K_L}(B_L)\). This is a direct product because \(B_L\) is centerless. Projection of \(B_U\) onto the soluble centralizer is trivial, giving \(B_U\le B_L\). If the labels differ, their simples differ because \(HB_U=K_U\) and \(HB_L=K_L\). ◻

Proof. The proper full stabilizer contains \(K_U\), so its label lies between \(K_U\) and \(K_{W'}\). It is therefore a subflat \(X\) of \(U\) containing \(W'\). If \(X\subseteq W\), its subgroup contains \(K_W\), contrary to the earlier exclusions; for \(W'=W\), properness gives the same conclusion. Such an \(X\) is nonempty and lies in the trunk chart containing \(U\), so 3 applies. Finally \(E<K_U\le K_X\). ◻

Proof. Choose a fixed model for the horizontal increment and many unused vertical coordinates, with coordinate space \(V_{\rm vert}\). For nonempty \(W\), take an origin in \(W\) and graph linear maps from its \(s\) new horizontal directions to the vertical directions; these maps vanish on \(W\). For empty \(W\), use \(s-1\) horizontal affine dimensions and graph affine maps instead. Every graph is an increment-\(s\) test flat.

Color each map by its witness’s failure color and the pullback of \(X(U)\) into the fixed horizontal model. There are finitely many colors for the fixed horizontal data. View the maps as words whose letters are individual-coordinate linear or affine functionals. Multidimensional Hales–Jewett (Hales and Jewett 1963) gives a monochromatic family with fixed coordinates and any prescribed number \(k\) of disjoint nonempty variable batches. Thus the pulled-back witness is one fixed flat \(X_{\rm hor}\not\subseteq W\).

Choose a horizontal functional \(\lambda\) nonzero at some point of \(X_{\rm hor}\); in the nonempty case it vanishes on \(W\). Restrict the functional in each variable batch to a scalar multiple of \(\lambda\). Let \(\iota:\mathbb F_q^k\to V_{\rm vert}\) send each parameter coordinate to the sum of the vertical basis vectors in its batch. The disjoint nonempty batches make \(\iota\) injective. The witness family is \[ X_t=\{(x,f_0(x)+\lambda(x)\iota(t)):x\in X_{\rm hor}\}, \qquad t\in\mathbb F_q^k, \tag{20}\] where \(f_0\) vanishes on the variable batches and records the fixed graph coordinates elsewhere. At a point where \(\lambda\ne0\), distinct parameters have different vertical coordinates, so the witness flats are distinct. The span of two graph flats adds exactly the vertical direction \(\iota(t'-t)\), and therefore contains the witnesses on the parameter line. More generally, \[ \operatorname{rk}\langle X_{t_0},\ldots,X_{t_h}\rangle =\operatorname{rk}X_{\rm hor}+ \dim\operatorname{aff}\{t_0,\ldots,t_h\}. \tag{21}\] The chart join rule converts these spans, when proper, into the asserted subgroup intersections and covers. Pair spans are proper because the trial ranks are below \(m<n-2\). The anchor inclusion follows from the round setup. Across private charts the witness flats lie outside \(W\) and have no common proper primal upper bound, so their reverse meet is the base \(H\).

Taking sufficiently many parameter directions supplies the first-round family of more labels than the interval height. Only pair spans and the bounded independent lists used later must fit the truncation; the span of the whole family need not be proper. In subsequent rounds the number of parameter directions is bounded independently of \(n\). ◻

Proof. Use a simultaneous initial round for all initial classical subspace exclusions relative to any classical ancestors (the ones allowing a more-than-height requirement); their pattern proofs need no earlier anchor. The remaining categories of the structure arguments can then be ordered as there, requiring only bounded line/orbit or independent-list bounds: e.g. direct line systems and then other direct systems, prime fields then ordered factors for linear type, partial ordered factors then prime fields then paired idempotents and remaining ordered factors for isometry types, elementary subgroups then tensor systems; in alternating ancestors subsets and block partitions then affine, Cartesian and diagonal structures. Subcases rendered impossible by an already forbidden structure need no round of their own. There are only boundedly many categories/ancestor instances altogether for bounded path length. We thus obtain a pruned flat of final increment \(r\) from \(W\); all labels at proper extensions within it avoid the exclusions, giving the needed next almost simple socles by the structural arguments, and giving transitivity or natural full spanning as applicable relative also to earlier ancestors. Selection of which extension to take (additional closure-field tensor/root protections within a same-characteristic run) uses 26. If the current parent instead is of bounded Lie rank, use 23 (same graph pattern with a bounded independent list and no prior anchor, relative just to this parent). ◻

Proof. Here is one parameter order, always finite. Fix the high thresholds, the resulting bounded-type constants, and the absolute bound on protected high-type path lengths from the group arguments. Choose \(D\) exceeding this bound by several stages (allowing the displaced point start and a stage for the bounded-type contradiction). Choose \(q\), a power of 7 for instance, above all absolute line-count requirements. Choose \(r\) as in the final flag selection, large for bounded-type requirements and the exponential-gap union bound over at most \(D\) steps, so the path ranks are at most \(Dr\). Choose larger intermediate trial increments backwards from \(r\), the increment after the first round thus a constant independent of \(m,n\): each preceding increment merely must allow the Hales–Jewett bound for the next bounded-slot round, at any path rank up to \(Dr\). Choose \(m\) above the ranks of these trials, then \(n\) sufficiently larger, including for residual interval length below all of them and bounded required spans. Now \(N\) can allow the first-round Hales–Jewett bound including the larger initial slot requirement depending on \(n\), and any no-anchor bounded-type trial. Finally take \(g\) sufficiently large for the failed-chart bounds at every stage. Thus the separate arguments imply contradiction on this finite lattice: either a bounded-type parent gives the direct contradiction, or too long a protected path is forced. ◻

Proof. For properness throughout, the high-rank covering matrix group of \(S\) is perfect and absolutely irreducible. It cannot preserve the initial subspace, signed-flag or individual-projector structures below (its own action is linear), nor even an adjoint-exchanged complementary pair of projectors up to swap in isometry type (perfection). It cannot normalize a proper direct-sum system below: irreducibility would force a uniform split by transitivity, then apply the direct-split order comparison below on \(V\) itself. Similarly use the tensor order comparison for ordered tensor factors or tensor systems; normalizing a field would force centralizing it by perfection, and a nontrivial elementary abelian subgroup inside \(S\) cannot be normal there. Thus when an earlier exclusion is invoked for a structure found using an anchor, its full stabilizer is indeed proper. ◻

Proof. First suppose linear type, without anti-actions. Exclude subspace stabilizers of nonzero proper \(U\). In each such label the shears from \(V/U\) to \(U\) are in the normal unipotent radical, hence in \(H\). Any two subspaces in the pattern must therefore be comparable. Stabilizers of a descending list along that chain have strictly descending intersections (special linear maps inside the smallest previous space can move a new proper subspace). The total interval height bounds their number, contradiction.

If \(M\) in linear type contains anti-actions, so does \(H\); fix one, \(a\in H\). Exclude the signed flag stabilizers for \(0<I\le J<V\) (automorphisms fix the two terms, anti-actions exchange them). For labels containing \(H\), \(J=aI\). Again the normal unipotent radical of the corresponding parabolic in \(S\) (preserved also by signed anti-actions) contains the shears from \(V/I\) to \(I\), and is in \(H\). Thus the \(I\)’s form a chain, the \(J\)’s the reversed chain above it. Intersections are strict taking descending \(I\)’s, just as before.

Also exclude stabilizers of individual nontrivial proper idempotents \(e\) (in the linear case with anti-actions). The two supported special linear groups on the complementary spaces are individually normal, commute projectively, and the quotient over their product is soluble. Thus exactly one is retained by (\(*\)); designate the retained space \(U\), with complement determined by \(a\) (anti-action on an idempotent sends its image to the subspace-dual transform of its kernel). In a pattern with \(\dim U\ge2\), the supported group acts irreducibly nontrivially there and fixes the complement pointwise. Any other retained space, being \(H^+\)-invariant, projects to zero along the complement or contains \(U\) (use differences under the supported group). By mutual irreducibility and distinctness, all retained spaces are pairwise in one another’s complements. Their projectors are thus orthogonal idempotents in the ring sense. Intersecting the stabilizers strictly decreases until possibly the last one, by taking special linear maps on the unsplit remainder. Again this is height-bounded.

The retained line case. For the color with retained dimension 1, take a cross-chart pair. There is a subspace \(B_0\) of codimension at most 4 in both designated complementary hyperplanes (that is, contained in both and of codimension at most 4 in \(V\)), disjoint from the span of both image lines; take a complement \(C_0\) containing both lines. The supported \({\rm SL}(B_0)\), fixing \(C_0\), belongs to \(K_i\cap K_j=H\). Thus every designated line in these patterns lies in \(C_0\) and every corresponding hyperplane contains \(B_0\), by large-dimension irreducibility. Both the total line span and the total span of dual functional lines have bounded dimension. Now a boundedly long list of independent pattern points already contradicts strict descent of intersections. Indeed descent must be strict also on intersection with \(S\), since \(HS=M\) and all groups contain \(H\). Track the growing two spans along the list; they each increase only boundedly often. Between increases, invariance is already imposed on the current spans, and further constraints are linear conditions on their endomorphism spaces (restrict matrices to the first and their transposes to the second). Each strict descent shrinks the solution subspace, by testing a witness matrix above the descent in \(S\). This bounds the length absolutely.

These exclusions imply irreducibility of the collineation part: if it preserves \(U\), then with anti-actions also \(aU\); the full group in question preserves these two with signed exchange. Nonzero intersection gives a signed flag (intersection and sum), and zero intersection a stabilized idempotent. ◻

Proof. For isometry types first exclude stabilizers of nonzero totally singular subspaces \(U\) (totally isotropic for unitary or symplectic). The standard unipotent radical is again in \(H\). Here are details of the resulting comparability. Use Witt coordinates \(U\oplus Z\oplus U^-\) with last coordinate \(\lambda\) paired dually with \(U\). The radical acts by \[(u,z,\lambda)\mapsto (u+f(z)+h\lambda,\ z+g\lambda,\ \lambda)\] with arbitrary \(f\) and accordingly adjoint \(g\), and a form-preserving correction \(h\). Such corrections exist by representing the quadratic correction as the diagonal of a bilinear pairing (orthogonal), filling an alternating correction (symplectic), or by surjectivity of quadratic-field trace on the diagonal in the hermitian case. With \(f=g=0\) the available \(h\) include the symmetric shears in symplectic type, skew-hermitian in unitary type, alternating in orthogonal type. These unipotent groups are inside the defining special groups (the additional determinant/spinor restrictions have order prime to the characteristic in the relevant unitary or odd-characteristic orthogonal cases). In orthogonal characteristic two the Dickson rank parity is even: the rank of the displacement is \(2\operatorname{rank}g\) plus the rank induced by \(h\) from \(\ker g\) to its dual \(U/{\rm im}\,f\), and the latter is alternating by the quadratic isometry condition.

Another invariant totally singular space within \(U^\perp\), if not within \(U\), must contain \(U\) by the arbitrary \(f\). A vector in such an invariant space with \(\lambda\ne0\) outside \(U^\perp\) is impossible in unitary/symplectic type by the pure \(h\)-shears (one can have nonzero pairing with \(h\lambda\)). In orthogonal type it forces the space to contain the annihilator hyperplane of \(\lambda\) in \(U\); also \(Z=0\), else choose a shear with nonsingular \(g\lambda\) and take the difference vector. Thus the only incomparable possibility consists of two split maximal singular spaces adjacent in a hyperplane. At most two distinct maxima can so occur (through their common hyperplane there are just two; a third adjacent to both and not through it contains two vectors with nonzero pairing modulo that hyperplane). The rest are in a chain. Intersections down a descending chain of singular spaces are strict by elementary special linear transvections inside the smallest previous singular space preserving the larger flag, extended dually to isometries (also in \(S\)). This again contradicts the pattern size.

Next exclude stabilizers of proper nonzero nondegenerate subspaces in isometry types. On the two orthogonal parts there are commuting normal supported special groups with soluble quotient over their product, so designate the unique retained side. For sufficiently large retained dimension its supported derived classical group acts irreducibly there, hence distinct retained spaces are pairwise orthogonal by the same mutual argument as above. Intersections strictly decrease until possibly the last (use the large irreducible supported group on the remaining orthogonal space). For bounded retained dimension use a cross-chart pair. Their two small spaces embed in a nondegenerate subspace of bounded dimension (complete the radical of their span by dual partners); the supported large irreducible classical group on its complement lies in \(H\). All the small spaces in the patterns thus have bounded total span, and strict descent for independent pattern points is bounded by the growing-span linear-constraint argument above. Use this same small-space argument for the stabilizers of nonsingular radical lines in orthogonal characteristic two. These exclusions ensure ordinary semilinear irreducibility: an invariant degenerate space has an invariant radical, totally singular except possibly in that quadratic case, where the kernel of the quadratic form on the radical is totally singular and otherwise the radical is a nonsingular line.

All initial categories needing more than height-boundedly many labels required no previously good anchor. ◻

Proof. Here and below we use an orbit bound. Suppose a group has two permutation actions, each preserving a partition into at most two classes and inducing on the points at least the independent alternating groups on its classes. Then the number of orbits on ordered pairs of pairs (one index from each action per pair) is bounded absolutely. Indeed choose a size gap among the at most four class sizes, with all small sizes bounded and every large size at least 8 with alternating order exceeding the product of the quotient-index bounds from the small factorials, class permutations and parities (choose five successive absolute thresholds rapidly increasing, or more to leave a gap). Intersect the joint action with the product of all large alternating factors. This is normal of bounded index, projecting onto each large alternating factor by the index comparison and simplicity (one can restrict for this check to the preimage of the single alternating factor on its action side, whose image there is onto by hypothesis). By the subdirect description these actions split as independent diagonals, with identifications natural up to relabeling (automorphisms of alternating groups in large degree). The orbits on the specified positions are then bounded by equality patterns and small-class choices. Bounded equivariant fibers keep the bound absolute. ◻

Proof. First exclude line systems, \(k=1\). In linear type the determinant-one diagonals in \(S\) are normal soluble in a label, thus in \(H\). A line of another system with support \(s\) on this one has diagonal orbit size \((|F|-1)^{s-1}\) (divide by \(\gcd(v,|F|-1)\) for full support), at most \(s\) by independence. For \(|F|=2\) the monomial collineations would preserve the all-ones vector line contrary to irreducibility. Thus distinct systems force \(|F|=3\) and support two throughout (\(v\) large and use \(E^+\)-transitivity). They then pair the lines of the first system by a perfect matching, putting both the sum and difference lines on each pair, uniquely. Indeed each support pair has orbit spanning its full plane, so distinct pairs are disjoint. In unitary type use determinant-one norm-one diagonals similarly; the base \(\sqrt{|F|}+1\ge3\) allows only support one. In odd-characteristic orthogonal type use sign flips even in each norm-square class, lying in \(\Omega\), again a normal soluble group. Orbit size at least \(2^{s-3}\) forces \(s\le5\); each occurring type is large, allowing arbitrary support signs, so actually \(s\le2\). Distinct systems again give matchings, uniquely per matching since the two lines on a pair must be orthogonal: they are generated by \(x+y,x-y\) from the two first-system lines with equal nonzero norms, forcing matching within one square class and the relative identification up to sign. There are no nondegenerate lines for alternating polar form. In the matching cases fix two pattern points \(i,j\) on a pattern line. Their intersection group contains alternating permutations of matched configurations independently in each of the (at most two) isometry types, so has a subgroup preserving both with absolutely bounded number of orbits on ordered pairs of first-system indices. Every other system on the pattern line gives a matching invariant by that subgroup. This bounds the number absolutely, contradiction for large lattice \(q\). ◻

Proof. We explain the supported inner group choices for \(k\ge2\). Use products across blocks of their natural special groups, actually supported (for orthogonal spaces we can take \(\Omega\) in the sense of the determinant-one spinor kernel in odd characteristic, and the kernel of the Dickson parity in even characteristic; thus in split four-space over two we use the group with the two independent natural \({\rm SL}_2\) tensor actions); replace \({\rm Sp}_4(2)\) by its simple derived group. Denote the resulting projective product by \(Y\trianglelefteq K\), and include its normal block-center subgroup among the soluble normals retained by (\(*\)). Except for soluble cases, modulo these centers \(Y\) is a direct product of isomorphic nonabelian simple groups, one per block, or two per block in orthogonal split dimension four. We recall details to control small cases. The standard natural special modules are absolutely irreducible in these dimensions except the orthogonal planes. This follows also from the usual natural modules in defining characteristic (orthogonal \(4^-\) uses the two Frobenius-conjugate natural \({\rm SL}_2\) factors tensoring after scalar extension, \(4^+\) the external tensor; orthogonal odd dimension here is in odd characteristic). For \({\rm Sp}_4(2)'\cong A_6\), restriction stays irreducible by normality in \({\rm Sp}_4(2)\) and the orders too large for \({\rm GL}_3(2)\); the end field cannot enlarge by the order in \({\rm GL}_2(4)\) either. Soluble low exceptions include natural \({\rm SL}_2\) (also \({\rm SU}_2\) or \({\rm Sp}_2\)) at parameters 2,3, \({\rm SU}_3(2)\) (its cycle and determinant-one cube-root diagonals act absolutely irreducibly), \(\Omega_3(3)\), and \(\Omega_4^+\) at 2,3; the indicated irreducibility still holds. For an orthogonal plane \(\Omega\) is the rotation group of order \((|F|\mp1)/\gcd(2,|F|-1)\), acting semisimply without fixed space or invariant nondegenerate line when that order exceeds two.

Small orthogonal planes. For order at most two in those planes: split over two gives a canonical nonsingular vector per block, impossible by irreducibility; split over three gives a canonical orthogonal nonsingular line refinement (the two nonsingular lines in the split plane), now excluded. The remaining cases (split over five and nonsplit over three) have individual supported signs \(z_a=-1\) on block \(a\) in \(H\). Treat patterns of these cases separately. If two systems are compatible (commuting projectors), distinctness and transitivity give a common orthogonal line refinement, impossible. Otherwise some sign of one moves blocks of the other (if they all fix them the projections split the blocks). By rank two it transposes precisely two and is identity on the rest. Those two planes must be graphs \(\pm f\) of an isometry from the sign block \(U_a\) to a plane in its orthogonal complement: their sum contains \(U_a\), they project isomorphically in both directions there, and they are orthogonal. Every other sign preserves the graph pair by commutation with \(z_a\), so the partner plane is precisely some \(U_b\) (each sign outside \(a\) acts on it as a scalar). Transitivity gives a perfect matching and a graph-isometry choice on each edge. The intersection group for two fixed systems contains alternating permutations of identically aligned matched configurations (all are isometric). Every further system on the pattern line must give a matching invariant by this group, boundedly many possibilities, and boundedly many isometry choices because the field here is bounded and only one choice per edge orbit needs specifying. The edge orbit bound again follows from bounded rank on the block indices. This contradicts large lattice \(q\). ◻

Proof. In the remaining nonexceptional direct systems, there exist for every block \(a\) actually supported matrix subgroups \(R_a\) projectively in \(E^+\), normalized semilinearly by the block stabilizer in \(E^+\), of order greater than two acting nontrivially there. For soluble \(Y\) or retained \(Y\), this follows from the full supported choices above. If \(HY=K\) is missing, \(H\) induces the full top permutation image (on blocks and simple sites), since \(Y\) acts trivially there. If two simple-site orbits occur (two per block), at most one orbit product is missing modulo centers, and the other supplies supported groups as required. Otherwise there is a transitive orbit, and we have the exact interval over \(P\) in the \(H\)-invariant subgroups of the simple power \(Y/(\text{block centers})\), where \(P\) is the base intersection image. In fact after the centers have been quotiented this transitive normal simple power is minimal normal, hence meets the base core trivially (being missing) and maps to the monolith in the core quotient. By 6, either \(P\) is a nontrivial product and all overgroups here are products, or all nonbottom labels are subdirect. The latter contradicts the height: strict comparable subdirect groups give strict refinement of invariant ties, whose chain length is absolutely bounded by the top orbit bound (simple sites have fibers of size at most two over blocks). Thus \(E>H\) gives an intersection product with all projections strictly larger than the corresponding nontrivial projections of \(P\). Lifting with the retained centers supplies the required singly supported \(R_a\) (one can take the full available preimage on that block in the specified supported group, including both sites if they are permuted there). Indeed all centers lost in passing from the supported choices in \(Y\) to the simple-site images have projective images in \(H\); so independent tuples in this intersection product lift with their supported representatives projectively in \(E\). Conjugating by a semilinear collineation of \(E^+\) preserving the block preserves the supported matrix choice as well, acting on representatives which are the identity on the other blocks. In all cases the block module for \(R_a\) is semisimple with no trivial constituent, since its module socle and its fixed space are invariant under the semilinearly irreducible block stabilizer. ◻

Proof. Any two systems in such a pattern must be compatible. If all \(R_a\)’s of one fix the other’s blocks individually, each projection onto \(a\) of such a block lies within the block (use differences and absence of a trivial quotient), giving compatibility. Thus incompatibility would force movement each way by supported matrices. A supported matrix moving blocks has displacement rank at most its support-block dimension and at least the moved block dimension. Hence both dimensions are equal and equality holds. On the moved system it then transposes just one pair of blocks with square the identity, and acts identically elsewhere (from the fixed dimension). Its displacement has full rank on the support block, so an involution there is impossible in characteristic two and can only be \(-1\) in odd characteristic. But a group of order greater than two cannot have only one element inducing a nonidentity block permutation, contradiction. ◻

Proof. Separate retained and missing colors. For retained systems, any two compatible systems must refine one another in some direction: incidence degrees are constant on each block side by transitivity, and if both exceed one, each touching block extends outside, hence is fixed setwise by the other’s supported group there (fixed outside points). This gives an invariant proper nonzero block intersection, nondegenerate when relevant, impossible by the retained actions above. A coarse block of large dimension cannot admit the resulting uniform finer split under its full supported special group by orders. Uniformly, with base size \(s_0=|F|\) (\(\sqrt{|F|}\) in unitary type), and \(c=1\) in linear/unitary, \(c=1/2\) in orthogonal/symplectic, the large \(m\)-dimensional special group order is at least \(s_0^{c m^2-O(m)}\), whereas the isometry (or linear) stabilizer of a corresponding uniform \(t\)-block direct split is at most \(t!s_0^{c m^2/t+O(m)}\) by standard orders, smaller uniformly for large \(m\) and \(2\le t\le m\). Bounded coarse dimensions permit only boundedly many comparable refinements. This excludes the retained patterns.

For missing systems take all points of a pattern line. Compatibility gives their common refinement into nonzero fine cells. Fix two systems \(i,j\) and group the fine cells by their pair of block indices. Each aggregate cell has a supported special group inside \(K_i\cap K_j\). If its dimension is large, no further system can split it: a splitting block extending outside the cell would give a proper invariant intersection under the supported group, whereas a block contained in the cell would give a uniform finer split contrary to the same order bound. A small aggregate has at most its dimension many nonzero fine cells. Thus the number of fine cells over each occupied pair is bounded by an absolute constant \(d\).

The map from a fine cell to its block-index pair is \(H\)-equivariant. Indeed \(H\) stabilizes both systems and permutes their common refinement; anti-actions also permute products of commuting projectors. The two rich top actions and 10 give an absolute bound \(C\) on the orbits of ordered pairs of occupied pairs. Each original system is recovered by grouping the fine cells and taking their direct sums. Distinct full stabilizer labels therefore give distinct \(H\)-invariant partitions of the fine cells. By (22) there are at most \(2^{Cd^2}\) such partitions, contradicting the chosen pattern-line size. This completes direct-system pruning. ◻

Proof. We next treat subfields and ordered tensor structures. All algebras below inside \(\mathcal A\) are unital \(F\)-algebras (“prime” extension or field algebra in these exclusions refers to prime relative degree, not to the characteristic field). Recall that a central simple matrix subfactor \(A\) gives \[V=X\otimes_F Y,\qquad A={\rm End}(X)\otimes1,\qquad A'=1\otimes{\rm End}(Y)\] with \(A'\) its commutant (central simple algebras over finite fields split). For \(A\) proper and nonscalar both factor dimensions are at least two. Linear normalization acts by tensor matrices up to scalars, by the matrix algebra automorphism theorem; similarly with field automorphisms. If \(A=A^*\), both factors have defining forms, with \(b\) their tensor form up to a scalar: use the restrictions of the matrix involution to the two factors. These are both hermitian for unitary type, and each symmetric/alternating bilinear otherwise. Semilinear similarities preserving the ordered tensor act by semilinear similarities on the factors. In bilinear characteristic two, an ordered factor normalized by such an irreducible \(E\) cannot have nonalternating form (the diagonal contraction has an invariant hyperplane, then tensor with the other space). With both alternating, any orthogonal quadratic form in question must be the canonical one for the polar tensor, vanishing on pure tensors. Such a quadratic form exists (impose zero on basis tensors and expand, using alternation on both factors: for a pure tensor the potentially nonzero pair terms have both indices different and cancel by interchanging which second indices are matched to the two first indices); it transforms under \(E\) like the given one, and a nonzero difference, the square of a linear functional, contradicts irreducibility. We will use these facts whenever there is a common previously good anchor. ◻

Proof. Useful also is the uniform order obstruction: a full supported perfect natural classical group on a space of sufficiently large dimension \(m\) cannot normalize a proper nonscalar central simple tensor factor, or a tensor system splitting the space into at least two nontrivial slots even up to permutations. Its order (also modulo scalars) is at least \(|F|^{m^2/2-O(m)}\); the linear tensor-system order is bounded by \[t!\, |F|^{\sum_j x_j^2},\qquad \prod_{j=1}^t x_j=m,\quad t\ge2,\quad x_j\ge2 ,\] which is uniformly smaller (\(t\le\log_2 m\), \(\sum x_j^2\le m^2/4+4\), e.g. by merging factors). Use the linear action of the covering matrix group for this comparison. This proves also properness of such stabilizers in the large ambient group. Properness for field algebras follows directly from perfection (fix them pointwise if normalized by the socle) and absolute irreducibility on the natural module. ◻

Proof. Exclude stabilizers of field algebras \(D_i/F\) of prime degrees \(r_i\). Their soluble normal projective torus \({\bf P}(D_i^*)\cap S\) is in \(H\), of order \[\frac{|F|^{r_i}-1}{|F|-1}\ \big/\ \frac{\gcd(v,|F|-1)}{\gcd(v/r_i,|F|-1)}\] by the norm determinant. This exceeds \(r_i\) except possibly \(|F|=3,r_i=2\), when it is at least two. If a pair of fields do not commute, each torus injects by normalization into the other’s relative field Galois group (a nontrivial torus element has lifts generating its prime field extension). This forces the exceptional case on both sides. If two fields commute, their generated algebra is commutative semisimple, necessarily a field since the primitive idempotent system would otherwise contradict the direct-system exclusion at \(E\). Distinct commuting fields therefore have distinct prime degrees. In a nonexceptional line take their compositum; the intersection group contains the natural determinant-one group over it, perfect if module dimension is at least two, which forces any third prime field on the line into the compositum by centralization (use elementary matrices). If dimension one, pairwise commutation alone does the same. This contradicts distinctness. In a line of the exceptional type, distinct fields must be noncommuting. They have generators \(J_i^2=-1\) with projective class in the indicated torus (norm one), and pairwise normalization gives anticommutation. Two generate a split quaternion algebra \(M_2(F)\); their intersection group includes the large determinant-one group on its tensor complement. Every line field lies in that bounded pair algebra by centralization, impossibly many. We separate the types by color here. ◻

Proof. Now exclude ordered proper nonscalar central simple factors \(A\). The projective supported special linear groups on both tensor spaces are commuting normals in the label, even allowing anti-actions, with soluble quotient. Exactly one is retained; designate its factor algebra. Its natural projective group is simple except dimension two over two or three. Over two, the characteristic order-three subgroup would give an invariant quadratic field, impossible. Over three, the \(A_4\) has a unique Klein four subgroup, normal and spanning the factor algebra by its quaternion lifts. Treat this last color separately from the simple case. For a pattern of designated algebras the groups are normal in \(H\); intersection nontrivial gives equality (or equality of the Kleins in the exceptional case), hence equality of algebras, since their lifts span. Otherwise they commute projectively and also linearly on special-group lifts (perfection, or no scalar characters for \({\rm SL}_2(3)\)). Thus distinct designated algebras commute elementwise. For a pair on a line the new algebras lie in the centralizer matrix factor, on whose space the intersection contains the full supported special linear group. In large dimension this prohibits any proper new tensor subfactor there by orders (the whole centralizer factor then permits at most one distinct new designated algebra); in bounded dimension only boundedly many distinct commuting nonscalar central simple factors can fit there anyway (their tensor-product matrix algebra embeds there by simplicity, so the product of degrees is bounded). Contradiction. ◻

Proof. First try to exclude ordered proper nonscalar selfadjoint central simple factors \(A\), leaving two possible remaining types temporarily. In the stabilizer the natural projective special groups supported on \(X,Y\) are commuting normals with soluble quotient. Again use \({\rm Sp}_4(2)'\) in its small case. The supports indicated lie in \(S\). Besides perfection in the usual cases, note unitary determinant one, and in orthogonal type: in characteristic two both factors are alternating as above and the tensor action from a factor has even displacement rank; in odd characteristic \(\Omega\) on a symmetric factor lies in \(\Omega\) on the tensor (diagonalize the other symmetric factor and use orthogonal sums; scaling does not change the spinor norm on determinant one). If both factors alternating in odd characteristic, the symplectic supports also lie projectively there by perfection or absence of a 2-quotient for \({\rm SL}_2(3)\). Quotient solubility follows by the two tensor projections and the standard similarity groups. Exactly one support is retained, and we designate its factor algebra.

We exclude the labels for which this retained type is neither a bilinear orthogonal plane in odd characteristic nor a symplectic plane over two. Here are the details of the supported projective groups within the designated algebra:

• Usually it is a single simple group with perfect special lift and absolutely irreducible spanning natural action. In split orthogonal dimension four (odd characteristic here) there are two projective \({\rm PSL}_2\) components on symplectic slots, with their matrix algebras tensoring to the factor. They are permuted by the label, even at parameter three: in the determinant-form model the singular rank-one quadric and its two rulings give these two tensor slots up to exchange (the maximal linear rank-one spaces are precisely of the two fixed-line kinds). Indeed the endomorphisms preserving each member of a ruling individually form the matrix algebra on the other coordinate (use two independent first-coordinate lines and their sum-vector line). Nonsplit four has the standard simple projective group with absolutely irreducible tensor-twist action instead.

• The degree-two symplectic or special unitary groups at parameter three (also the symplectic slots just described at three) give projective \(A_4\) with Klein lifts spanning the slot and special lift \({\rm SL}_2(3)\) having no ground-field scalar characters. Indeed the Klein preimage in the rank-one special group at three is quaternion of order eight with commutator \(-1\) acting as negative identity in degree two, thus absolutely irreducible there (semisimplicity for this 2-group, no linear constituent possible), and the abelianization of \({\rm SL}_2(3)\) has order three whereas the ground scalar group here has order two or eight. A degree-two unitary factor at parameter two is impossible: its characteristic order-three subgroup in the projective special has two orthogonal nondegenerate eigenlines over four, giving a forbidden nondegenerate direct decomposition on tensoring. An orthogonal 3-space over three is likewise impossible: for the form up to similarity a sum of three squares, the three norm-one lines are the axes preserved as a set even by similarities, again a direct decomposition.

• For the special unitary 3-space over four (parameter two) the projective special \(C\) of order 72 has a unique normal Sylow 3-group \(D_9\) elementary of order nine, contained in every nontrivial normal subgroup, and its lifts span the slot. Indeed in an orthonormal basis use the cycle and \({\rm diag}(1,\omega,\omega^2)\) for a cube root \(\omega\ne1\). Their commutator generates the scalar special center; projectively they give \(D_9\). In the projective full unitary group of order 216, the Fourier matrix and \({\rm diag}(1,1,\omega)\) normalize it inducing generators of \({\rm SL}_2(3)\) (rotation and shear). Thus the whole 216 normalizes it with centralizer the nine. The special image contains the nine and induces order eight acting irreducibly there (a line stabilizer in \({\rm SL}_2(3)\) has order six). Thus the nine is minimal normal and a normal subgroup disjoint from it must centralize it, proving the assertion. Its linear special lift has no character to the scalars of order three by this action and the center commutator.

These lists follow from the same natural classical groups as for direct blocks, now with odd characteristic on symmetric factors. The individual component matrix algebras here are selfadjoint.

Color according to simple components, \(A_4\) components, or the single \(C\). The projective components lie in \(H\), which permutes at most two of them in each system. In a simple or \(A_4\) pattern they cannot swap the other’s pair (no quotient of order two), so components across systems mutually normalize. A nontrivial intersection, normal in each, gives equality or common Klein, hence the same component algebra; otherwise they commute projectively and linearly on special lifts as above. Likewise for the \(C\) pattern, using a common \(D_9\) if intersection is nontrivial (it contains the normal Sylow on each side, which must then be the same). Thus all the distinct component algebras are commuting tensor slots. For two labels \(i,j\) on a line their components (at most four) have a tensor centralizer factor, itself normalized by \(E\); any new component lies there. In the pair intersection is the large supported perfect classical group on that centralizer space if of large dimension, by the tensor form observations. It must stabilize individually the at most two components for each further point of the line. The order bound prohibits a proper nontrivial central simple subfactor there being new (the whole factor itself would add only one); if of bounded dimension only boundedly many commuting new ones fit. Thus the list of distinct component algebras on the whole line is absolutely bounded (at most four from the original two labels plus those new ones). Each designated algebra consists of one or two such components tensoring and determines the stabilizer (it and its tensor commutant determine the ordered tensor). Only boundedly many labels then fit on the line, contradiction. Consequently any remaining previously good-normalized proper ordered selfadjoint factorization must have the retained side one of the two remaining plane types. ◻

Proof. For a normalized field (degree possibly composite), let \(\delta=*|_D\). There is a nondegenerate transferred form \(B\) given by \[b(dx,y)={\rm Tr}_{D/F}(d B(x,y)).\] It is linear over \(D\) on the first slot and \(\delta\)-semilinear on the second. It is symmetric/skew or hermitian/skew-hermitian as appropriate (skew-hermitian may be rescaled). In particular if \(b\) is alternating and \(\delta=1\) then so is \(B\), also in characteristic two by testing all square scalars. For orthogonal characteristic two and \(\delta=1\) transfer \(Q(dx)={\rm Tr}_{D/F}(d^2 Q_D(x))\), valid since the left side is additive in \(d\) (the cross term vanishes by alternation of \(B\)) and scales by squares over \(F\), so gives a linear function of \(d^2\) over \(F\); testing the trace against these same squares gives \(D\)-quadratic homogeneity of \(Q_D\) and polar \(B\). For orthogonal characteristic two and \(\delta\ne1\) use instead \[Q(x)={\rm Tr}_{D_0/F}(B(x,x)),\qquad D_0=D^\delta .\] Indeed the right side has polar \(b\) and transforms the same way under any previously good anchor normalizing \(D\). The difference from \(Q\) has zero polar form, hence is the square of a linear functional over the finite, therefore perfect, ground field. If nonzero, its kernel would be a proper nonzero invariant subspace for that anchor. Irreducibility therefore makes the difference zero. All form transformation assertions follow from trace uniqueness under semilinear maps normalizing \(D\). For instance for a similarity \(u\) with multiplier \(\mu\) on \(b\) and induced field power \(\tau\) on \(D\), testing at \(d^\tau\) gives \(B(ux,uy)=\mu B(x,y)^\tau\). Thus the stabilizer acts projectively by corresponding \(D\)-semilinear similarities (extra scalar kernel soluble), and the standard \(D\)-linear special group, when perfect, lies inside \(S\). ◻

Proof. If this normalizer is nonsoluble and total ground-field dimension large, we can use a perfect such classical group \(P_D\) on the natural \(D\)-module, absolutely irreducible over \(D\). Indeed either module dimension or extension degree/field size is large, avoiding the small parameter exceptions; in the permanently soluble form dimensions nonsolubility of the normalizer would be impossible. Its centralizer in ground-field endomorphisms is exactly \(D\), except possibly the unitary module of dimension two over \(D\) with \(\delta\ne1\) fixing \(F\). To see this, extending from \(F\) to algebraic closure splits the module into copies twisted by the different embeddings of \(D\) over \(F\). They are irreducible and pairwise inequivalent since matrix traces generate \(D\) over \(F\): symplectic type uses an \({\rm SL}_2\) block; orthogonal type can use split 4-space with tensor trace there an arbitrary product, or norm traces from the \(4^-\) tensor twist, and in dimension three the symmetric-square traces (odd characteristic). Unitary type has traces generating at least \(D_0\) via \({\rm SU}_2\), and in dimension at least three a norm-one diagonal with entries \(a,b,(ab)^{-1}\), where \(a,b,ab\ne1\), has trace differing from inverse trace, giving all of \(D\). In dimension two, \(D_0F=D\) if \(\delta\) is nonidentity already on \(F\). Here on a split nondegenerate orthogonal 4-space use the two \({\rm SL}_2(D)\) tensor actions, supported there (available also within higher dimensions of Witt index at least two); in the nonsplit dimension-four case the trace on the tensor twist ranges over norms of arbitrary \({\rm SL}_2\)-traces, and in dimension three the values \(u^2-1\) from the symmetric square generate the field in odd characteristic. For \({\rm SU}_2\) the natural action over \(D\) realizes the split rank-one traces over \(D_0\). The indicated three-entry unitary diagonal exists even when the norm-one group has order three (take \(a=b\)); the difference of its trace from inverse trace, multiplied by \(ab\), is \(-(a-1)(b-1)(ab-1)\). Thus the embedding distinction (and absolute irreducibility in each copy, as for natural classical groups above) gives scalar centralizer on each over closure whenever claimed, hence the centralizer assertion over \(F\). In particular the exception does not arise for prime degree in sufficiently large ground-field dimension (it would require degree two and module dimension two). ◻

Proof. We will also use the following paired situation (still with the prime-degree \(D_i/F\) and high total dimension). Suppose two central or commuting projections to be used below are exchanged by \(*\) (a complementary pair \(e,e^*=1-e\) normalized as a pair by \(E\)), and commute with \(D_i\). Their spaces are dually totally isotropic, and totally singular also for \(Q\) when relevant (in characteristic two each quadratic restriction with zero polar is a linear square, and the sum of the two kernels would be a proper nonzero invariant space if a restriction were nonzero; the dimensions are large). The full simultaneous stabilizer of \(D_i\) and the pair thus includes paired \({\rm SL}_m(D_i)\) on these spaces, acting by inverse adjoint on the second, where \(m\) is their \(D_i\)-dimension; if \(m=1\) the stabilizer would be soluble. For \(m\ge2\) this group is perfect inside \(S\). Every endomorphism centralizing its matrices commutes with \(D_i\). Indeed over closure the constituents are natural at embeddings \(\tau\) over \(F\) and dual at \(\tau'\delta_i\) with \(\tau'\) over \(F\) (by the transferred pairing); the field scalars act there by \(\tau,\tau'\), respectively. Natural twists are distinguished by traces, as are dual twists; a natural and dual identification could only have \(\tau=\tau'\delta_i\) by traces on an \({\rm SL}_2\) block. It cannot occur for \(m\ge3\) (diagonals \(a,b,(ab)^{-1}\) again; \(|D_i|\ge4\)). If \(m=2\), either the two identified constituents have the same field-scalar action, or \(\delta_i\ne1\) fixes \(F\), forcing prime degree two and bounded total dimension. This proves the claim. ◻

Proof. A larger center field. If the center of \(R\) is a field larger than \(F\), take a central prime extension \(D'\) invariant under \(*\) and \(T\). If \(K_i\le N_M(D')\), then \(P_{D_i}\) forces \(D'\subseteq D_i\) by centralization, so \(D_i=D'\) commutes with \(D_j\). Otherwise the covering property gives \[T=K_i\cap N_M(D')\] exactly. Consider the commutative semisimple \(L_f=F[D_i,D']\). If a field, the degrees must be distinct primes with compositum degree their product, and the perfect transfer group over it is available in \(T\) (the subfields are unique, and \(T\) is nonsoluble since it contains \(E\)). This perfect group centralizes \(D_j\), forcing commutation with \(D_i\) by the centralizer assertion. In the exceptional unitary dimension two case here, the compositum degree is \(2s\) with \(s\) a large odd prime (total dimension \(4s\)). Write \(a=|F|\) and identify the compositum with \(C=\mathbb F_{a^{2s}}\). Work in the actual scalar group \[U=\{z\in C^\times:z^{a^s+1}=1\},\qquad |U|=a^s+1.\] The subgroup \(U_S\) whose projective classes lie in \(S\) has index at most two in \(U\): these are actual determinant-one isometries, with at most the spinor or Dickson restriction still to impose. Their classes belong to \(T\). Let \(U_0\le U_S\) be the scalar preimage of the kernel of their action on \(D_j\). Since the matrices are \(F\)-linear, this action has order at most \([D_j:F]\). That prime degree divides \(4s\), so it is either \(2\) or \(s\), and \[|U_0|\ge\frac{a^s+1}{2s}.\] The two maximal proper relative subfields of \(C/F\) are \(\mathbb F_{a^s}\) and \(\mathbb F_{a^2}\). Their intersections with \(U\) have sizes at most \(2\) and \(a+1\), respectively: the first satisfies \(z^2=1\), while on the second the oddness of \(s\) turns the norm-one condition into \(z^{a+1}=1\). For sufficiently large \(s\), the displayed lower bound exceeds \(2+(a+1)\). Hence \(U_0\) contains a generator of \(C/F\). Ground-field scalar changes do not affect conjugation on \(D_j\), so this is a count in the scalar preimage throughout. Such a generator centralizes \(D_j\), forcing all of \(C\), and in particular \(D_i\), to commute with \(D_j\).

A split compositum. If \(L_f\) here splits, there are exactly two exchanged components, and \(D_i,D'\) must have the same prime degree \(r\) (tensor over \(F\)). The possible primitive components are indexed by relative identifications of the two degree-\(r\) fields; \(E\) acts on these by translations of a cyclic \(r\)-set (the two field automorphisms extend the same base automorphism), transitively on the occupied subset. Thus \(r=2\). Now apply the paired situation to the two components: the indicated paired special linear group also centralizes \(D'\) (on each component that field acts through \(D_i\)). Hence it lies in \(T\), centralizes \(D_j\), and gives commutation with \(D_i\).

Two exchanged central components. If the center of \(R\) instead has two exchanged components, use their projection-pair stabilizer in place of \(N_M(D')\). It cannot contain \(K_i\) by perfection and the centralizer of \(P_{D_i}\), so its full intersection with \(K_i\) is \(T\), by covering. The paired situation applies directly, yielding the same commutation.

Scalar center. It remains to consider center \(F\). Here \(R\ne\mathcal A\), else \(T\)’s actions on the two fields have trivial projective kernel and soluble image. Thus \(R\) gives a proper ordered selfadjoint factorization, necessarily with one of the retained remaining plane factors by the partial exclusion. If \(R\) itself is that plane matrix algebra, then for bounded field size the intersection contains the high-dimensional perfect classical group on its tensor complement, forcing every line field into bounded \(R\), contradiction. For an odd orthogonal plane factor at large field size the projective supported rotations (from \(\Omega_2\)) have order greater than four, and lie in \(H\) as a normal soluble group for the stabilizer label of \(R\) (which contains \(E\)). Normalizing both \(D_i,D_j\) (quadratic within \(R\)), some nonscalar rotation centralizes both. In the plane matrix algebra this gives their equality (the centralizer of a nonscalar matrix there has dimension two), impossible. If instead \(R'\) (the matrix commutant of \(R\)) is the remaining plane algebra, \(K_i\) cannot preserve \(R\), since then \(P_{D_i}\) would act trivially on \(R'\) by its soluble similarity projection, contrary to its field centralizer. Thus again \(T\) is the full intersection \(K_i\cap N_M(R)\). On the large \(R\)-factor space apply transfer to \(D_i\) there, selfadjoint for its factor form. Its field normalizer there must be nonsoluble (otherwise both factor projections of \(T\) soluble). We get its supported perfect transfer group in \(T\), with centralizer in \(R\) exactly \(D_i\); in characteristic two here use the alternating factor form, the total tensor preserving its given orthogonal quadratic if necessary as explained earlier. This group centralizes \(D_j\subset R\), impossible.

Completion of the line argument. Consequently the fields on a pattern line all commute unless we already obtained a contradiction. Two distinct ones of the same prime degree would give a split commutative algebra with exactly two exchanged components as above, degree two, and the two adjoint restrictions must differ (to exchange the relative-identification idempotents). Use homogeneous color of the adjoint type (identity or not on the prime-degree field algebras) to prevent this. Thus two have distinct primes and generate a field. Its transfer group in the nonsoluble intersection forces any third into the compositum, impossible by prime degrees. There is no unitary dimension-two exception here: if the involution fixes \(F\) but not the compositum, the two prime fields have different adjoint types. This completes prime-field pruning. ◻

Proof. Exclude stabilizers of unordered pairs \(e,e^*=1-e\) of proper idempotents. For two such labels, choosing idempotents \(e,f\), the generated selfadjoint algebra is again semisimple. A center field larger than \(F\) is now forbidden. If there are two central components exchanged, their projector pair commutes with each of the given pairs. Distinct commuting such pairs are impossible: they would give all four projector cells (or already be the same pair, using adjoint exchange), with forbidden orthogonal direct sums on adjoint orbits. For center \(F\), note \((e-f)^2\) is central, and the algebra is spanned over this scalar algebra by \(1,e,f,ef\), since the relation rewrites \(fe\) in terms of \(e,f,ef\) and scalars and the generators are idempotent. It is therefore \(M_2(F)\), which must have remaining plane type. An orthogonal plane has at most one such pair. For the bounded symplectic plane case, the high perfect tensor-complement group in the intersection fixes individually each further pair on the line, forcing them into the bounded plane algebra. Thus all paired stabilizers are excluded.

Finally the remaining ordered factors themselves cannot occur: an odd orthogonal plane factor has the unique pair of isotropic lines over the closure, giving a forbidden projector pair if split, or a forbidden quadratic field (the diagonal algebra descended to \(F\)) if nonsplit, selfadjoint and normalized by all the factor similarities. A symplectic plane over two gives a normalized selfadjoint quadratic field via its characteristic order-three subgroup. This completes the stated field and ordered-factor exclusions. ◻

Proof. After the preceding exclusions we have the following full spanning property: any nontrivial projective subgroup \(P\le S\) normalized by a previously good anchor has \[F[\text{lifts of }P]=\mathcal A .\] Indeed the radical vanishes by anchor irreducibility, and in isometry type the algebra is selfadjoint. Multiple center components are now impossible by the direct-system exclusions (and the exchanged pair exclusion for adjoints). Center-field enlargement and a proper nonscalar central simple factor are also excluded. Semilinear anti-actions in linear type respect the algebra as well (their group action on lifts uses inverse, harmless here). ◻

Proof. Exclude full normalizers of nontrivial elementary abelian subgroups \(P\le S\). In a pattern each such subgroup is contained in \(H\le E\) by (\(*\)). Its scalar commutator pairing from linear lifts is nondegenerate by full spanning. Thus it identifies \(P\) with its ground-field character dual (nondegeneracy already forces the exponent prime to divide \(|F^\times|\)). Lifts are linearly independent by their distinct simultaneous conjugation characters, and their linear span is the generated algebra. Hence \(|P|=v^2\) and its projective linear centralizer is itself (match the scalar conjugation character using the pairing, then use full spanning). There cannot be two distinct ones here normal in \(E\): a nontrivial intersection would itself satisfy the same properties, and a trivial intersection would give projective commutation. This excludes the pattern. ◻

Proof. Now exclude stabilizers of tensor systems of at least two nontrivial matrix factors tensoring to \(\mathcal A\), allowing permutations of slots, with factors individually selfadjoint in isometry type. For systems under consideration the anchor acts transitively, else an orbit product contradicts ordered-factor exclusion. Thus we have \(t\) slots of common dimension \(k\). The tensor forms in isometry type can be aligned to identical forms on the factors up to scalars by transitivity/similarity (finite-field form classes). In bilinear characteristic two all are alternating (by transitivity and alternation on \(V\)); again any orthogonal quadratic here is the canonical one zero on simple tensors by irreducibility. Consequently the stabilizer label \(K\) induces at least \(A_t\) on the slots by taking commutators of actual tensor-slot permutations inside the full linear or isometry group.

We can reduce to the case of a normal supported projective product \(Y=B^t\) inside \(K\cap S\), one nonabelian simple group per slot (embedded projectively), whose linear lifts span each slot matrix algebra. Use the natural special groups as above, including \({\rm Sp}_4(2)'\). Here is the treatment of other possibilities. Small degree-two parameter cases at two and three in the linear, symplectic and unitary families give canonical nontrivial elementary abelian projective groups per slot (order three or the Klein, characteristic in the projective special). Their product is elementary abelian projectively inside \(S\), now forbidden; the same applies for \({\rm PSU}_3(2)\) using \(D_9\). Containment uses the same special supports as for ordered factors. \(\Omega_3(3)\) again gives direct orthogonal line cells from the axes. Odd-characteristic orthogonal planes of split type give pairs of singular lines per slot, whose tensor cells paired with their duals give a forbidden nondegenerate direct decomposition. In nonsplit type use the canonical quadratic fields per slot; their tensor algebra has \(2^{t-1}\) components, each central projector selfadjoint (conjugation on all the fields simultaneously), again forbidden. Split orthogonal four-space factors canonically refine into pairs of symplectic plane tensor slots as above, giving a normalized tensor system of the latter instead. Thus it suffices to exclude the single-simple-per-slot systems (first treating just these labels; the refinement observation then handles the four-space case). ◻

Proof. For a pattern with \(Y\) retained, these products are minimal normal subgroups of \(E\) by slot transitivity. Two must be equal (commutation impossible by perfection and spanning), which recovers the same system by the lifts of the simple factors. For missing patterns \(HY=K\), \(H\) induces the full top action, transitive and containing the natural alternating group. Exactly as for missing direct simple-site products, the interval over \(H\cap Y\) in \(Y\) of invariant subgroups must be the nontrivial product case by the reductions and subdirect tie-chain bound. In particular on a pattern line, for each system \(i\), \[N_i=E\cap Y_i=\prod_a R_{i,a}\trianglelefteq E,\] a slotwise projective product strictly above the base intersection. ◻

Proof. Write bars for the quotient by \(\operatorname{core}_E H\), and let \(W\) be the nonabelian simple-power monolith of \(\bar E\). Each \(\bar N_i\) contains \(W\). The simple factors of \(W\) partition among the \(\bar R_{i,a}\): their commutators with \(W\) generate \(W\) by perfection, lie in the respective intersections with \(W\), and are normal there and mutually commuting.

We need these assignments to give nontrivial intersections in the actual group, not only compatible images in the core quotient. Put \(N=\bigcap_iN_i\). Since the \(N_i\) are normal in \(E\), an iterated commutator of them lies in \(N\); its image contains \(W\) by perfection. Thus \(\bar N\supseteq W\). Fix a simple factor \(C\) of \(W\), and let \(a_i\) be its assigned slot in system \(i\). Because \(R_{i,a_i}\trianglelefteq N_i\) and \(N\le N_i\), we have \[[N,R_{i,a_i}]\le N\cap R_{i,a_i}.\] The image of this commutator contains \(C\). Each \(N\cap R_{i,a_i}\) is normal in \(N\), so a second iterated commutator shows that \[M_\gamma:=\bigcap_iR_{i,a_i},\qquad \gamma=(a_i),\] has image containing \(C\). In particular it is nontrivial.

Now take the intersections \(M_\gamma\) for all slot tuples, allowing trivial intersections. They are permuted by \(E\) and generate a nontrivial \(E\)-normalized subgroup of \(S\), whose lifts span \(\mathcal A\) by 20. For distinct tuples the lifts commute linearly, because they lie in distinct tensor slots of at least one original system. Their lift algebras therefore commute and jointly generate \(\mathcal A\). The center of each algebra centralizes all of \(\mathcal A\), hence is scalar. Its radical, multiplied by the commuting remaining algebras, would give a nilpotent ideal of \(\mathcal A\), hence vanishes. The non-scalar lift algebras are thus central simple; their tensor product maps isomorphically onto \(\mathcal A\).

This also recovers the original embedded factors. Group the tuple algebras by their assigned slot in any one system. Each grouped algebra is contained in that slot, and the product of all grouped algebras is still \(\mathcal A\). Comparing dimensions in the original tensor decomposition forces equality in every slot. We have obtained a common tensor refinement, not merely abstract isomorphic factors.

The fine factors are selfadjoint when appropriate and are permuted transitively by \(E\), by the ordered-factor exclusion. Thus the same form and quadratic observations apply. In particular in bilinear characteristic two the fine-slot forms are all alternating, by transitivity and alternation of the total form. Grouping them into two nonempty batches still gives alternating forms on both. The orthogonal quadratic, if present, is zero also on pure tensors for every such two-batch grouping: its canonical quadratic with the given polar form exists by alternation and is zero on fine pure tensors too. The two quadratics therefore agree on a basis and, having the same polar form, agree everywhere. ◻

Proof. Fix two systems \(i,j\) and send each nontrivial fine factor to its pair of original slot indices \((a_i,a_j)\). Grouping factors with the same pair gives an aggregate matrix factor. On a large aggregate space, \(K_i\cap K_j\) contains the full supported perfect classical group. No further system on the line can split this aggregate: its fine factors would give a tensor system normalized by that group, contrary to 16. Thus a large aggregate contains only one nontrivial fine factor. On a small aggregate of degree \(h\), there are at most \(\log_2h\) such factors, because their degrees multiply to \(h\) and each is at least two. The fibers of the displayed map are therefore bounded absolutely.

The map is \(H\)-equivariant, since \(H\le E\) stabilizes both original systems and permutes all tuple intersections; discarding scalar factors preserves this action. By 10, the two rich top actions give an absolute bound on ordered-pair orbits of occupied pairs, hence on ordered-pair orbits of fine factors. Each original system is recovered by grouping those embedded factors, as proved in 23. Distinct stabilizer labels therefore give distinct invariant partitions. The bound (22) contradicts the pattern-line size and completes tensor-system exclusion. ◻

Proof. Finally consider a tested descendant \(K\) with prepared interval, good for these exclusions (or containing the final good anchor). It has \(K\cap S\ne1\) since \(K\) is nonsoluble. Take a minimal \(K\)-normal subgroup there. It cannot be elementary abelian, and if a power of nonabelian simples, their lift algebras commute linearly by perfection, and jointly span fully. As in the refinement argument they are central simple tensor factors, individually selfadjoint when relevant; multiple ones would be a forbidden tensor system. Thus there is an actual simple \(B\trianglelefteq K\) in \(S\), absolutely irreducible projectively. Its projective linear centralizer is trivial by perfection and spanning, so \(C_K(B)\) injects in the soluble outer quotient and lies in \({\rm core}_K H\) by (\(*\)). Also \(B\) is not retained (else base-core quotient soluble), hence \(HB=K\). Quotienting its centralizer gives the required next almost simple group with socle \(B\), keeping the exact interval. ◻

Proof. We use the following orbit bound for two finite set actions on sizes \(s,t\), each inducing at least the alternating group on that set. The number of orbits on ordered pairs of pairs (one coordinate from each set per pair) is absolutely bounded. Indeed the joint action is subdirect on the two induced groups. In large degrees either the product of the alternating groups is contained, or the two alternating simples are linked diagonally, hence the same degree and natural identification up to relabeling. This follows by the normal kernels of subdirect projections and simplicity (normal subgroups not containing the alternating simple must centralize it); use also the usual automorphism theorem for alternating groups in high degrees. If only one degree is large the large alternating factor acts independently by the same argument. Equality patterns and the bounded small sizes give the assertion. ◻

Proof. First exclude stabilizers of proper nonempty subsets. For a cross-chart pair, the intersection contains supported alternating groups on at most four cells. Thus \(H\) has boundedly many point orbits, and every subset whose stabilizer is a witness is a union of these orbits. Only boundedly many subsets, and hence witness labels, are possible.

Next exclude nontrivial equal-block-size partitions, which suffice for primitivity once transitivity is known. Write \(s\) for the number of blocks and \(k\) for their common size. In a witness label, the intersection \(Y\) of \(M\) with the base block group is normal. If \(Y\) is retained and \(k\ge3\), every individually supported \(A_k\) lies in \(H\). Its primitivity makes the restriction of any other tested \(H\)-invariant equivalence to that block either universal or discrete. In the discrete case a point has an equivalent point outside the block, since the other partition also has block size at least two. The supported transitive group fixes that outside point and would make it equivalent to every point of the original block, a contradiction. Consequently each original block lies in a block of the other partition. Applying this in both directions to two retained systems with block sizes at least three forces their equality. For retained \(k=2\), the even simultaneous flips on any two blocks lie in \(H\). A distinct matching fails invariance by flipping a block while leaving one of its points’ outside partner fixed, using the high degree.

If \(HY=K_i\), then \(H\) induces the full symmetric top action on the blocks. For a cross-chart pair of missing partitions, 24 bounds the orbits on ordered pairs of block cells. Moreover \(H\) contains the supported alternating groups on the points of each cell, so each such orbit has boundedly many point-pair orbits above it, including when both points lie in the same cell. Thus \(H\) has boundedly many orbits on \(\Lambda^2\). Every invariant equivalence is a union of those orbits, giving the absolute partition bound used in (22). This is too small for all the distinct pattern labels. Here and below the chart field size exceeds these absolute bounds, after coloring by the indicated finitely many cases. ◻

Proof. Exclude next normalizers of elementary abelian regular groups. Such groups lie in \(A_v\) (parity of translations, high degree) and would thus be retained soluble normals. They would be minimal normal in the primitive previously good \(E\), by transitivity of every nontrivial normal subgroup and regularity. Distinct ones must commute, impossible for regular self-centralizing groups. ◻

Proof. Next exclude stabilizers of nontrivial Cartesian decompositions of the point set (product coordinates \(\Lambda\simeq\prod_{a=1}^s \Lambda_a\), \(s\ge2,\ |\Lambda_a|\ge2\); stabilize their system, i.e. equality-of-coordinate partitions, allowing slot permutations). Primitivity of \(E\) gives slot transitivity (an orbit of slots otherwise gives an invariant nontrivial equivalence), and thus common slot set size \(k\), say on \(s\ge2\) slots. Sizes \(2,3,4\) would give canonical elementary abelian regular groups (\(S_2,A_3,V_4\), respectively, multiplied over slots). For \(k\ge5\) let \(Y_i=A_k^s\) acting slotwise, inside \(A_v\) and normal in the label \(K_i\). The induced slot action of \(K_i\) contains \(A_s\) by commutators of true slot permutations. These stabilizers are proper (Cartesian coordinate agreement prevents full 2-transitivity). If \(Y_i\le H\), it is minimal normal in \(E\), and has trivial centralizer on \(\Lambda\) (point stabilizer in \(Y_i\) self-normalizing). Thus all retained ones in the pattern must coincide, determining the slots themselves by the orbits of factor subproducts. ◻

Proof. Here and in the next exclusion we use a simple consequence of 6. If \(HY_i=K_i\) for a direct normal power of nonabelian simples and \(H\)’s action on the factors is primitive, then \(P=H\cap Y_i\) is a nontrivial coordinate product and all groups between \(H,K_i\) intersect \(Y_i\) in coordinate products. Indeed \(Y_i\) is minimal normal, missed by the base so embeds as the minimal normal modulo core, and the exact interval is that of invariant subgroups above \(P\). In the earlier reductions, the subdirect case and the all-nonbottom-subdirect case are now both impossible: ties must be factor-index invariant, so primitivity allows no chain of comparable distinct proper nontrivial subdirect groups. This leaves precisely the asserted product case. ◻

Proof. For missing Cartesian labels \(HY_i=K_i\), \(H\) induces the full top image, transitive and containing \(A_s\), hence primitive on slot indices. On a line of these labels write \[N_i=E\cap Y_i=\prod_a R_{i,a}\trianglelefteq E\] a slotwise product strictly above \(H\cap Y_i\). In \(\bar E=E/\operatorname{core}_E H\), each \(\bar N_i\) contains the nonabelian simple-power monolith \(W\). A simple factor \(C\) of \(W\) lies in one of the \(\bar R_{i,a_i}\) for each \(i\): the commutators of those slot groups with \(W\) generate \(W\), are normal in \(W\), and commute for distinct slots.

Apply the two commutator steps from the proof of 23. First, since the \(N_i\) are normal in \(E\) and their images contain the perfect group \(W\), their intersection \(N=\bigcap_iN_i\) still has image containing \(W\). Next, \([N,R_{i,a_i}]\le N\cap R_{i,a_i}\) has image containing \(C\). The subgroups \(N\cap R_{i,a_i}\) are normal in \(N\), so iterating commutators among them shows that the actual tuple intersection \[T_\gamma=\bigcap_iR_{i,a_i},\qquad \gamma=(a_i),\] has image containing \(C\).

Take these intersections for all slot tuples. They are permuted by \(E\) and generate a nontrivial normal subgroup, which is transitive because \(E\) is primitive. Distinct tuple groups commute, since they act in distinct slots of at least one original system. Consider their action through the abstract product \(\prod_\gamma T_\gamma\) and fix a point. If a product of tuple elements fixes that point, retain only the components assigned to one prescribed slot \(a\) of an original system \(i\). This partial product still fixes the point: it acts only on coordinate \(a\), where its action agrees with that of the original fixing product. Repeating this restriction through every original system isolates one tuple component at a time. Thus the stabilizer in the abstract product is exactly the product of the individual point stabilizers.

Transitivity now identifies the point set with the product of the individual coset sets. Each original coordinate depends only on the tuple components assigned to its slot. The maps from these separate batches to the original coordinate sets have a bijective product map, so each is itself a bijection. Grouping the fine coordinates therefore recovers the original equality-of-coordinate partitions. Discard the trivial coset sets; the remaining coordinate directions are still permuted by \(E\). ◻

Proof. Fix two distinct systems \(i,j\) and group the fine coordinates by the occupied pairs of their original slot indices. On an aggregate set of size \(u\), the supported \(A_u\) lies in \(K_i\cap K_j\), hence in every label on the pattern line. If another system split that aggregate, then on a slice with all outside coordinates fixed its nonconstant coordinate partitions would give a nontrivial Cartesian decomposition invariant under the natural \(A_u\). Such a split has \(u\ge4\) and distinguishes pairs agreeing in some coordinate from pairs agreeing in none, contrary to 2-transitivity. Thus there is only one nontrivial fine coordinate over each occupied pair.

The map from fine coordinates to occupied pairs is \(H\)-equivariant: \(H\le E\) permutes the tuple groups and their coordinate directions, and stabilizes both original systems. By 24, the two slot actions bound the number of ordered-pair orbits on this set. Each original system is recovered by an \(H\)-invariant grouping of the fine coordinates, as proved in 27. Distinct full stabilizer labels therefore give distinct invariant partitions. The bound (22), with fibers of size one, contradicts the number of labels on the pattern line. ◻

Proof. Recall a short primitive-group observation after these exclusions. A minimal normal \(P=C^u\) of a primitive group \(E_0\) here is transitive with \(C\) nonabelian simple (an abelian one would be regular), \(E_0=P(E_0)_x\) at a point \(x\). Its point stabilizer \(Q=P_x\) has no proper intermediate increase towards \(P\) normalized by \((E_0)_x\). Taking its coordinate product closure, \(Q\) is either a product or subdirect. In the product case \(u>1\) would give a Cartesian coset product; in the subdirect case the same holds for multiple tie parts (uniform sizes by factor transitivity of \((E_0)_x\)). Indeed the point stabilizer of \(E_0\) is transitive on factors (minimal normality, with \(P\) itself not permuting them). In the product situation, including when taking batches given by multiple tie parts, the cosets \(P/Q\) accordingly split as a product of coset sets. Their sizes are uniform and nontrivial, \(P\) respects this product coordinatewise, and \((E_0)_x\) respects it by its conjugation/transport of the factors or batches with their respective stabilizers in \(Q\). Thus for \(u>1\) we have just a full diagonal, possibly with twists, and \(v=|C|^{u-1}\). For \(u=1\) with nontrivial \(C_{E_0}(P)\), that centralizer must be transitive so both are regular opposite simple groups, again giving a transitive power \(C^2\) with full diagonal point stabilizer. Consequently any non-almost-simple case here normalizes a transitive \(Y=B^s,\ s\ge2\), \(B\) nonabelian simple, with full diagonal point stabilizer. ◻

Proof. Finally exclude the normalizers \(K_i\) of all such \(Y\). They are proper since \(Y\le A_v\) by perfection, and induce at least \(A_s\) on factors by diagonal-coordinate factor permutations and commutators. For retained \(Y\le H\), transitivity of \(E\) on factors, when present, makes \(Y\) minimal \(E\)-normal with trivial centralizer (the diagonal is self-normalizing), leaving no distinct such system. If instead \(E\) has multiple factor orbits, their products are each transitive so use at least \(s-1\) factors each, forcing just two opposed regular simple groups, the only minimal normals in \(E\), again determining \(Y\). If \(K_i\)’s top action is intransitive, likewise \(s=2\), and at least one individual factor is retained by (\(*\)); such factors are regular simple minimal normals of \(E\), determining \(Y\) by taking also their opposite regular group (distinct such minimal normals commute so determine the same pair).

The missing transitive case. It remains that \(HY=K_i\) and \(H\) induces a transitive top action containing \(A_s\). By the interval product consequence, \(N=E\cap Y=\prod R_a\) is a nontrivial slot product with each projection proper (since \(E<K_i\)). Choose a minimal \(E\)-normal \(P'=C^u\) inside it. Each factor of \(P'\) is contained actually in one \(R_a\), since \([P',R_a]\le P'\cap R_a\), normal in \(P'\), and those commutators generate \(P'\) by perfection. This assignment is \(E\)-equivariant with some common positive number \(h\) per slot, so \(u=sh\). The primitive observation applied to \(E\) gives \(Q=P'_x\) a full diagonal. If \(h>1\), take inside \(P'\) the product of the projections of \(Q\) along the factor blocks of the assignment. It is a proper intermediate increase, normalized by \(E_x\), impossible. If \(h=1\), point degrees give \(|C|^{s-1}=|B|^{s-1}\), contrary to containment in the proper \(R_a\). ◻

Proof. These arguments use only successively primitive, non-affine, and non-Cartesian prior anchors as needed and finite case subdivisions. Thus after the exclusions the tested descendant with its prepared interval is primitive almost simple, with an actual simple transitive socle in \(A_v\), supplemented by \(H\) via (\(*\)) (it cannot be retained, since then factoring by the base core leaves a soluble quotient by Schreier, inconsistent with the prepared interval), as desired. ◻

Proof. Two useful uniform comparisons follow. Write \(b=\dim G\). For \(P\) large, \(B(\Psi)=(G^\Psi)'\) is simple of order at least \(P^b(1-o(1))/z\) inside \(G^\Psi\) of order \(P^b(1+o(1))\), with index at most a center bound \(z\) from the root system. In particular \(|B(\Psi)|\) exceeds the entire fixed-group order at any smaller positive height, since \(z<(\text{unit})^b\) by the usual center sizes. This applies uniformly to comparison with smaller heights: below parameter \(\sqrt P\) even the rough order bounds suffice, and otherwise the two product formulas both have small relative error. Also \(B(\Psi)\) lies in no proper closed subvariety of bounded degree, for \(P\) sufficiently large for that bound. On any irreducible component of dimension \(e<b\) and bounded degree, the fixed-point equations of degree \(O(P)\) cut a finite set of size \(O(P^e)\), by affine Bézout (combine the equations successively generically to cut to dimension zero). Hence a point map of positive height fixing \(B(\Psi)\) pointwise must be a power of \(\Psi\): divide its height by the given height and compose off that power on points. Positive remainder is impossible by orders, zero-height nonidentity by the bounded equations of its fixed locus.

For the last assertion, if \(\alpha\) has height \(q\operatorname{ht}(\Psi)+r\) with \(0\le r<\operatorname{ht}(\Psi)\), the point map \(\Psi^{-q}\alpha\) still fixes \(B(\Psi)\) pointwise. If \(r>0\), its fixed group is too small. If \(r=0\), a nonidentity algebraic automorphism has a proper fixed locus with uniformly bounded equations, contradicting the second assertion. Hence \(r=0\) and \(\alpha=\Psi^q\). ◻

Proof. Consider a proper \(K\) in the interval, above a tested large-index label, and \(\Gamma=K\cap S\ne1\). If insufficiently general for the sandwich, it lies in a proper bounded-degree-equation closed \(X\subset G\) by Larsen–Pink as used above. Take the subgroup of our point-map group preserving \(S\) and inducing \(M\)-elements on \(S\); it maps onto \(M\) by the automorphism description. Let \(\tilde K\) be the full preimage of \(K\) there (so each such map normalizes \(\Gamma\)). Then \[Z=\bigcap_{\alpha\in\tilde K,\ \gamma\in\Gamma} \alpha(\gamma X)\] contains \(\Gamma\), is proper closed with equations of uniformly bounded degree, left-\(\Gamma\)-invariant and \(\tilde K\)-invariant. Even for the whole intersection one can just span all the equations occurring in the fixed bounded-degree space. Its point-map stabilizer (within that preimage of \(M\)) thus defines a label \(K'\ge K\), since the stabilizer before projection contains the kernel there (already in \(\tilde K\)). Identifying \(S\) with its inner actions on itself, \(K'\cap S=N_G(Z)\cap S\) by using the actual inner lifts to \(G\). The conjugation normalizer here has bounded-degree equations (test the equations of \(Z\) on all conjugates of its points by \(g,g^{-1}\)). It is proper since otherwise all conjugates of \(\Gamma\) left-stabilize \(Z\), while their normal closure is Zariski-dense by simplicity and trivial center. The degree comparison for \(S\) thus makes \(K'\) proper.

Here \(N_G(Z)\) denotes the conjugation stabilizer, not the left stabilizer. To justify its properness explicitly, if every \(g\in G(k)\) normalizes \(Z\), left invariance under \(\Gamma\) implies left invariance under every \(g\Gamma g^{-1}\). The Zariski closure of the group generated by these conjugates is a nontrivial closed normal subgroup of the simple adjoint group \(G\), hence is \(G\). Since \(Z\) is closed and nonempty, left invariance by this dense subgroup forces \(Z=G\), a contradiction. The nonidentity of \(\Gamma\) suffices because the adjoint group has trivial center. ◻

Proof. At the current parent flat \(W\), test a sufficiently large fixed increment \(t\) in a private chart. We first find a flat \(U\) of that increment for which every \(K\in[K_U,M]\) admits the sandwich for \(K\cap S\). If every trial failed, 30 would supply full stabilizer witnesses above the trial labels. The affine-pattern construction gives an independent parameter list whose successive label intersections decrease strictly for more steps than the dimension of the relevant polynomial space. Descent remains strict after intersection with \(S\), since \(M=HS\) and every overgroup of \(H\) is determined by its intersection with \(S\). Each strict descent must add a defining polynomial outside the span of the preceding equations. But all conjugation-normalizer equations lie in one bounded-dimensional space, a contradiction. This step needs only a bounded independent list, no prior anchor, and tests relative to the current parent.

Put \(E=K_U\) and \(\Gamma_K=K\cap S\) for \(E\le K\le M\). We now have \[B(\Psi_K)\le\Gamma_K\le G^{\Psi_K}.\] The reserved interval length below \(E\) makes every parameter here large enough for 29. Indeed, if that residual height is \(h\), then \(|\Gamma_K|\ge|\Gamma_E|\ge[E:H]\ge2^h\), whereas all fixed groups below a prescribed parameter have bounded order for the fixed type list. Write \(B_E=B(\Psi_E)\). Perfection and the abelian quotient of a fixed group by its derived subgroup give \(\Gamma_E'=B_E\).

Since \(B_E\le\Gamma_K\), the map \(\Psi_K\) fixes \(B_E\) pointwise. Rigidity therefore gives \(\Psi_K=\Psi_E^{a_K}\) for a positive integer \(a_K\). The ambient map \(F_A\) likewise fixes \(B_E\), so \(F_A=\Psi_E^d\). It also fixes \(B(\Psi_K)\) pointwise, and rigidity applied to that simple group makes \(F_A\) a positive power of \(\Psi_K\). Consequently \[F_A=\Psi_E^d,\qquad \Psi_K=\Psi_E^{a_K},\qquad a_K\mid d.\] These are equalities of point maps, so each exponent determines an actual embedded fixed group.

Every lift of \(E\) normalizes \(\Gamma_E\) and hence \(B_E=\Gamma_E'\). Conjugating \(\Psi_E\) by such a lift gives a map of the same height fixing \(B_E\) pointwise; rigidity makes it equal to \(\Psi_E\). Thus the lifts commute with \(\Psi_E\). For each divisor \(a\) of \(d\), put \(B_a=B(\Psi_E^a)\). This group lies in \(S\): its elements are fixed by \(F_A\), and perfection puts it in \((G^{F_A})'\). It is normalized by \(E\), so \(E B_a\) is an overgroup label. Moreover \[(E B_a)\cap S=\Gamma_E B_a\subseteq G^{\Psi_E^a}, \qquad E B_1=E,\qquad E B_d=M.\] If \(a\mid b\) with \(a<b\), then \(B_a\le B_b\), and 29 gives \(|B_b|>|G^{\Psi_E^a}|\). Hence \(E B_a<E B_b\). A saturated divisor chain therefore gives \(\Omega(d)\) strict steps in the exact height-\(t\) interval \([E,M]\), where \(\Omega(d)\) counts prime factors with multiplicity. Thus \[\tau(d)=\prod_{\ell^a\parallel d}(a+1) \le2^{\Omega(d)}\le2^t.\]

For a fixed exponent \(a_K\), the sandwich leaves only boundedly many choices for \(\Gamma_K\): its quotient by \(B_{a_K}\) is a subgroup of the bounded-order group \(G^{\Psi_E^{a_K}}/B_{a_K}\). If \(z\) bounds that order, \(C=2^z\) bounds the number of choices. Since \(K=H\Gamma_K\), no additional choices of outer automorphisms occur. We conclude that \(|[E,M]|\le C2^t\).

The reverse-flat interval, however, contains at least \(q^{\lfloor(t-1)^2/4\rfloor}\) intermediate subflats, by the ordinary linear-subspace count; this bound also covers an empty starting flat. For sufficiently large fixed \(t\) it exceeds \(C2^t\), already using \(q\ge5\). This is the required contradiction, uniform in characteristic and parameter, whenever a bounded-type parent has a stage remaining. ◻

Proof. Any nontrivial irreducible projective representation of \(S\) in characteristic \(p\) has dimension at least a positive absolute constant times \(v\) in high rank: its highest-weight Weyl orbit already has at least that size (permutations of a nonconstant weight word in type A; signed permutations, possibly with even sign changes, in B,C,D). This bound suffices here and also bounds any nontrivial projective representation, using perfection if the lift has only trivial composition factors. In the last case a lift with all composition factors trivial has unipotent, hence soluble, image. A perfect group has no nontrivial soluble image, so a nontrivial representation must have a nontrivial composition factor. ◻

Proof. Any alternating subsequence of the path has bounded length. Suppose there are three socles \(A_n>A_m>A_e\) as subgroups (\(n>m>e\)), with the transitivity for each ancestor pair. For large \(e\), \(m\ge\binom e2\). In fact an intransitive point stabilizer in natural \(A_e\)-coordinates giving index \(>e\) has index at least \(\binom e2\) (if within a point group \(A_{e-1}\) properly, use simplicity for minimum proper index at least \(e-1\)). Proper transitive subgroups of \(A_e\) have index at least exponential in \(e\) for large \(e\), by Bochert’s Theorem in the primitive case, and the block partition count in the imprimitive case. For instance with \(b\) blocks of size \(a\), both at least two, pair the blocks disjointly and interchange corresponding positions arbitrarily in each pair, giving at least \(2^{(a-1)\lfloor b/2\rfloor}\) partitions. Thus \(|A_e|=\exp(o(m))\). If \(D\) is the stabilizer in \(A_m\) on the \(n\)-set, \(A_m=A_eD\) by transitivity, so \(D\) cannot be transitive on the natural \(m\)-set by the same index bound. Nor can it have just orbits of sizes 1 and \(m-1\): the bound on proper transitive subgroups on \(m-1\) would force \(D=A_{m-1}\), contrary to \(n>m\). It thus stabilizes an \(h\)-set, \(2\le h\le m-2\), making \(A_e\) \(h\)-homogeneous on \(m\). This implies 2-homogeneity: the count inside an \(h\)-set of edges from any one pair orbit is constant; comparing sets consisting of \(i\) or \(j\) and \(h-1\) other points shows the adjacency rows at \(i,j\) agree away from both (all such sums of differences vanish). Thus there is just one edge orbit. Even order then gives 2-transitivity by an involution flipping an ordered pair. By the classification of finite 2-transitive groups, alternating simple groups of sufficiently large degree have no such nonnatural action. This proves the bound. The permutation-group inputs used here are the primitive subgroup bound and the classification of finite doubly transitive groups; see Dixon and Mortimer (1996). ◻

Proof. For high classical pairs in distinct characteristics, absolute constants give \(v_i\ge P_j^{c v_j}\) for ancestor \(i\), descendant \(j\), with \(c>0\). Here is a direct bound. The high classical \(S_j\) contains images of paired special linear groups \({\rm SL}_h(P_j)\), \(h\ge c' v_j\), using a subspace in linear type or paired maximally isotropic coordinates in the form types (one may restrict scalars of matrix entries in the unitary coordinates to \(\mathbf F_{P_j}\)). These act nontrivially including on column shear elements. The restricted projective representation lifts linearly on that \({\rm SL}_h(P_j)\), centrally closed for such high \(h\). Its abelian column shears of size \(P_j^{h-1}\) are diagonalizable here and have some nontrivial character occurring; the \({\rm SL}_{h-1}\) action is transitive on all their nontrivial characters. This proves the assertion. Thus we cannot have classical nodes \(i<j<l\) (indices increasing downwards) with \(p_i\ne p_j,\ p_j\ne p_l\), since then \[\log|S_l|=O(v_l^2\log P_l)=O(\log^2 v_j)\] whereas \(v_i\) is at least exponential in \(v_j\). But \(|S_l|\ge v_i^2\) by its absolutely irreducible projective action in \(i\): lift that action to its perfect central cover. Absolute irreducibility implies that its matrices span the full \(v_i\times v_i\) matrix algebra. Central lifts of one projective element are scalar multiples, so at most \(|S_l|\) matrices are needed in this span. This contradicts the two exponential lower bounds for sufficiently high rank.

Hence there are boundedly many interruptions of consecutive high-classical runs in the same characteristic, on paths avoiding bounded types. It suffices to bound those runs. We now work in such a run, notation \(S_i=B_{W_i}\). ◻

Proof. To verify this, in any tensor factorization into nontrivial slots with \(B\) fixing the factor algebras individually, lift each projective action to \(\widehat B\). They tensor to the original lift by perfection, and are all irreducible. Regard them as restrictions of rational simples with restricted highest weights for the finite-group parameter of \(B\). Their tensor is then irreducible rationally as well. By Steinberg decomposition, even the subproducts of the contributing digit modules at any one exponent (untwisted) must be irreducible.

Recall the top Levi face of a rational simple \(L(\lambda)\) for nodes \(I\), using weights differing downwards from highest only by \(I\)-roots, is a simple highest module for the Levi derived group with the corresponding coefficients. Indeed it is generated by the highest vector there by distribution-algebra triangular decomposition (PBW); a nonzero Levi submodule on that face has a highest unipotent-fixed vector, fixed also by outside raising groups since they go off the weights. In the simple module this forces the highest line. The face of a tensor product with the sum as highest weight tensors the corresponding faces. Now at any one digit, no two modules from different slots can have active coefficients in the same simple-bond connected component of the Dynkin diagram. Otherwise take a shortest simple-bond path between active nodes of two different slots, possibly just one node (minimize over such pairs in a single simple-bond component). For the two modules the faces on that path have endpoint highest weights \(a\omega_1,b\omega_\ell\) of type \(A_\ell\), \(1\le a,b<p\), with no other supports there by minimality. These are symmetric powers of natural and dual modules (the same natural type for \(A_1\)), simple for these exponents. For example simplicity follows using torus monomials of distinct weights and elementary root transfers of powers with nonzero coefficients at degree less than \(p\). Their tensor is reducible by contraction (one derivative on each side paired), a nonzero equivariant map to a strictly smaller space. This contradicts the face property.

Thus coefficients are disjoint with no carry, and the digitwise sum highest weight is restricted and must be exactly the given tuple for \(V\). Take one atom per nonzero digit except for the following possible refinement in B,C: if the regular chain and the node beyond the multiple bond both contribute and can be separated in a factorization, split them. Indeed in such a separation their two rational pieces tensor irreducibly to the digit module, so all such splits at all digits can be made together. This gives a factorization, and any other one as above has its slot representations given by grouping the atoms (by the coefficient argument and uniqueness of the simples). This is true of the actual matrix subalgebras: equivalent factorizations with fixed slot representation classes have their tensor identifications differing, after intertwiners, by a scalar since \(V\) is absolutely irreducible for \(\widehat B\). In more detail, separated active components at a digit give actual rational factor modules by taking the digit pieces of the slot modules of that separation; their subproduct, untwisted, is irreducible as above, with the individual component weights and their sum all forced. Thus it realizes the indicated split even as rational modules of that digit, independently of the other digits. This proves the claim. Any semilinear or anti-action on the algebra normalizing the projective \(B\)-action (using inverse with an anti-action) permutes the atoms, by the same intrinsic description. ◻

Proof. The map \(\sigma\) permutes the atoms because it fixes the projective \(B\)-action, using inversion with the anti-action in unitary type. We first specify lifts of the ancestor automorphisms. Use finite-field projective semilinear representatives, similarities in the form types, and also graph anti-actions in linear type. Lift their field exponents by integral powers of split Frobenius on \(k\). These lifts commute with \(\sigma\): in unitary type this is the hermitian-similarity identity extended to \(k\), and otherwise it follows from the finite-field matrices.

The finite projective representatives are faithful on the ancestor simple group. Indeed, if a collineation with coefficient exponent \(0\le a<\log_p|F|\) centralizes it, perfection identifies its natural finite lift with the coefficient twist of highest weight \(p^a\omega_1\). Restricted-weight uniqueness forces \(a\equiv0\pmod{f_i}\). In unitary type it also rules out \(a=f_i\), since wrapping produces the distinct diagram-dual natural weight. With the field part removed, absolute irreducibility leaves only a scalar. The same weight comparison rules out a linear-type duality in the kernel. Thus closure lifts differ only by arithmetic ground-field Frobenius powers. A system stable under those powers as a collection has a well-defined full stabilizer in the finite ancestor. In unitary type the representatives here are collineations; \(\sigma\) itself is the semilinear adjoint anti-action and \(\sigma^2\) is arithmetic Frobenius for \(F\).

We next check the forms needed for descent. In bilateral types the perfect lift of \(B\) preserves the ambient form. Restricted-weight uniqueness makes its digits self-dual, and any pieces separated across the B or C multiple bond are self-dual as well. Their invariant forms tensor to the ambient form by uniqueness, so the atom algebras are individually selfadjoint. In characteristic two each factor form is alternating: a nonalternating symmetric invariant form would give the nonzero invariant functional \(x\mapsto\sqrt{b(x,x)}\) on a nontrivial irreducible module.

If a quadratic form is present in characteristic two, every grouping into at least two aggregate factors has the canonical quadratic with the given polar form and value zero on simple tensors. To construct it, prescribe zero on tensor-basis vectors. On a simple tensor, polar terms with a repeated index vanish and the remaining terms cancel in pairs by exchanging one index. This quadratic is invariant under the lifting group. Its difference from the given invariant quadratic is the square of an invariant linear functional, hence zero.

Suppose \(\sigma\) has more than one orbit on atoms, and tensor together the atoms in each orbit. The aggregate algebras descend under arithmetic Frobenius to central simple \(F\)-factors, which split over the finite field. In unitary type first descend under \(\sigma^2\); stability under \(\sigma\) then gives selfadjointness for the hermitian adjoint. Bilateral selfadjointness was just proved. Candidate lifts commute with \(\sigma\) and therefore permute these aggregate factors. This is exactly a ground-field tensor system forbidden by the preceding exclusions.

Its full stabilizer is proper. Otherwise \(S_i\) acts trivially on at most \(\log_2v_i\) slots, by simplicity and comparison of its order with the corresponding factorial. Absolute irreducibility makes each slot action nontrivial. A smallest slot has degree at most \(\sqrt{v_i}\), contrary to 31 at the fixed high threshold. There is therefore one \(\sigma\)-cycle. Its atoms have equal degree \(u\), and \(u^t=v_i\). ◻

Proof. For each prime \(s\mid t\), group the atoms by their residue classes modulo \(s\) around the \(\sigma\)-cycle. Candidate lifts commute with the cycle, so they normalize this canonical coarsening \(\mathcal P_s\). Its full stabilizer is well defined by 35. If the entire socle \(S_i\) preserved it, its permutation action on the \(s\) slots would commute with an \(s\)-cycle. That image is cyclic, hence trivial by perfection. A slot of degree \(u^{t/s}\le\sqrt{v_i}\) would then carry a nontrivial projective action, contradicting 31. Thus each coarsening gives a proper containing label.

Let \(d\) be the product of distinct primes already imposed. Their common refinement is \(\mathcal P_d\), with \(d\) slots of degree \(h=u^{t/d}\). In \(G_i(k)\) take the product \(X_d\) of the supported connected projective groups on these slots: \(\operatorname{PGL}_h\) in ambient type A, and the connected projective classical groups of the factor forms in bilateral types. Put \(X_1=G_i(k)\) itself. In particular, at \(d=1\) we keep the orthogonal ambient group in characteristic two. For \(d>1\) the tensor construction embeds these products in the connected ambient group; the canonical-quadratic observation in 35 justifies the characteristic-two orthogonal case.

The map \(\Phi_i\) cycles the \(d\) factors through \(\sigma\), using inversion on units in the anti-case. On one factor \(\Phi_i^d\) is an ordinary positive Steinberg map of parameter \(P_i^d\): after this coefficient twist, its matrix-algebra action is an isomorphism or anti-isomorphism preserving the factor group. The fixed group \(X_d^{\Phi_i}\) therefore projects isomorphically to that one-factor fixed group. While \(h\) is sufficiently large, its derived group gives a full high-rank natural simple \(Y\) on the chosen slot. It lies in \(S_i\) by taking commutators and preserves every previously imposed coarsening.

For a new prime \(s\), suppose \(Y\) also preserves \(\mathcal P_s\). Perfection and commutation with the induced \(s\)-cycle make it fix those slots individually. Inside the chosen \(d\)-slot, the two coarsenings intersect in \(s\) tensor factors of degree \(h^{1/s}\), by the Chinese remainder description of their atom indices. At least one factor action is projectively nontrivial, since their tensor is the natural action of \(Y\) on that slot. Its degree contradicts 31. Thus the next stabilizer makes the intersection strictly smaller.

Interval height bounds the primes added while \(h\) is high. At stopping, \(h\ge2^{t/d}\) and \(t/d\) is divisible by every unused distinct prime. Bounded \(h\) leaves only absolutely many unused primes. This gives the claimed additive constant \(c_1\) in the actual interval, whose height is unchanged by the centralizer quotient. ◻

Proof. The inner centralizer of \(B\) is trivial by projective absolute irreducibility and perfection. Modulo inner automorphisms the point-map group is abelian, with at most two diagram choices at any height. Consequently two maps of the same height and diagram choice fixing \(B\) agree, and all fixing roots commute. Conjugating a root by a candidate lift changes no outer class and still fixes \(B\) pointwise. The same argument shows that the lift commutes with the root and normalizes \(S(\Theta)\).

An inner element normalizing \(S(\Theta)\) likewise commutes with \(\Theta\), by applying this centralizer argument to the natural absolutely irreducible fixed simple. Since \(G_i\) is adjoint, that element itself lies in \(G_i^\Theta\). The root has height \(f_i/s<f_i\), so the order comparison makes its normalizer proper in the ancestor. We have thus obtained actual proper labels containing the candidate.

Choose one root \(\Theta_s\) for each occurring prime. For a product \(d\) of primes already chosen, select integers \(a_s\) with \(\sum_{s\mid d}a_s d/s=1\) and put \[\Theta_d=\prod_{s\mid d}\Theta_s^{a_s}.\] The roots commute, and \(\Theta_s^s=\Phi_i\) gives \[\Theta_d^{d/s}=\Theta_s,\qquad \operatorname{ht}(\Theta_d)=f_i/d.\] For example, raising the product to \(d/s_0\) gives \(\Theta_{s_0}^{a_{s_0}d/s_0}\) times \(\Phi_i^{\sum_{s\ne s_0}a_s d/(ss_0)}\), which is exactly \(\Theta_{s_0}\). This verifies the identity also with diagram components. For the empty product set \(\Theta_1=\Phi_i\).

The group \(S(\Theta_d)\) is fixed by each previous root and lies in all their normalizer labels. If it normalized the fixed simple for a new prime root \(\Theta_l\), the preceding inner-normalizer argument would make it fixed by \(\Theta_l\) as well. It would then lie in the fixed group of the combined root \(\Theta_{dl}\), contradicting \[|S(\Theta_d)|>|G_i^{\Theta_{dl}}|.\] Each new prime therefore makes the label intersection strictly smaller. There are at most as many primes as the interval height, and at most two roots at each associated height. The number of normalizer labels is consequently at most twice that height. ◻

Proof. We record why root-free protection precludes equal-natural embeddings within one root series. If the action of \(B\) on \(V_i\) is projectively its own natural module (allowing Frobenius twists, also the diagram natural/dual twist in A), both groups in the same classical series and dimension, then \(B=S(\Psi)\) inside \(G_i\) for a standard Steinberg map up to conjugacy. Indeed identify the natural spaces over \(k\) by the representation equivalence (split Frobenius twists preserve the standard finite images); forms match up to scalar by uniqueness, also the quadratic when needed by square-difference uniqueness on the finite group. This identifies the corresponding algebraic adjoint groups. Divide the height of \(\Phi_i\) by that of \(\Psi\); after composing off that power of \(\Psi\) we still fix \(B\) pointwise, thus cannot have positive remainder by the order comparison. At height zero, an inner centralization is trivial; so is a possible D-diagram centralization (acting in \({\rm PGL}(V_i)\)); and a non-inner A-graph centralization would give projective self-duality of the finite natural representation, impossible by the distinct restricted weights and perfection. So \(\Phi_i=\Psi^z\), \(z>1\) by strict inclusion, and some prime root fixes \(B\). ◻

Proof. Work inside a successful pruned flat of extension rank \(r\) at the current node \(W\). All its comparable candidate labels have the prepared properties and actual nested simples, by 5. Color its proper extensions with at most \(c\) colors, recording in particular whether their socle continues the current same-characteristic high-classical run. Consider a uniform random flag at extension ranks \[\ell,2\ell,\ldots,(c+1)\ell,\qquad r=(c+1)\ell.\] Every flag has an equal-colored pair. Hence some two ranks \(a<b\) have matching probability at least \(\binom{c+1}{2}^{-1}\). Conditioning on the \(b\)-extension gives a particular \(U\) for which at least that fraction of the \(a\)-extensions \(L\subset U\) have its color. If this color does not continue the run, any such \(L\) needs no additional protection.

Otherwise apply the two counts to \(B_U\) in each ancestor \(i\) of the current run, including \(W\) itself. These lists detect every failure of protection at a matching \(L\). Indeed, if \(t(B_L)>1\), a prime congruence coarsening of its atoms has a stabilizer containing \(K_L\). Since \(B_U\le B_L\) fixes its factors individually, each factor groups atoms of \(B_U\). The partition is invariant under the single Frobenius cycle; a cycle has a unique invariant partition into \(s\) congruence classes. It is therefore one of the coarsenings counted for \(B_U\). A prime root fixing \(B_L\) likewise fixes \(B_U\) and occurs on its root list.

None of the counted labels contains \(K_W\). For an earlier ancestor, such containment would make \(B_W\) fixed by the root, or individually fix the coarsening slots: its slot permutation commutes with the cycle and is trivial by perfection. Either conclusion contradicts the protection already imposed on \(W\). For the current ancestor, properness of the counted labels gives the same noncontainment. Thus each bad label is \(K_Z\) for a subflat \(Z\subseteq U\) with \(Z\not\subseteq W\).

For a uniform \(a\)-extension \(L\) in \(U\), the probability of \(Z\subseteq L\) is at most \(2q^{a-b}\): it suffices to require one fixed point of \(Z\setminus W\) in the ordinary affine-flat count. Here \(q\) is the chart field size. The two counts give \(O(1+\operatorname{ht}[K_U,K_{W_i}])\) bad labels per ancestor. Reverse rank differences bound this height by \(Dr\) over at most \(D\) path steps. The total number is therefore at most \(CD(1+Dr)\) for an absolute \(C\) after the colors and thresholds are fixed. Since \(b-a\ge\ell\), their union occupies a fraction at most \[2CD(1+D(c+1)\ell)q^{-\ell}\] of those extensions. This tends to zero with \(\ell\), whereas the matching-color fraction is at least \(\binom{c+1}{2}^{-1}\). Choose \(\ell\) large enough also for the earlier fixed requirements. A matching \(L\) outside the union has one atom and no fixing prime root relative to every ancestor in the run. It retains the structural exclusions because it lies in the successful pruned flat.

There is no circular choice of parameters: the following path bound concerns paths already satisfying these group protections. Its absolute constant is obtained first, then \(D\) is fixed, and only then are \(\ell\) and \(r\) chosen. ◻

Proof. Put \(\mathcal G=k[J]^{J^\Phi}\), with right finite invariance and left translation. It suffices to show \[(D'\otimes(\mathcal G/k))^J=0.\] Indeed, a finite-fixed vector splitting an extension of \(k\) by \(D'\) defines the equivariant function sending \(g\) to its translate of that vector. It is right finite invariant, and its image in the quotient module is constant. Modulo constant functions it therefore lies in the displayed invariant space. Vanishing makes the function itself constant, giving a rationally fixed lift.

The Lang map \(g\mapsto g\Phi(g)^{-1}\) identifies the right quotient \(J/J^\Phi\) with \(J\) (Lang 1956; Steinberg 1968) (Milne 2022, Definition 17.95 and Theorem 17.96). It is surjective and étale since \(d\Phi=0\), hence faithfully flat. Its fibres are exactly the right cosets of the finite reduced fixed-point scheme: \(h=gx\) with \(x=\Phi(x)\). Descent of regular functions identifies \(\mathcal G\) with the regular function algebra carrying the twisted left-right action.

The Donkin–Koppinen filtration begins with constants and has tensor pairs of induced highest-weight modules as its sections (Mathieu 2000, Theorem 4.3 and its preceding construction) (Bendel et al. 2012, sec. 2.4). Pulling this two-sided filtration back to the twisted diagonal gives an exhaustive filtration of \(\mathcal G/k\) with sections \[\nabla(\nu)\otimes\nabla(\nu^*)^{[\Phi]}, \qquad \nu\ne0\ \text{dominant}.\] Here \(*\) denotes highest-weight dual, \([\Phi]\) pullback, and \(\Delta\) below is a Weyl module. A nonzero invariant after tensoring a section by \(D'\) gives a nonzero map \[\Delta(\nu)^{[\Phi]}\longrightarrow L(\xi)\otimes\nabla(\nu).\] The source is still generated by its highest vector, since \(\Phi\) is surjective on algebraic-group points. The map is nonzero there, so \[P\tau(\nu)\le\xi+\nu\] in dominance order. The source is also trivial on the \(f\)-th Frobenius kernel \(J_f\). Its nonzero image therefore gives \(\operatorname{Hom}_{J_f}(L(\xi)^*,\nabla(\nu))\ne0\), with kernel and invariance understood as group schemes.

Since \(\xi\) is \(P\)-restricted, the restricted-simple theorem and Steinberg decomposition force a rational composition factor of \(\nabla(\nu)\) whose highest weight has the form \[\xi^*+P\gamma\le\nu,\qquad \gamma\text{ dominant}.\] Indeed, every rational simple restricts to copies of the \(J_f\)-simple for its \(P\)-restricted part, and those restricted simples are pairwise distinct. Pairing the two inequalities with \(\rho^\vee\) gives the diagram- and dual-invariant height comparisons \[(P-1)\operatorname{ht}(\nu) \le\operatorname{ht}(\xi)\le\operatorname{ht}(\nu).\]

Only \(P=2\) and equality remain. Then \(p=2\), \(f=1\), and \(\xi\) has an active coefficient one. Equality gives \(P\tau(\nu)=\xi+\nu\), so the image of the source highest vector lies on the unique tensor highest line. Let \(v_\xi\otimes v_\nu\) span it and choose \(\alpha\) with \(\langle\xi,\alpha^\vee\rangle=1\). The lowering operator has value \[(f_\alpha v_\xi)\otimes v_\nu +v_\xi\otimes(f_\alpha v_\nu).\] The first summand is nonzero by the simple-root bracket, and the first-factor weights \(\xi-\alpha\) and \(\xi\) differ. The summands cannot cancel, including in characteristic two. The line is not \(J_1\)-fixed, contradicting the kernel-trivial source action. Thus no section has an invariant. Every invariant would occur in the exhaustive filtration and have a nonzero successive image, so the required vanishing follows.

The restricted-kernel and regular-bimodule results are the simply connected theorems in Jantzen (2003, pt. II, Chapter 3 and II.4.20); see also Lin and Nakano (2006, sec. 1.4) and Mathieu (2000, Theorem 4.3). The pullback to the ordinary diagram-twisted diagonal uses no complete reducibility. ◻

Proof. Untwist \(2^y\) to \(L_J(\eta)\); split Frobenius permutes \(J^\Phi\) because \(\Phi\) is a standard pinned graph/field map. Exact invariance under the chosen finite linear lift holds by perfection: its similitude scalar is a character. Rational invariance of the polar form makes \[c(g)=gQ-Q\] a regular cocycle with values in the subspace of squares of linear forms. We will take its coefficientwise square root to obtain a cocycle in \(L_J(\eta)^*\).

The target Lie action on squares is zero, so the differential of \(c\) at the identity kills commutators. The Lie algebra of simply connected type D is perfect also in characteristic two: simple coroots span the torus part and are opposite-root brackets, and every D-root pairs oddly with some coroot. Thus this differential, and by translation every differential of \(c\), vanishes.

Each coordinate of \(c\) is consequently a square in the function field, using the kernel of differentials over a perfect field. Its square root is regular by normality. Taking roots of the coefficients in a fixed dual basis therefore gives a regular linear-form cocycle, zero on \(J^\Phi\). Additivity, injectivity and equivariance of squaring on \(k\)-points give its cocycle identity.

The square-root cocycle takes values in the nontrivial restricted dual module after untwisting. By 36, its class is zero, so it has the form \(g\ell-\ell\) for a linear functional \(\ell\). Its vanishing on \(J^\Phi\) makes \(\ell\) finite-fixed. The restricted finite module is nontrivial and irreducible, so \(\ell=0\). Thus the square-root cocycle, and therefore \(gQ-Q\), is identically zero. The perfect-Lie-algebra argument above is used here only for type D in characteristic two. ◻

Proof. Consider path nodes \(i<j<l\) in one protected run, of the same root series A, B, C or D. Use consistent finite-group identifications at each node (actual simple \(B_{W_j}\), its projective natural realization, and simply connected cover, for both its incoming and outgoing edges). Thus the projective restriction along \(i\to j\to l\) is just via inclusion of the same nested simples as \(i\to l\); the weight colors below also respect diagram changes. Write \(G_s=\widehat G_j\) and \(J=\widehat G_l\) temporarily for the simply connected algebraic groups. Choose pinned coordinates with standard graph/field Frobenius on each (conjugating the finite realizations accordingly if necessary). The edges \(ij,jl\), evaluating natural modules on descendants projectively, have highest weights \(p^x\lambda,p^y\eta\) respectively for the lifts, \(\lambda,\eta\ne0\) \(p\)-restricted, since each has one atom. We claim \(L_J(p^y\eta)\) on the natural space for \(j\) gives a morphism \(J\to (G_s)_{\rm ad}\) inducing the given projective action on finite points. This is immediate in A. In the remaining types the finite representation preserves the bilateral form (perfection); by restricted-weight uniqueness it is rationally self-dual, and the invariant form rationally therefore agrees up to scalars. This suffices except for D in characteristic two, where 37 supplies rational invariance of the quadratic form as well.

The morphism to the adjoint group (in form types one can first map by the given linear representation to the connected classical isometry group, using connectedness of \(J\), then take its adjoint quotient) lifts to \(G_s\) by simply-connectedness. For example pull back the central isogeny; the reduced identity group in the pullback maps surjectively to \(J\) (finite surjection, finitely many components), by a central isogeny with kernel of multiplicative type; its radical has trivial image and hence dimension zero, so it is semisimple and maps isomorphically by the simply connected root datum of \(J\). Its natural action agrees linearly with the above one (no scalar characters). On finite points the lift differs at most by central elements from lifts to the usual finite covering group at node \(j\). Indeed both give the prescribed projective matrices on the natural space (the image of the finite cover surjects projectively onto the given socle), and the kernel of the projective natural map on simply connected algebraic-group points is central. Thus composing \(L_{G_s}(p^x\lambda)\) yields on restriction to \(J^\Phi\) projectively the direct-edge action \(il\). The rational composition is irreducible by finite irreducibility. Its highest weight \(\mu\) for \(J\) is again a single \(p\)-digit times a power of \(p\), by Steinberg decomposition and the protected individual tensor indecomposability on finite points. Indeed multiple rational digits here would give multiple nontrivial-dimensional factors fixed individually by the finite cover, hence also as matrix factor algebras by the finite projective \(S_l\)-action (and this projective action already agrees as just explained). Moreover the digit shape agrees up to diagram with the direct edge’s: reduce its Frobenius twist exponent on the finite group using the defining graph/field map and apply restricted-weight uniqueness (linear lifts agree by perfection). ◻

Proof. Map a maximal torus of \(J\) into one of \(G_s\). The restricted natural multiset is \(p^y\Omega\). By irreducibility of the composition and the highest-weight convex hull property, the maximum pairing on the orbit of \(p^x\lambda\) is the linear function given by \(\mu\) throughout the positive cocharacter chamber of \(J\). Choose a maximizing orbit weight at a strictly interior cocharacter. Its restriction must equal \(\mu\) as a vector, so that same Weyl translate \(w\lambda\) maximizes everywhere in the chamber.

It also maximizes each active fundamental summand. For any one cocharacter a common Weyl translate maximizes all fundamental summands; equality in their positive sum therefore forces equality for each. Two distinct ambient orbit weights cannot both restrict to the composed highest weight, whose multiplicity is one.

Suppress the positive scales \(p^x,p^y\) for the cut description. In type A an active upper node \(a\) selects \(a\) natural coordinates at \(w\lambda\). Exchanging one selected and one unselected coordinate shows, throughout the lower chamber, that every selected restriction dominates every unselected restriction. An equal-vector tie would give another maximizing ambient orbit weight across a strict coefficient boundary, contradicting highest multiplicity one.

In B and C the analogous choice takes \(a\) entries of the full signed multiset, also at the last node up to the B spin scale. In D the same argument applies before the spin nodes when \(\lambda\) has no spin support. Sign changes show that selected restrictions lie in the positive cone; in D compensate by an unused coordinate. A zero-vector restriction would permit a sign change in the full maximizing orbit weight, using a zero-coefficient coordinate for D parity, and again contradict multiplicity one. Thus the selected values are nonzero. Signed coordinate exchanges compare them with every unused value, including negative choices and the extra zero in B. An equal nonzero tie across the cut also changes the ambient orbit weight; parity can be preserved by changing both directed signs when needed. These cuts are therefore strict as asserted.

For D spin support, take an active spin fundamental summand instead. Its maximizing signed choices \(u_i\) satisfy the stated pair-sum condition, because flipping any two choices cannot increase the pairing on the chamber.

These comparisons concern all entries of \(\Omega\), including nonextremal weights and repeated entries. A cut of its Weyl orbit alone is only a necessary consequence. The single-root multiplicity assertion follows because the corresponding weight space of the highest-weight module is generated by the one divided-power monomial in that root; it does not assert multiplicity one for arbitrary weights. ◻

Proof. Take \(J\) of type \(A_{e-1}\). Its orbit on \(\eta\) consists of permutations of a descending nonconstant word, ordered for dominance by prefix sums (coordinate sum zero). In a universal cut of \(\Omega\), the upper orbit words form a nonempty proper subset \(U\), since the orbit contains the full highest and lowest weights. We may study the smaller orbit side, replacing the full cut by the longest-element reversal of its complement when needed (this is again an upper cut of the weight multiset, now of complementary size) so \(|U|\) on the orbit is at most half. Then \(U\) is an initial segment both for lexicographic decreasing order and for lexicographic increasing order starting from the right. This follows by the first differing coordinate comparisons from strict cone domination and equality of total sums.

Here is the description needed of such segments. Let \(M\) be the largest letter, multiplicity \(b\), and \(m\) the smallest, multiplicity \(c\). If \(U\) contains a word not starting with \(M\), it contains all starting with \(M\). If \(b\ge2\) this includes one also ending with \(M\), forcing all ending with lower letters; the two fractions (and extra words) contradict the half bound. Thus \(b=1\). With second-largest letter \(T\), a starting-\(M\) word ending in \(T\) forces in all ending below \(T\). If there are at least two such lower letters counting multiplicity, these include one also starting below \(T\), forcing all starting at or above \(T\). Again the fractions contradict the bound. Namely if \(l\) letters are below \(T\), the proportions ending below \(T\) and starting at or above \(T\) are \(l/e\) and \(1-l/e\); in the indicated situation \(U\) contains each such class entirely and strictly. We obtain only a natural two-value shape with singleton extreme against constant, or \(M,T^{e-2},m\). Symmetrically these are the exceptions if some selected word does not end with \(m\).

Otherwise all words in \(U\) start with \(M\) and end with \(m\). Until the first word not ending with \(m\), left lex order fixes the first \(e-c-1\) coordinates of the descending word: the greatest word not so ending already has those coordinates. Right lex order confines changes to the first \(b+1\). Only positions in \([e-c,b+1]\) can therefore change. This interval has at most one position unless \(b+c=e\), the binary case; then its two positions permit only the single boundary swap. Thus, outside the exceptional shapes, \(U\) consists of \(\eta\) alone or of \(\eta\) and that swap.

For \(M,T^{e-2},m\), use positions \(a,d\) of the largest and smallest letters. Both orders begin with the half \(a<d\), one prioritizing \(a\) increasing there and the other \(d\) decreasing. A proper nonempty common prefix of the half has only \((1,e)\) (taking \(a>1\) forces \((1,2)\), then the full half; similarly for \(d<e\)). Dominance across the cut consisting of the half requires \(M-T=T-m\), by comparing words with both defects early and both defects late, in both choices of early side. At a prefix separating the two defects of one word (both before the boundary) from those of the other (both after) this compares \((M-T)+(m-T)\) to zero, in both directions. Thus \(\eta=z(\epsilon_1-\epsilon_e)\) in that case (\(\epsilon_i\) standard coordinates). In the full weight set each selected weight then has all prefix sums nonnegative (dominance over negative scaled simple roots on the other side), and each unselected one has them nonpositive. If \(z>1\), the weight \(\eta-\alpha_1\) occurs by simple-root lowering (\(z<p,\ \alpha_1=\epsilon_1-\epsilon_2\)), and can be permuted to mixed prefix signs. So this half exception requires \(z=1\). ◻

Proof. Take the first three of four homogeneous nodes, with the notation of 27. For each color we first constrain the extremal orbit cut, then use actual full weights or their multiplicities.

Proof. Now \(J\) is B,C or D of rank \(e\), and dominance in particular requires the prefix comparisons in the usual ordered signed coordinates. The orbit of \(\eta\) is antipodal by self-duality; make its dominant magnitudes nonnegative using the D diagram if necessary. First suppose no D spin nodes active (case of spin activity handled separately). For a universal fundamental cut above in \(\Omega\), upper orbit words \(U\) include \(\eta\), and each has all prefix sums nonnegative and is nonzero. Write \(h\) for the number of nonzero magnitudes in a word, and \(O^+\) for all unsigned permutations.

If some unsigned word is outside \(U\), comparison of total sums gives \(U\subset O^+\). If \(h<e\), we have all choices of signs (even in D, by zeros). Before the earliest negative word in decreasing lex order the first \(h-1\) positive coordinates are fixed, hence \(|U|\le e-h+1\). Indeed in this case all negative words are outside \(U\), and one can put the largest \(h-1\) magnitudes first positive, with the final nonzero one negative later; left prefix domination gives the lex bound. For \(2\le h<e\) in high rank this is at most half the unsigned words (there are at least \(\binom eh\)). In this range apply the A-word prefix description among those equal-total words. We obtain at most two upper words (the adj-shaped half for a word \(M,T^{e-2},m\) of three distinct magnitudes would be too large by \(e-h+1\), and the remaining binary singleton-extreme exception in this range has a singleton zero, already giving bound two by \(e-h+1\)). For a binary shape in that range with equal nonzero coordinates they are \(\eta\) and possibly \(s_h\eta\), also when \(h=e-1\) by left lex order. If instead \(h=e\), here only needed for B,C, the first \(e-1\) positives are fixed before the first negative word, giving just \(\eta\).

Suppose all unsigned words are selected. If \(h<e\), every signed word with positive first coordinate is forced in: otherwise it fails the first-prefix comparison with a selected unsigned word starting with zero. For \(h\ge3\), put a smallest nonzero magnitude first positively and two others negatively; the third prefix is negative. For \(h=2\) with unequal magnitudes, the smaller positive followed by the larger negative gives the same contradiction. Thus only \(h=1\) or two equal nonzero magnitudes can occur.

For \(h=e\) in B or C, suppose the smallest and largest magnitudes are \(P<M\), and let \(S\) be the sum of all \(e\) magnitudes. The signed word with \(M\) first and all other entries negative must be selected, by comparison with a selected unsigned word starting with \(P\). The word starting with \(-P\) and having all other entries positive is unselected, since its first prefix is negative. Domination of their total sums would require \[2M-S\ge S-2P,\qquad\text{hence}\qquad S\le M+P,\] impossible with at least three positive coordinates. All magnitudes are therefore equal. A selected word cannot have two or more negatives, since total-sum comparison would then force in every single-negative word, including one with negative first prefix. Left lex order before the first two-negative word now leaves only the all-positive word and the words with one negative in either of the last two positions. Thus \(|U|\le3\). ◻

Proof. Use the triple notation of 27, and consider the possible common edge color.

Proof. Within a protected classical run, first color vertices by the four series A, B, C, and D. The edge palettes above have an absolute number of colors and are compatible with diagram changes. Finite Ramsey theory (Ramsey 1930) therefore bounds the number of nodes in the run: a sufficiently long run contains four same-series nodes whose pair edges are homogeneous, contrary to [w:a-obstruction,w:signed-obstruction]. All pairs of the selected nodes remain protected because the protection was imposed for every ancestor–descendant pair, not just consecutive nodes.

By 32, at most two high alternating nodes occur. By 33, there cannot be two successive changes of characteristic among the classical nodes after ignoring the alternating interruptions. Thus there are only absolutely boundedly many same-characteristic classical runs. Applying the run bound to each gives \(D_0\). ◻

Completion of the proof of 4. Use the almost simple reduction of 5 and the prepared path construction of 5. Choose the high thresholds and \(D_0\) first, then take the number \(D\) of reserved generations greater than \(D_0\), with room for the initial displacement and one remaining bounded-type stage. Choose the lattice field size \(q\) above the finitely many absolute line requirements. Next choose the terminal chart increment \(r\) using 26 and 23. Choose the earlier trial increments backwards from \(r\) as in 5, then \(m\) above their ranks and \(n\) large enough for all residual interval lengths and bounded spans. Finally choose \(N\) for the initial pattern demand and enough private charts \(g\) for all finite losses.

At each stage, [sec:classical,sec:alternating] supplies a pruned chart with the ancestor properties of 6. A bounded-order socle is excluded by the residual interval length. A bounded-type Lie socle is impossible by 23. In every remaining case the next high-type node can be selected, with 26 adding the protections whenever the current classical run continues. This produces more than \(D_0\) high-type nodes, contradicting 9. It proves the required finite large-family nonrepresentation statement. Together with 3, this gives a finite minimum strictly greater than seven. Its explicit size ceiling and the description of \(L_0\) are supplied in [sec:quantitative,sec:minimum] after the uniform-decision argument. ◻

Proof. We first show that a normal subgroup can give only an endpoint label. The unique nonabelian monolith will then follow from nonmodularity. Suppose \([H_0,G_0]\) is such an interval, and let \(D\in[X,A_n]\) label \(H_0N\), where \(N\trianglelefteq G_0\). This label satisfies the left-modular identity, in either orientation. Moreover, projecting \(H_0N\) to \([L,K]\subseteq[H_0,G_0]\), by meeting and joining in either order, gives \[L(N\cap K),\qquad N\cap K\trianglelefteq K.\] The identity \[ z\vee(d\wedge y)=(z\vee d)\wedge y\qquad(z\le y) \tag{23}\] is self-dual. Consequently neither \(D\) nor any of these projections can admit a test \[ D\cap y\le z<y\le\langle D,z\rangle. \tag{24}\] We show that \(D=X\) or \(D=A_n\).

Choose \(37\) points in distinct \(X\)-orbits. Their \(X\)-translated \(37\)-tuples are pairwise disjoint. Let \(T\) be the product of the alternating groups supported on these tuples. Thus \(X\cap T=1\), and \([X,XT]\) identifies with the \(X\)-invariant subgroups of \(T\). For \(L\le A_{37}\) let \(T_L\) denote its product in the matched coordinates. Put \(D_0=D\cap T\), and write \(V\le A_{37}\) for a coordinate projection of \(D_0\). The following pair of conditions is impossible whenever \(B_0\le a<c\le A_{37}\): \[ \langle V,B_0\rangle\cap c\le a, \qquad c\le\langle a^{\langle V,a\rangle}\rangle. \tag{25}\] Indeed project once more to \(D'=\langle D_0,T_{B_0}\rangle\). The first condition gives \(D'\cap T_c\le T_a\). Conjugating independently supported copies of \(a\) by elements whose site projection belongs to \(V\) gives \(T_c\le\langle D',T_a\rangle\), contrary to (24).

Suppose first that \(V\) is proper and transitive. Burnside’s prime-degree Theorem (Müller 2005) and the classification of finite two-transitive groups imply that \(V\) lies in the affine normalizer of a regular cyclic group of order \(37\); in particular it contains a \(37\)-cycle. Here the non-affine possibilities at degree \(37\) all contain \(A_{37}\). For completeness, \(37\) is not a projective point number \(1+s+\cdots+s^j\) for a prime power \(s\): for \(j=1\) one would have \(s=36\), for \(j=2\) the adjacent possibilities \(s=5,6\) give \(31,43\), and for \(j\ge3\) the small remaining possibilities likewise miss \(37\). The other classical and exceptional two-transitive degrees give none at \(37\); we use the permutation-group classification in Meagher et al. (2016, Table 1).

Choose a cyclic \(37\)-group \(a\) so that \(c=N_{A_{37}}(a)\) intersects \(V\) trivially. To justify the choice, both \(V\) and \(c\) have order at most \(37\cdot18=666\). A nonidentity affine permutation fixes at most one point. From its cycles one can choose nine disjoint directed moved pairs. A commuting permutation is determined on those eighteen points by the ordered images of the nine sources, and has at most \(19!\) choices on the rest. Its centralizer in \(S_{37}\) therefore has order at most \((37\cdot36\cdots29)19!\), so its \(A_{37}\)-class has size at least \(28!/(2\cdot19!)>666^2\). Averaging the intersections of \(V\) with conjugates of \(c\) now gives a conjugate with no nonidentity intersection. The groups \(V\) and \(a\) contain distinct regular cyclic \(37\)-groups; a proper transitive subgroup would be affine, where the Sylow \(37\)-group is unique. Hence \(\langle V,a\rangle=A_{37}\). Its simplicity gives (25) with \(B_0=1\), a contradiction.

Suppose next that \(V\) is nontrivial and intransitive. Choose complementary invariant sets \(A,F\) as follows: if there are at least two singleton orbits, use two of them as \(F\); if there is exactly one, use that one; if there are none, use a smallest orbit. In all cases \(|A|\ge19\). Remove two points from \(A\) when \(|F|\ge2\), and three when \(|F|=1\), leaving a set \(Z\) meeting every orbit in \(A\). In the first case with two fixed points removed to form \(F\), this is possible because a nontrivial even action has \(\sum_O(|O|-1)\ge2\) on its other orbits. In the remaining cases the orbits of \(A\) are nonsingletons, so the required removals are possible. Set \(B_0=A_Z\). The \(V\)-conjugates of \(B_0\) generate \(A_A\): their supports cover \(A\) and have large overlaps.

Take a \(4\)-cycle on the removed points and enough points of \(F\) to make four. When two points come from each set, arrange the two from \(A\) consecutively and the two from \(F\) consecutively. Multiply this cycle by a disjoint transposition in \(Z\), obtaining an even permutation \(g\). Set \[a=\langle B_0,g^2\rangle,\qquad c=\langle B_0,g\rangle.\] No element of \(c\setminus a\) preserves \(A\), whereas \(\langle V,B_0\rangle\) does. Thus the first condition in (25) holds. The normal closure of \(a\) in \(\langle V,a\rangle\) contains \(A_A\) and its conjugate by \(g^2\). Their overlapping supports generate the alternating group on \(A\) with the used points of \(F\) added. This group contains \(c\), proving the second condition and yielding another contradiction.

The two exclusions show that every site projection \(V\) is \(1\) or \(A_{37}\). We next rule out diagonal ties in the latter case. Then \(D_0\) is subdirect and in fact equals \(T\). Otherwise the description of subdirect simple powers as products of twisted diagonals gives a tied pair of sites. Choose \(a=C_2<c=C_6\le A_{37}\). Independent nontrivial supports from \(T_a\) break all ties, so \(\langle D_0,T_a\rangle=T\). Modularity would give \[T_c=\langle D_0\cap T_c,T_a\rangle.\] On a tied pair, quotient each \(C_6\) coordinate by its \(C_2\) subgroup. The image of \(D_0\cap T_c\) has size at most \(6\), whereas that of the left side has size \(3^2=9\). The displayed equality is impossible. Thus every transversal product \(T\) is either contained in \(D\) or has trivial intersection with \(D\).

If one such \(T\) is contained in \(D\), then all are. Change one point of a transversal at a time, using spare \(X\)-orbits when needed. Consecutive transversals share supported three-cycles, so the alternative of trivial intersection is impossible at the next one. The supported \(A_{37}\)’s on the transversals themselves have connected, overlapping supports covering the letter set. They generate \(A_n\), and hence \(D=A_n\).

Otherwise \(D\cap T=1\) for every transversal product, also for shorter transversals after extending them to length \(37\). We will show that every element of \(D\) then belongs to \(X\). First suppose \(d\in D\) sends a point outside its \(X\)-orbit. Consider the directed non-loop moves on the set of \(X\)-orbits. If three have six distinct endpoint orbits, use a three-cycle \(u\) on their sources; its \(d\)-conjugate \(u'\) is supported on their destinations. If not, endpoints of a maximal disjoint matching comprise at most four designated orbits, and every non-loop move meets one of them. Since each designated orbit has size \(|X|\), only boundedly many orbits, in terms of \(|X|\), are touched. On sufficiently many untouched orbits, two points in different orbits have the same \(X\)-offset under \(d\). Multiplying \(d\) by an element of \(X\le D\), make it fix those two points. Use a three-cycle \(u\) on them and one point moved outside its orbit, and again put \(u'=u^d\). In either case one transversal product \(T\) contains \(u,u'\), and \[z=XT_{\langle u\rangle}<y=\langle z,u'\rangle\le XT, \qquad D\cap y=X,\qquad y\le\langle D,z\rangle.\] This contradicts (24).

It remains to exclude \(d\in D\setminus X\) preserving each \(X\)-orbit. Some transversal five-tuple has its matching of \(X\)-translated tuples disrupted by \(d\); preservation of every such matching would force all offsets to be one common element of \(X\). On this matching use \[z=XT_{C_5}<y=XT_{A_5}.\] Inside \(\langle D,z\rangle\), take the supported five-cycles on the original matching, their \(d\)-conjugates, and all \(X\)-translates. The connected components of their support hypergraph on the five orbits have size at least ten: the original matching partitions these orbits into five-tuples, a disrupted tuple joins at least two, and the \(X\)-action is transitive on the original matching tuples. On each component the generated group is primitive. Indeed a supported prime cycle cannot permute nontrivial blocks: if it moved blocks of size at least two, it would move at least ten points. Its support must therefore be within a single block, and connectedness of the support hypergraph leaves no nontrivial block system. Jordan’s prime-cycle Theorem (Jones 2014, Theorem 1.1) now supplies the full alternating group on each component, independently, because the generating cycles are independently supported. It follows that \(y\le\langle D,z\rangle\), again contradicting (24). Thus \(D=X\) or \(A_n\), proving the normal endpoint assertion.

Pass to the quotient by \(\operatorname{core}_{G_0}(H_0)\). Every nontrivial minimal normal subgroup supplements the base. An abelian one would identify the interval, by intersection, with an invariant-subgroup lattice of an elementary abelian group, hence a modular lattice. But \([X,A_n]\) is nonmodular, as already witnessed inside the transversal product labels by the intransitive test above. If two distinct minimal normals occurred, they would commute. Using one as the supplement identifies the interval with its invariant subgroups; supplementation by the other means that the base induces all inner actions on the first. All these intersections would then be normal subproducts of its simple factors, again giving a modular lattice. The same contradictions hold for the reversed lattice. The core quotient therefore has a unique nonabelian minimal normal subgroup. ◻

Proof. A proper subdirect label cannot contain a nontrivial invariant product \(P\): factor transitivity makes every base projection nontrivial, and its independent supports normally generate each full simple site. For \(b=1\) the assertion is automatic. A coatom therefore has proper product closure and, by maximality, equals that closure. A subgroup of \(A_v\) generated by prime-order elements is a join of atoms in parameter order; its reversed label is a meet of coatoms, hence a product. In particular a supported \(A_n\) is a product label. It can be used as the new base; we keep the name \(P\).

Identify \(S\) with its inner automorphisms and set \(I=\Lambda\cap S\). A proper coatom product projects to a nontrivial proper group \(T\) normalized by \(I\), maximal in the local proper invariant interval. Since \(TI=S\) would make \(T\trianglelefteq S\), we have \(TI<S\), whence \(TI=T\). Intersecting these coatoms gives \(I\le Q\), since the corresponding prime-order groups generate \(A_n\). Consequently an overgroup of \(H\) in \(M\) is determined by its intersection with \(S\); that intersection transports equivariantly to every site. Conversely each invariant site projection containing \(Q\) gives such an overgroup. Coordinate products are closed under intersection and generation, proving the interval assertion.

Fix \(X\) as in the statement and put \(Z=K_X\cap S\). The exact invariant interval from \(P\) to its product label is an ordinary interval with endpoints \[ P\rtimes\Delta,\qquad (\textstyle\prod Z)\rtimes\Delta, \tag{26}\] in matched-site notation; replace \(\Delta\) by its finite acting image if necessary. It has reversed parameter lattice \([X,A_n]\). A \(K_X\)-normal subgroup \(N\le Z\) transports well-definedly, by site-stabilizer invariance, to a normal product at the upper endpoint of (26). By 41, unless \(N\le Q\), it supplements the lower endpoint. Thus \(QN=Z\), and \(HN=K_X\).

We also need actual product coatoms with meet \(H\). On three points in distinct free \(X\)-orbits, the product of translated supported \(C_3\)’s is elementary abelian and disjoint from \(X\). Coprimality \(q\ne3\) splits it into irreducible \(X\)-modules. Adjoining one nonzero irreducible summand to \(X\) covers \(X\), remains strictly below \(A_n\), and is prime-generated. The joins of all these groups contain every supported transversal three-cycle, hence generate \(A_n\). On reversing, their product labels are strictly intermediate coatoms whose meet is \(H\). Using disjoint triples of orbits gives arbitrarily long chains of such prime-generated product labels.

Now let \(R\trianglelefteq K_X\) satisfy \(R\cap Z\le H\). Then \([R,Z]\le H\cap Z\). Therefore \(R\) normalizes the intersection with \(Z\) of each indicated coatom. If \(R\) is not contained in one of them, it and that coatom generate \(K_X\), so the coatom’s inner intersection is normal in \(K_X\). That intersection properly contains \(H\cap Z\), but its product with \(H\) is precisely the proper coatom. This contradicts the preceding normal test for normals within \(Z\). Hence \(R\le H\).

Every missing normal thus has a missing intersection with \(Z\). Two missing normals commuting modulo the base core, including a single one abelian modulo the core, would transport their inner intersections to missing normal products in (26). They still commute modulo the transported product of the core’s intersection with \(Z\). That product is normal at the upper endpoint and lies in its base. This contradicts the unique nonabelian monolith from 41. All assertions of property \((*)\) follow. Factoring a core or centralizer kernel already in the base preserves the exact intervals, the coatom meets, and these conclusions. ◻

Proof. At a stage over \(W\) take a replicated regular action of \[ W\times\operatorname{PSL}_2(\mathbf F_{q^u}). \tag{27}\] Assign different projective points \(e\) to the private charts, and embed their new coordinates as \(N\)-dimensional \(\mathbf F_q\)-subspaces in the corresponding unipotent root groups. For distinct \(e,f\), unipotent determinant-one lifts with coordinates \(x,y\) have product trace \[2+c_{ef}xy,\qquad c_{ef}\ne0.\] Choose the embeddings so that, for every pair of nonzero chart vectors from different charts, this trace has square outside \(\mathbf F_q\). Such a choice exists: independent random linear maps from the fixed-dimensional chart spaces to \(\mathbf F_{q^u}\) are injective with probability tending to one. Nonzero input vectors in different charts have independent uniform images. For each fixed nonzero \(c_{ef}\) the probability that \(2+c_{ef}xy\in\mathbf F_q\) tends to zero, uniformly in the chosen points. There are only finitely many tests. Since \(u\) is odd, an element of \(\mathbf F_{q^u}\) whose square belongs to \(\mathbf F_q\) already belongs to \(\mathbf F_q\); thus the stated squared-trace avoidance follows.

The Dickson subgroup Theorem for \(\operatorname{PSL}_2\) (Giudici 2007, Theorem 2.2) now implies that the two translations generate the full projective group. Their distinct fixed points exclude the point-stabilizer alternative; their order \(q\) excludes the dihedral groups, whose orders are \(q^u-1\) and \(q^u+1\), and \(q>5\) excludes the small groups \(A_4,S_4,A_5\). Since \(u\) is odd, the square-field \(\operatorname{PGL}_2\) alternative cannot occur. Since both \(q\) and \(u\) are prime, the only remaining proper subfield group is \(\operatorname{PSL}_2(q)\). Squared traces of determinant-one lifts in a conjugate of that group belong to \(\mathbf F_q\), contrary to our choice. The square is unchanged by changing the lift projectively.

Choose \(n\) divisible by all orders needed for this finite recursive construction and sufficiently large for every residual-height test. At a new attachment, the given semiregular \(W\)-action and the restriction of the replicated regular action (27) are both disjoint unions of regular \(W\)-sets of the same total size. An equivariant bijection identifies them, extending the existing action. The actions lie in \(A_n\): \(W\) has odd order, and the projective factor is perfect, so neither has a nontrivial sign character. No disjointness between later subtrees is needed. The recursive construction is finite, and the orders and sizes can all be accommodated by the final choice of \(n\). ◻

Proof. All intermediate subgroups between comparable elementary chart parameters are elementary and prime-generated. By 42, their labels are products, so these intervals are exact even after passage to the ordinary site interval. The same is true between an ancestor and a comparable trial extension, and after the indicated ancestor core or centralizer quotients. Thus full stabilizers witnessing failure have labels \(K_{X_i}\) with all prepared-link normal tests available.

Within an available extension of \(W\), choose horizontal and vertical complements from the new root coordinates. Color graphs of linear maps by the failure’s ancestor, category, bounded case data, and horizontal witness subspace. The multidimensional Hales–Jewett Theorem supplies arbitrarily many affine parameter slots with all this data fixed (Hales and Jewett 1963). The fixed horizontal witness is not contained in \(W\). Choose a horizontal functional nonzero on its image outside \(W\), and vary each slot by multiples of that functional. Distinct parameters give distinct witness subspaces \(X_i\). The span of two includes all witnesses on their parameter line and adds exactly one dimension; an affinely independent list adds a dimension at every step. Upon reversal, pair intersections lie below every label on that line, are covered by the two labels, and bounded independent lists give strictly decreasing successive intersections. In particular all pairs of distinct labels on one parameter line have the same intersection \(E\), and \(E\) lies in every label on that line.

For two private charts attached at \(W\), write \(L_0,L_1\le W\) for the projections of their witnesses to \(W\). These projections are constant within their respective patterns, because only the vertical root coordinates vary. Put \(G=\operatorname{PSL}_2(q^u)\) for the attachment factor. Any cross pair generates a subgroup of \((L_0+L_1)\times G\) projecting onto \(G\), by 43. Its commutator subgroup lies in \(G\) and projects onto \([G,G]=G\), since \(W\) is abelian. Hence the generated subgroup is exactly \((L_0+L_1)\times G\). It is independent of the chosen pair, contains every witness parameter in both patterns, and is prime-generated. Its ordinary reversed label \(J\) is therefore the common cross-pair intersection, and \[H\le J\le K_{X_i}\] for every witness in either pattern.

A witness need only contain the subspace of its tested ancestor and extend outside the current \(W\); it need not contain \(W\). The fixed \(W\)-projections suffice for the preceding argument. We now verify the uses of \(J\) in the initial exclusions. For subset stabilizers, \(J\) contains the supported alternating groups on the at most four intersection cells, so it has boundedly many point orbits; every subset in either pattern is \(J\)-invariant. For missing equal-block partitions, supplementation still gives \(H\) its rich actions on both block lists. Thus \(H\) has an absolute bound on orbits of ordered pairs of intersection cells. The supported alternating groups within those cells lie in \(J\); a cell-pair stabilizer therefore has an absolute bound on point-pair orbits, including coincident cells and cells of size at most two. Hence \(J\) has boundedly many point-pair orbits and boundedly many invariant equivalences. This gives the contradiction in 19; the retained-block cases use the original normal tests. In the classical small-space cases, the supported large group on the common complement lies in \(J\) and preserves every structure in both patterns. The bounded-span and linear-constraint arguments of [c:linear-subspaces,c:isometry-subspaces] therefore apply to a bounded independent list.

The remaining initial classical cases need only a parameter line. For a descending list of distinct subspaces or signed flags, the intersection of the first two stabilizers fails to preserve the third, by the elementary maps in 9. This contradicts \(E\le K_{X_3}\). For singular spaces, first discard the at most two exceptional incomparable maximal spaces, then choose three members of the remaining chain. For mutually perpendicular retained large spaces or orthogonal idempotents, choose four distinct members. After the first two structures are imposed, the third cannot exhaust the unsplit remainder because the fourth nonzero space remains. The supported classical or special linear group on that remainder gives the same strict third decrease. Thus a fixed number of labels on a line replaces every height-sized initial comparison. ◻

Proof. Color every two-dimensional subspace, with its canonical reduced row-echelon basis \((b_1,b_2)\), by the finite table \[\bigl(\phi_{a b_1+b b_2}(c b_1+d b_2)\bigr)_{a,b,c,d\in\mathbf F_q}.\] The number of colors depends only on \(q,h\). The finite-vector-space Ramsey Theorem gives a homogeneous subspace of dimension \(m+2\) (Graham et al. 1972, 1973). Let \(e_1,e_2,\ldots,e_{m+2}\) be its canonical basis, and put \[A=e_1+\langle e_3,\ldots,e_{m+2}\rangle.\] Reduced row-echelon coordinates are compatible with this passage to a subspace: multiplying a reduced row-echelon matrix in these coordinates by the reduced row-echelon matrix of the containing space again gives reduced row-echelon rows in ambient coordinates. For \(i\in A\) and \(w\in\langle e_3,\ldots,e_{m+2}\rangle\), both \((i,e_2)\) and \((i,e_2+w)\) are such canonical plane bases. Their colors agree, so \(\phi_i(e_2)=\phi_i(e_2+w)\). Linearity gives \(\phi_i(w)=0\), which proves the claim. ◻

Proof. Write \(Y=Y_i\) temporarily, so \(HY=K_{X_i}\). Every intermediate overgroup is \(H\) times its intersection with \(Y\). Take the coordinate-product closure of this intersection and multiply by \(H\). For \(X_i\le Z\le B\) this is an overgroup \(K_{g_i(Z)}\), where \[X_i\le g_i(Z)\le Z,\] and \(g_i\) is monotone. The resulting group’s intersection with \(Y\) is the product closure itself. Therefore \(g_i(Z)=X_i\) forces the intersection at \(K_Z\) to be subdirect in \(Y\). All intermediate intersections from \(K_Z\) to \(K_{X_i}\) are then subdirect. Their tie relations are unions of \(H\)-orbits on ordered pairs of sites; a strict inclusion forces strict refinement of ties, because comparable subdirects with the same ties coincide. The pair-orbit bound gives an absolute \(h\) with \[g_i(Z)=X_i\quad\Longrightarrow\quad\dim(Z/X_i)\le h.\] Choose a complement \(C\) of \(g_i(B)/X_i\) in \(B/X_i\). Monotonicity gives \(g_i(C)\le g_i(B)\cap C=X_i\) (using the preimage of \(C\) in \(B\) in this notation), so \[ \operatorname{codim}_B g_i(B)\le h. \tag{28}\] Exactness between comparable elementary labels ensures that every closure label used here really is indexed by the indicated subspace. The same calculation can be made modulo a retained normal subgroup; taking its preimages does not alter the intervals.

In the graph pattern, \(X_j\) differs from \(X_i\) in the single vertical direction determined linearly by the parameter difference \(j-i\). The preimage of \(g_i(B)\) in these vertical directions has codimension at most \(h\), by (28). Choose a map \(\phi_i\) to \(\mathbf F_q^h\) with that preimage as kernel. Apply 45, asking beforehand for enough Hales–Jewett slots. In the resulting affine subpattern, \(X_j\le g_i(B)\) for every \(i,j\). Let \(U\) be the span of its witnesses and put \(E=K_U\). Then \(U\le g_i(B)\) for all \(i\), and consequently \[H<K_{g_i(B)}\le E<K_{X_i}.\] Here \(K_{g_i(B)}>H\) because \(g_i(B)\le B<A_n\). The label \(E\) is above \(K_B\), so it inherits all previous-good exclusions, and 42 gives property \((*)\) for \([H,E]\).

The intersection \(K_{g_i(B)}\cap Y_i\) is a coordinate product not contained in the base; otherwise supplementation would give \(K_{g_i(B)}=H\). Each of its single-site components already lies independently in \(E\cap Y_i\). Let \(R_{i,a}\) be the full subgroup of \(E\cap Y_i\) supported on site \(a\) and put \(N_i=\prod_aR_{i,a}\). Conjugation by \(E\) permutes these support subgroups, so \(N_i\trianglelefteq E\), and \(N_i\not\le H\). Site transitivity and \(E<K_{X_i}\) imply that each factor is proper: one full simple factor would force all of them and then \(E=K_{X_i}\). The factors are mutually conjugate under \(H\), and if they were soluble, \(N_i\) would be a soluble missing normal, contrary to property \((*)\). Thus they are nonsoluble. The image of \(N_i\) contains the common nonabelian monolith of \(E/\operatorname{core}_E(H)\), again by property \((*)\).

We spell out the scope needed in the structure exclusions. For direct systems calculated modulo block centers, each block carries at most two simple sites, and the at-most-two block classes have independent alternating top actions induced already by \(H\) through supplementation; in linear type there is one list. These give the required absolute pair-orbit bound. In the case of several site orbits one instead uses the supports of a retained orbit as in 13. Lifting independent factors on a block to its natural supported matrix group gives projective image in \(E\), since the lost centers were retained. The lift is preserved by the semilinear block stabilizer in \(E^+\) and supplies the groups of order greater than two required in the block-module argument. If needed take products of the supported factors over one block.

For tensor and Cartesian systems the \(Y_i\) already have independent simple supports. The common-monolith and iterated-commutator arguments of [c:tensor-refinement,c:cartesian-refinement] apply to these \(E\)-normal products \(N_i\), without requiring \(N_i=E\cap Y_i\). Their tuple intersections still generate a nontrivial \(E\)-normalized subgroup. It spans in the tensor case, and is transitive in the Cartesian case, by the previous-good exclusions inherited from \(K_B\). In the diagonal primitive case choose a minimal \(E\)-normal subgroup inside the nontrivial product of proper supported factors. This is exactly the containment needed by that argument. Retained centers, when present, are normal in \(E\) and lie in \(H\), so these conclusions survive their removal and reinstatement. ◻

Proof of 10. Prepare the charts in 43, and use 42 for every elementary parameter encountered. At every node the current almost simple quotient is the one from 6; initially it is \(M\). The normal, residual-height, and exact comparable-link hypotheses have all been verified. 44 supplies the line, independent-list, and cross-chart configurations for the pruning rounds relative to every ancestor. 46 replaces the interval-product consequence exactly at its uses for missing direct, tensor, Cartesian, and diagonal systems. Thus each finite round either loses only the bounded number of charts allowed by the matched failure arguments or produces the next actual almost simple socle, with all ancestor-relative exclusions. These are the structural conclusions of [c:classical-conclusion,c:alternating-conclusion].

If a parent has bounded type, its bounded-order case is excluded by residual height. In bounded Lie rank, the bounded-equation pattern argument of 8 applies unchanged with a bounded independent list: strict subgroup descents are strict after intersection with the socle, whereas the spaces of equations have bounded dimension. Thus one obtains a sandwich throughout the interval over a selected increment \(t\). Its height is \(t\) and the same divisor count gives at most \(C2^t\) labels. The exact \(t\)-dimensional vector-subspace interval has at least \(q^{\lfloor t^2/4\rfloor}\) labels, a contradiction for sufficiently large fixed \(t\). The sufficiently-general input is used here in precisely the scope established in 8 (Larsen and Pink 2011).

For a high-type path, use linear flags in the final protection selection of 26. All bad containing labels belong to the exact subspace intervals just proved. The same exponential-gap estimate for flag proportions applies; hence the closure-field tensor and root protections hold simultaneously for the required ancestor comparisons. The absolute group-theoretic bound 9 on the length of such a protected path then contradicts a choice of depth \(D\) larger than that bound.

There is no dependence of the earlier finite choices on \(n\). First fix the high thresholds and the protected path bound, and choose \(D\) beyond it. Next choose a prime \(q\) above every absolute line bound, then \(r\) for the bounded-type count and protection estimates. Choose trial dimensions backwards through the finite pruning rounds, allowing at each round all Hales–Jewett and 45 slot requirements for the trial and bounded path ranks. The latter requirements depend only on \(q\), desired fixed dimensions, and absolute orbit bounds. The initial comparisons require only a line or a bounded independent list, as proved in 44; none requires comparison with \(|A_n|\). Choose \(N\) and the number of private charts large enough for the finite color and chart-loss bounds. Only then choose the fields of (27) and finally \(n\). All elementary groups \(X\) used are now bounded in size, so \(n/|X|\) can exceed all normal-test and residual-height requirements while \(n\) accommodates every replicated regular action. This gives one fixed forbidden degree \(n\). For every \(v\ge n\), the supported \(A_n\) restriction of 42 gives the same contradiction. ◻

Proof. Bounded order is immediately impossible by the length of an axis chain. In bounded Lie rank use the argument of [w:lp,w:height-rigidity,w:bounded-exclusion]. For a prescribed sufficiently large constant \(k\) test \(M_U\), \(|U|=k\). There must be some for which all ordinary labels above it admit the sandwich on intersection with \(S\): otherwise pick many disjoint such sets and take proper bounded-equation stabilizer witnesses as there. Each has omitted (nonabsorbed) set nonempty inside its tested set, hence intersections strictly descend (also ordinary products, strict on intersection with \(S\)). The linear spaces of equations cannot give such long descent. Now residual lengths under \(M_U\) are arbitrarily large and the same height and order comparison for the sandwich maps bounds the number of ordinary labels above it by \(C2^{h}\), \(h=O(k\log\log(k+3))\). But all partitions of \(U\) occur via intersections of cuts \(C_D\) (individual symmetric group on every part), giving at least \(\exp((1/2+o(1))k\log k)\) for example by pairings. This is impossible. In particular we can require arbitrarily high thresholds on the remaining types here or at any use with a large enough marker set. ◻

Proof. The full formal link \([g_D,r_D]\) is \(F_{T'\setminus D}\): every intermediate increase keeps \(V_D\) independent. For disjoint supports the label calculation gives \(g_{D_i}\le r_{D_j}\) and \(r_{D_i}\vee r_{D_j}=M_A\). The link \([g_*,J]\) is likewise pure on \(T'\setminus D_*\), with independent side group \(\prod_i V_{D_i}\). We prove the normal assertion for either pair of endpoints, denoted by \(g,r\).

Work before any ancestor-centralizer quotient, and let \(S\) denote the original simple site. Put \[L=K_g,\qquad R=K_r,\qquad Q=L\cap S,\qquad Z=R\cap S.\] Let \(P_g,P_r\) be the corresponding products at the original sites, with fixed-site projections \(Q,Z\). They have these projections because \(\Lambda\cap S\) lies in the original base. With \(\Delta\) replaced by its finite acting image if necessary, the full formal link has the exact ordinary realization \[[P_g\rtimes\Delta,\ P_r\rtimes\Delta].\] Indeed its intermediates all contain \(\Delta\), and hence correspond to precisely the \(\Delta\)-invariant subgroups between the two products. The normal test applies to this full interval.

First let \(N\trianglelefteq R\) lie in \(Z\). Since \(\Lambda\le L\le R\), its transport to the other sites is well-defined. The resulting product is normal in \(P_r\rtimes\Delta\). By 48, either \(N\le Q\) or \(QN=Z\); in the second case \(LN=R\). Thus every missing normal inside the original site supplements the ordinary lower endpoint.

Now let \(N\trianglelefteq R\) satisfy \(N\cap Z\le L\). Then \([N,Z]\le Q\), so \(N\) normalizes the intersection with \(S\) of each singleton-omission axis coatom of the link. These coatoms are products, are strictly intermediate, and have meet \(L\). If \(N\) escaped one of them, adjoining it to that coatom would generate \(R\); the coatom’s inner intersection would therefore be normal in \(R\). It strictly contains \(Q\), but its product with \(L\) is only the proper coatom, contradicting the preceding paragraph. Hence \(N\le L\). Together the two paragraphs prove the normal endpoint assertion for every normal subgroup of \(R\).

Finally put \(C=\mathop{\mathrm{core}}_R(L)\). Two missing normals commuting modulo \(C\), or one missing normal abelian modulo \(C\), would transport their intersections with \(Z\) to missing normal products in the exact semidirect interval. Their commutators lie in the transport of \(C\cap Z\), a normal product contained in its base. This contradicts the unique nonabelian monolith supplied there by 48. The ordinary interval therefore has property \((*)\). Quotienting an ancestor centralizer preserves it, since that kernel already lies in \(L\). ◻

Proof. Indeed, all positions outside \(D_1\cup D_2\) are already absorbed at \(u\). The part of \(D_2\) still not absorbed there stays separate also from \(D_1\) since \(u\le r_{D_1}\), and is preserved by \(V_{D_2}\) since \(J\le u\); if this part were empty one would already have \(r_{D_1}\). ◻

Proof. There are two sets of coordinates in this argument: the original \(b\) simple sites, and the simple factors of \(Y/Z\) inside each of them. We first obtain an exact interval at the new sites, then show that its axis groups contain independent supports.

Lift \(Y,Z\) inside the actual \(B_A\le S\), before any ancestor quotient, and transport them across the original sites. Since \(K_{g_D}Y=K_{r_D}\), we also have \[(K_{g_D}\cap S)Y=K_{r_D}\cap S.\] Thus the transported product of copies of \(Y\) supplements the full lower endpoint, while the transported product of copies of \(Z\) is retained. Every intermediate subgroup is its lower endpoint times its intersection with that normal product. After factoring the retained product, \([g_D,r_D]\) is therefore an exact invariant-subgroup interval in \((Y/Z)^b\). The operators are the original action together with conjugation by the label at \(g_D\); they act transitively on the new simple sites.

Consider a singleton axis \(g_D\vee X_{\{i\}}\), with \(i\) outside the used supports. Its intersection with the new power remains a product across the original \(b\) sites. It has nontrivial projections at the new sites, by strict increase from the lower endpoint and transitivity. These projections cannot be full. Otherwise every subgroup on a long axis chain above it would be subdirect. Strict axis increases give strict intersection increases by supplementation; inside an original site these would require strict refinements of \(K_{g_D}\)-invariant tie partitions. The bounded number of orbits on ordered pairs bounds the length of such a chain, as in 10, whereas the residual marker set is arbitrarily large.

The filter above this singleton axis is again an exact pure-marker interval. Its bottom has nontrivial proper projections to the new sites, so 49 makes it a coordinate product there. 50 then makes all subsequent axes products at those sites as well. Applying 51 to this exact filter forces its new site simple to have arbitrarily high type. This uses the bounded-type exclusion, not the conclusion of the forward theorem being proved.

Now put \(E=K_J\). The axis \[g_D\vee X_{T'\setminus D_*}\le J\] supplies independent new-site supports in \(E\), modulo \(Z\). Inside one original site take the product of all individual simple-factor intersections of \((E\cap Y)/Z\). This product is \(E\)-normal. It is missing relative to \(K_{g_*}/Z\): the axis intersection just obtained is a product contained in it, and supplementation recovers that axis label by multiplying its lift by \(K_{g_D}\). That axis is not below \(g_*\).

By 52 on \([K_{g_*},K_J]\), the supported normal product has image containing the nonabelian monolith modulo the base core. It is nonsoluble, and transitivity makes every supported factor nonsoluble. If \(J<r_D\), each factor is proper: a full one would, by the transitive \(K_{g_D}\)-action and supplementation, give \(E=K_{r_D}\). This proves precisely the supported-product conclusion needed below; it does not identify that product with all of \((E\cap Y)/Z\). In direct-system applications the block centers are retained, so lifting these supports with their centers gives the required actual block-supported matrices. ◻

Proof. The coordinate closure of \(U\) is \(Y\). Conjugating the independent copies of \(X_i\) inside \(U\), whose projections are all of \(Y_i\), therefore gives \[N=\prod_i\langle X_i^{Y_i}\rangle\le U.\] This is an invariant coordinate product with \(X\le N<Y\), so the product-cover hypothesis forces \(N=X\). Thus every \(X_i\) is normal in \(Y_i\). ◻

Proof. In particular, in the relative label lattice, the meet of cuts on disjoint nonempty \(D,D'\) absorbs everything outside \(D\cup D'\), with side group \(S_D\times S_{D'}\). The only nonempty sets available for coatom cuts above that meet are \(D,D',D\cup D'\) (unions of the unabsorbed orbits). The first two coatoms cover the meet, since inside one such cut one would have to absorb from the other set, hence absorb that whole set, for any strict increase. Up to the union cut instead we have the full Young subgroup interval in \(S_{D\cup D'}\). These are formal statements before the product selection (and within the ideal at \(M_A\)); in particular all three coatoms persist, and formal covers between products are still covers. In the other direction when needing an intermediate such as the formal wreath label, which need not itself be product, we will argue using product closure rather than just assuming persistence. ◻

Proof. Apply finite Ramsey homogenization to the bounded omission patterns. Concretely color bounded-size ordered index subsets by all the tested label predicates (Boolean, also subdivided into the boundedly many indicated cases) relative to their prefixes \(A\) whenever those have the required path-node normal simple in the ordinary product interval. It is canonical if it exists, by its soluble centralizer. Include type predicates and the bounded comparisons across subsets as used for selection. From a sufficiently large homogeneous set, for fixed \(A\) with fresh indices after it, any bad label with small nonempty omitted set after \(A\) thus gives replicas with every \(k\)-set of later fresh positions (\(k=|D|\)), including as many disjoint ones as needed. Choices of witness structures can also be further finitely colored and thinned. The exclusions are used successively: earlier ones for larger bounded sets as needed to make the intersections (which contain the axes omitting the support unions) common previous-good labels, just as properties there passed upwards from anchors. Thus they apply simultaneously relative to all indicated high-type ancestors. All bounds for combined supports in each round need only be finite in terms of subsequent round requirements, and never involve the ambient group dimensions or fields.

More explicitly one may use successively shorter initial segments of a homogeneous reservoir as rounds proceed. Assert each exclusion (conditional on absence of the point configuration) for all prefixes \(A\) of the allowed path sizes with the stated high simple data, and omitted sets in that round’s segment, \(A\) before the further omissions in index order. If a bad full stabilizer contains a tested axis label, its exact omitted support is a nonempty subset of the tested further omissions. Replicate this support at fixed \(A\) using the spared tail of the preceding segment; earlier-good properties are then available for each specified bounded union of the replicas by the result of preceding rounds on that segment. Thus choose earlier omission bounds and segment lengths large enough backwards (e.g. include combined supports of the finitely many simultaneous intersections needed in a later round, and allow extra finite Ramsey thinning there). Predicate colors on ordered subsets can include all splits/sublist choices at the finitely bounded sizes, recording actual path-simple existence even before descending (\(B_A\) is unique by its soluble centralizer), needed same-characteristic/type tests, and existence of a witness with exactly those support positions in the relevant categories. Not the unbounded field sizes, group orders, etc., only finitely many case/comparison predicates are needed. In a contradiction at a fixed prefix one can further choose witnesses and color them by finitely many tests, without losing earlier-good properties. All unions and links still refer to the original \(T\); in particular the residual marker counts for the normal/missing-site tests remain arbitrarily large. ◻

Proof. For subset stabilizers \(K_{r_D}\), among the two supported alternating groups precisely one side is missing relative to \(K_{g_D}\) by the normal property of 52 (soluble quotient over both). Use the retained side’s point subset \(U_D\); \(K_{g_D}\) acts transitively there by supplementation with the other side, even for small size. Thus retained point sets for disjoint supports are disjoint, by mutual containment of each smaller label in the other’s stabilizer and distinctness. In a homogeneous \(k\)-set family we can take their sizes each below a small fixed fraction of the whole (color, using arbitrarily many disjoint replicas). Their stabilizers are maximal: an increase would give transitivity and overlapping conjugates of the large complementary supported alternating. So \(r_D=C_D\) (relative labels; all relative coatoms are products by intersecting global cuts with \(M_A\)). The same maximality applies to the subset union for two disjoint \(k\)-sets. The union label contains \(C_D\wedge C_{D'}\), and therefore equals \(C_{D\cup D'}\), since the only coatoms above that meet are these three, and the union stabilizer moves the individual subsets (correct parity outside). Consequently the union itself is independent of the split by its small size and the stabilizer orbits.

Decomposing the retained subsets.

It follows that \(U_D\) splits into elementary disjoint subsets indexed by the individual \(i\in D\), the same in every use. Indeed for any point the indicator \(p_D\) has \(p_D+p_{D'}\) fixed by the union of disjoint index sets. Thus \(p_{F\cup i}-p_{F\cup j}\) on replacing one index is independent of the other \(k-1\) indices \(F\), by comparing with a common disjoint auxiliary \((k-1)\)-set. For \(k\ge2\) we obtain a sum of individual real weights plus a constant: the replacement differences obey the triangle identity by using the same \(F\) for three positions, so are differences of weights, and replacements connect all the \(k\)-sets. Weights differ only by zero or one up to sign; with our ample reservoir, two or more each of a high and a low weight would produce a difference of two between indicator sums. So either all indicators vanish (a constant one violates disjointness) or just one position is exceptional and membership is exactly when it occurs (the reverse choice again violates disjointness). For \(k=1\) use disjointness directly. At most one elementary subset can be empty by label distinctness. For any nonempty one its stabilizer label is \(M_{\{i\}}\), since it contains \(C_D\wedge C_{D'}\) for pairs meeting just in \(i\); two such meets with remaining positions disjoint already join to \(M_{\{i\}}\). Indeed each meet fixes position \(i\) and absorbs all outside \(D\cup D'\); use four sets all meeting pairwise just at \(i\). For \(k=1\) we already have the label \(C_{\{i\}}=M_{\{i\}}\) itself.

Equal sizes and singletons.

Unions of two or four of these nonempty elementary sets again have the corresponding cut labels by the coatom-meet argument. For disjoint pairs \(P_1,P_2\), the full interval from the two cuts’ meet \(J_0\) to the union cut \(F_0\) is that from \(S_2\times S_2\) to \(S_4\), with a proper wreath intermediate. If actual union sizes for the two differed, the ordinary interval would be a cover (project the full stabilizers to the symmetric group on the combined set, using outside parity correction). Apply 55 to the product endpoints in the original power. The formal wreath intermediate would force \(K_{J_0}\cap S\trianglelefteq K_{F_0}\cap S\) before the ancestor quotient. Passing to that quotient and intersecting with the actual \(B_A\) gives a contradiction: on the combined set the upper group induces the full symmetric group by outside parity correction, whereas the lower group induces exactly its two-part stabilizer. For unequal positive part sizes that stabilizer is maximal proper in the full symmetric group (an increase is transitive, merging overlapping larger parts via conjugates of their supported symmetric group), not normal. Thus pair-sums of sizes agree and all nonempty elementary sizes agree (compare pairs sharing one index against a common disjoint pair). For two elementary sets the analogous formal interval is just a cover, so their equal size must be one by block-swapping with parity correction. This gives the desired point labels. Henceforth we can use transitivity after this exclusion if success was not already obtained. ◻

Proof. For full stabilizers of nontrivial equal-block partitions (the imprimitivity test) let the sizes be \(a\) per block, \(b'\) blocks. If \(a\ge3\) with \(A_a^{\,b'}\) retained, the supported primitive groups force refinement into blocks of another system of the pattern (discrete restriction to a block is impossible by moving a point while fixing its equivalent outside neighbor), hence equality mutually. For \(a=2\) use instead retained even simultaneous flips of two blocks, precluding any other matching of this kind. Otherwise \(a\ge5\) and the alternating base product is missing. Here the full stabilizer is maximal by the supported \(A_a\)’s: an increase gives overlapping conjugate blocks, thus full alternating on merged components, and its primitive top on blocks forces total merging. Indeed supported alternating groups of these sizes on overlapping sets generate alternating on the union (conjugate three-cycles through a common point; even with just one common point this brings in three-cycles using one point of the other set and two of one’s own). The full top symmetric action is available, with internal parity correction. Take two disjoint failures, giving coatoms \(C_D,C_{D'}\) and a meet covered in each. The two block systems have common cell size \(h\) on occupied incidences by prior transitivity. They cannot coarsen one another by primitive top action, which is realized inside the intersection for both by supplementation relative to the \(g\) labels. If \(h\ge2\), the cell-refinement stabilizer intersects a row label properly (by its supported primitive block groups), strictly above the intersection because one can swap two cells within a row while fixing points elsewhere, with internal parity correction, breaking the columns. This contradicts covering. If \(h=1\), the intersection embeds subdirectly in \(S_{b'}\times S_{b''}\) via the two top actions, both block counts at least five by the touching block sizes. The kernel of projection to the first top action must still supply all even-total base sign patterns there (supplementation with \(A_a^{\,b'}\), which acts trivially on that action’s blocks and signs). But this kernel injects as a normal subgroup of \(S_{b''}\); its abelianization has size at most two, versus its required homomorphic sign image of size at least \(2^{b'-1}\). ◻

Proof. Using then common primitive previous-good labels \(E\), full normalizers of elementary abelian regular groups are excluded by 20: the regular groups are retained and hence mutually included in \(E\), where each is minimal normal; distinct such groups commute, whereas an abelian regular group is self-centralizing. For Cartesian stabilizers, 25 applies with mutual inclusion in intersections for disjoint supports; small slot sizes \(2,3,4\) are likewise precluded by the regular-group test. In the missing power case for two disjoint supports, the supported normal products in \(E\) are supplied by 54, so the tuple commutator and point-stabilizer factorization in 27 give a common Cartesian refinement with at most one nontrivial fine slot per pair of slot positions. Take \(E\) to be the two-label intersection. Its primitivity gives transitivity on occupied fine slots, say each of set size \(h\ge2\). The top action at each of the two original systems inside \(E\) is transitive containing the alternating group by supplementation, hence primitive, so grouping fine slots into rows and columns splits each slot position on both sides into at least two fine ones (refinement would be improper by that primitivity). Both original slot set sizes are large by the bounded-type consequence for the missing powers.

Let \(u\) be the intersection of the full row-system stabilizer with the fine-slot stabilizer. This is a proper decrease by the supported natural alternating action on each row slot, and it contains \(E\). It permits interchanging two fine slots in one row, leaving the others in place: if the parity of the interchange needs correction then \(h\) must be odd (at least three fine slots), so use an internal symbol transposition. Together with the full column-system stabilizer this generates the ambient group. Here are details of the latter assertion. Local alternating gates on two overlapping sets of Cartesian directions each of product size at least ten generate the alternating gates on their union. Split into differences and shared part of alphabet sizes \(a,c\) and \(b\), all at least two (proper overlap). On the union the generated group is two-transitive: ordered distinct pairs can be changed to any one fixed pair distinct in all three positions using two-transitivity on each of the two touching position-sets (first obtain a difference in the shared part on a position-set with some difference, then set the first and shared values as desired, then the shared and last). For \(b\ge3\) use three-cycle gates on the two position-sets whose lifted supports intersect in one tuple there (one shared value appearing once in each cycle and disjoint other shared-part values). Their commutator supplies a three-cycle, sufficient by primitivity. For \(b=2\), both \(a,c\ge5\). Flip the shared bit on two values of the first alphabet, and use arbitrary \(A_c\) conditional on bit one. Commuting these and commuting again with conditional \(A_c\) gives \(A_c\) conditional on bit one and membership in the chosen two values (by perfection). Indeed the first commutation applies an element of \(A_c\) on one bit value and its inverse on the other just for those two first-alphabet values; intersect the condition with bit one by commutators again. Two such two-value choices intersecting once and commuting their conditional actions likewise give supported \(A_c\) on a single fiber, again sufficient (a three-cycle there, then Jordan). All gates obtained act independently of unused coordinates, meaning the same union permutation repeated on all outside tuples. Now we have the column gates and those on the conjugate columns by the interchange. Indeed an altered column and its original overlap properly, and merging them extends the latter by a slot from a second column, next allowing merger with that entire column by the same observation. Align actual permutations of the columns preserving fine coordinates up to a permutation (uniform fine-slot alphabets and common fine-slot count per column); even column permutations so aligned are in the socle as well. Thus the merged directions connect all columns by primitivity (for two columns the one interchange already connects). The gate observation yields the whole alternating group, hence full join (the base already supplements it). This contradicts \(u\vee r_{D_2}<M_A\). ◻

Proof. By 28, every remaining primitive group that is not almost simple normalizes a transitive power \(Y=B^s\), \(s\ge2\), with a full possibly twisted diagonal point stabilizer and natural degree \(|B|^{s-1}\). This includes two opposed regular simple factors.

First consider retention. A retained power with transitive factor action is a minimal normal subgroup of the common primitive intersection \(E\), with trivial centralizer because its diagonal point stabilizer is self-normalizing. Distinct retained powers would therefore coincide. If the factor action is intransitive, each factor-orbit product is transitive; in the diagonal action a transitive subproduct uses at least \(s-1\) factors. Thus the only possibility is \(s=2\), the two opposed regular simple factors. A retained regular factor is minimal normal in \(E\), and its centralizer supplies its opposite regular group. Distinct minimal normals commute and consequently determine the same opposed pair. These are exactly the retained alternatives in 22, and exclude distinct disjoint-support witnesses.

It remains to consider a missing power with transitive factor action. Take disjoint labels and their proper previous-good intersection \(E\). By 54, \(E\) has a missing normal product \(R=\prod_{a=1}^s R_a\) of singly supported proper groups, with nonsoluble factors. This assertion does not require \(R=E\cap Y\). Choose a minimal \(E\)-normal subgroup \(P'=C^u\) inside \(R\). It is nonabelian by the affine exclusion, and transitive by primitivity of \(E\). Every factor of \(P'\) lies in one \(R_a\): the groups \([P',R_a]\le P'\cap R_a\) are normal in \(P'\), distinct slots commute, and these commutators generate \(P'\), since \(P'\le R\) is perfect. The assignment is \(E\)-equivariant. Supplementation relative to \(g_D\) makes \(E\) transitive on the \(s\) slots, so there is a common positive number \(h\) of factors over each slot and \(u=sh\).

The primitive-group argument of 28, applied to \(E\), says that the point stabilizer \(Q=P'_x\) is a full diagonal. If \(h>1\), the product of the projections of \(Q\) to the \(s\) assigned batches is a proper intermediate subgroup between \(Q\) and \(P'\), normalized by \(E_x\), contradicting primitivity. If \(h=1\), equality of point degrees gives \[|C|^{s-1}=|B|^{s-1}.\] But a factor \(C\) embeds in a proper subgroup \(R_a<B\), an impossibility. Thus the diagonal case is excluded, and the required descendants are primitive almost simple. These conclusions apply simultaneously at the eligible ancestors by 57. ◻

Proof. Soluble unipotent radicals are retained, thus each lies in all other labels on a disjoint family. For linear subspaces (no anti-actions), signed flags (linear with anti-actions), or singular subspaces in the isometry types, the shear arguments of [c:linear-subspaces,c:isometry-subspaces] give a chain (using the smaller space of each signed flag, the other determined in reverse by an anti-action of the base), up to the bounded singular-space terminal exception. Pick two in a many-member disjoint family there (below terminal dimensions with ambiguities in orthogonal type, e.g. below Witt index minus one in split type D). Their full stabilizers and intersection meet the ancestor socle in the corresponding parabolics of compatible flags. By the standard parabolic overgroup theorem for the natural \(BN\)-pair in the simple Lie-type group (overgroups of a parabolic are parabolic, with the standard-parabolic subset order; in the classical interpretation use the indicated subspace or singular flags, away from the split type \(D\) terminal fork here), the labels are maximal and their meet has only the two maximal increases to coatoms: overgroups intersect the socle in parabolics obtained by dropping flag terms, determining the overgroup since it contains the supplementing \(H\). In the signed case terms must be kept in duality-stable pairs (possibly a single middle term). This contradicts the three coatoms above the meet of two coatoms with nonempty disjoint supports. ◻

Proof. Indeed write the hyperplane of orthogonality to a line representative \(u\) as \(u^\perp\); the simple-group stabilizer contains products of two nonsingular symmetries with centers therein. An increase gives another such hyperplane \(w^\perp\); there is a common nonsingular partner for bridging the products (high dimension). Odd and even words of symmetries using these two hyperplanes then give by conjugacy all centers, so even products giving the simple group are contained. Indeed arbitrary even words from the two center sets together are available by inserting a common partner for mixed products; once all centers are obtained as symmetries by words (necessarily of odd length by Dickson parity), their pair products thus belong and give a nontrivial normal generating set projectively by simplicity. In detail, for nonsingular \(v\) outside both one can choose nonsingular \(z\in u^\perp\) whose symmetry takes \(v\) into \(w^\perp\), by choosing both polar pairings \(b(z,v),b(z,w)\) nonzero there, then ensuring \(Q(z)=b(z,v)b(z,w)/b(v,w)\) for the quadratic form \(Q\). One can adjust this value arbitrarily with a large nondegenerate subspace orthogonal to the indicated vectors. Neither functional vanishes identically since the lines are distinct and \(v\notin u^\perp\). This proves the generation assertion, also sufficient in the ambient quotient since the base supplements its socle. For two disjoint failures the pair span \(P\) must be nondegenerate, else its singular line would be preserved and has been excluded on the combined support. The pair intersection then agrees with the full intersection of either line stabilizer with the plane stabilizer (by covering). Indeed the latter cuts it properly, e.g. products of nonsingular symmetries in \(u^\perp\) move the plane (choose one center pairing nontrivially with \(w\) and outside the plane, correcting by one fixing the plane). But this full intersection on the \(u\) side includes the symmetry at \(u\) corrected outside \(P\), failing to preserve \(w\). This gives a contradiction. ◻

Proof. Let an overgroup properly contain \(K\). We show that the transported copies of the supported special group on \(W\) generate the whole ambient special group. First we find large summands close to \(W\) whose vectors span \(V\) in boundedly many summands. We then collapse those sums and recover the global elementary generators.

Throughout, supported groups use their natural matrices, equal to the identity off the indicated summand; their projective images are the groups in \(M\). Transporting a splitting transports these groups with it. In linear type an anti-action transports the special linear support by inverse transpose, and exchanges the corresponding primal and dual data.

Near splittings and bounded sums.

Any increase contains transported copies of the supported special on \(W\) at the largest summand of any moved projector. This copy cannot normalize the original projector: by irreducibility on its support and dimensions it would first force its image space to agree with \(W\), then its fixed complement to be \(U\). The high classical special groups here are generated by matrices of bounded displacement rank (ordinary elementary/root generators, also obtained by Witt reduction). For instance take all inner conjugates of a noncentral elementary generator (in an isometry type one can use a special linear shear on two totally singular basis directions extended dually and identically off the associated two Witt pairs); they suffice by projective simplicity and perfection (generation modulo the special group’s center suffices by perfection). Thus a near copy of the splitting with moved projector is attained, with new largest space \(W'\) differing from \(W\) by bounded codimension of intersection. Use the anti-action if necessary in linear type to get movement of the image (also to achieve kernel movement). Use these and their \(K\)-transports as near splittings (transports by anti-actions of these bounded-rank displacements remain near). Every vector is a sum of boundedly many from these large spaces. Indeed projections span \(U\) by semilinear irreducibility. If \(U\) is bounded-dimensional this suffices. Otherwise transport a nonzero such projection by the available classical actions to get bounded-sum generation there. In the linear or symplectic case use transitivity, and for a norm constraint use high Witt dimension: every vector \(v\) is a sum of boundedly many vectors of a given nonzero vector’s type (norm, including isotropic nonzero). For instance prescribe \(y\) and \(v-y\) both of the given norm by first imposing the corresponding polar pairing with \(v\ne0\) (a trace in hermitian type), then adjusting the norm in a large orthogonal subspace, avoiding zero vectors. Here we use the usual transitivity on nonzero vectors of given norm also under the high special by Witt extension with determinant, spinor or Dickson corrections: a large nondegenerate subspace orthogonal to the vector to be fixed supplies arbitrary such coset data (in unitary type norm-one determinants; in orthogonal type reflections/symmetries at nonsingular vectors, both norm-square classes available if needed). Likewise after prescribing the pairing to get two summands of the given norm, orthogonal adjustment can be made perpendicular also to a first solution of the pairing, using a large nondegenerate subspace representing every norm many times. Use \(W\) itself for the unprojected correction. In the linear setting dual generation holds also: under transport by a fixed anti-action at the original splitting, large-space images for near projectors correspond by a semilinear primal-to-dual identification to annihilators of the transported kernels. The family of these initial near splittings can be taken closed under that transport in both directions (bounded defect on both images and kernels). Thus these annihilators span the dual, with the same bounded-sum conclusion there; in isometry type use selfadjointness and primal spanning. In particular the common kernel of the available near projectors to the large spaces is zero.

Collapsing bounded sums.

One can collapse such bounded sums to one summand from some conjugate splitting, still within bounded codimension of \(W\). Write \(W_i\) for summand spaces used and \(p_i\) for their projectors, with bounded pairwise defect, so intersections also contain large nondegenerate spaces when relevant. If \(s=w_i+w_j,\ w_i\in W_i,w_j\in W_j\), has \(p_i(s)\ne0\), choose \(d\in W_i\cap W_j\) so \(p_i(w_j+d)\) is nonzero of matching type/norm as \(p_i(s)\). Use largeness of the intersection to solve the norm condition, with nonzero choices, and the special-group orbit transitivity with corrections established above. The complementary components agree: \[(1-p_i)(w_j+d)=(1-p_i)w_j=(1-p_i)s.\] The supported group on \(W_i\) can therefore send \(w_j+d\) to \(s\), mapping \(W_j\) to another near splitting. For sums of three or more, if all pair-collapse projections vanish, perturb two summands oppositely by an intersection vector to make another pair collapse. For example perturb the first and second by a common nonzero vector with opposite signs; the first projection for the first-plus-third sum is now nonzero. Defects stay boundedly large at most because only boundedly many steps are needed. More precisely, write \(\delta(E,F)=\dim E-\dim(E\cap F)\) for these equal-dimensional spaces. A transport \(g\) supported on \(W_i\) fixes its projector, so \[\delta(gW_j,W_i)=\delta(W_j,W_i),\qquad \delta(gW_j,W)\le 3d_0\] when both current defects from \(W\) are at most \(d_0\). The same calculation holds on the dual supported spaces in linear type, since \(g\) fixes that part of its splitting as well. Thus we maintain bounded defects for both images and kernels. Large nondegenerate parts of intersections in these steps follow since intersection with a near space has bounded codimension in the nondegenerate summand (and we may still take boundedly many orthogonality constraints). Finally if a nonzero pair sum lies in both projection kernels, take a fresh near \(W_3\) with nonzero projection of the sum. For a nonzero summand \(w_i\), choose in \(W_i\cap W_3\) a nonzero vector of its type and transport it to \(w_i\) by the supported special group on \(W_i\). This transport fixes the sum, since \(p_i(s)=0\). Its image of the \(W_3\)-splitting therefore covers \(w_i\) and still has nonzero projection of \(s\). Use that splitting to collapse the pair. The same construction holds dually in linear type, with both defect bounds maintained as above.

Global elementary generators.

Thus all centers are available near \(W\). Supported special transvections now generate globally in unitary and symplectic type. In orthogonal type use products of reflections/symmetries of same norm, bridging by a common partner in the large near-space intersections; such products generate (in particular supply global nontrivial normal generators). In linear type for \(v,f\) covered respectively in some \(W_i\) and some dual supported \(W_j^*\), \(f(v)=0\), take a commutator of supported transvections with tensors \(v\otimes h,\ z\otimes f\), with \(f(z)=h(v)=0,\ h(z)\ne0\), \(h\) supported in the first dual space and \(z\in W_j\). These choices exist by the large intersections relative to the first projection kernel (one can take \(z\) in \(W_j\cap W_i\cap\ker f\) independent of \(v\), then choose \(h\) on \(W_i\); its scaling gives arbitrary transvection coefficient): in the linear argument keep bounded intersection defects simultaneously on images in primal and dual. This holds for the initial near splittings by bounded displacement (also of the contragredients); when a supported group transports another near splitting in a collapse, it keeps its own splitting fixed, so both defects still grow only boundedly. This gives the required full transvections. This proves maximality. ◻

Proof. Now take isometry stabilizers of proper nondegenerate subspaces, or linear-with-anti individual proper nontrivial projector stabilizers. Designate by \(U_D\) the side with supported special retained by the normal property of 52 over \(g_D\); exactly one side is missing. Write \(e_D\) for the projector to \(U_D\). For a pattern with large retained dimensions, disjoint supports then have mutually orthogonal projectors as in [c:linear-subspaces,c:isometry-subspaces], by supported irreducibility and mutual preservation. For a pattern with bounded retained dimension (take it constant), the full stabilizers used are maximal by 32. We explain small exceptions/ordering in isometry type: the full individual isometry group on \(U_D\) is available in projection with corrections on the large complement. It acts irreducibly except possibly split orthogonal planes over two and three. Over two there is a forbidden nonsingular radical line. Over three there are two nonsingular axes individually invariant unless interchanged by a similarity giving semilinear irreducibility. Thus first run the nondegenerate line exclusion, then these other dimensions with invariant-line cases forbidden on the combined supports. There is no dependence of the line test itself on excluding these planes.

For two disjoint bounded-dimensional occurrences the sum of the small spaces is again complemented, by its orthogonal space or by the kernel intersection, via the prior singular/signed flag exclusions (also nonsingular radical lines when relevant). In linear type an anti-element of \(H\) fixing both projectors exchanges the sum with the kernel intersection, as do signed anti-actions of the pair group; a nonzero intersection gives a signed flag. Also images determine kernels in this case. The sum projector stabilizer cuts each individual stabilizer properly (large complementary supported group) and thus its full intersection with either equals the pair group by covering. Consequently all linear or isometric maps on one small summand, identity on the rest of the sum (split it using that summand’s prescribed complement) and corrected outside, preserve the other small space. The outside here is in both projector kernels, giving a large space for arbitrary special-group coset corrections as above. Distinct small spaces (complements determined and dimensions taken constant) cannot contain each other; this also explains strict cutting by the sum stabilizer, since a projection of the sum to an individual large complement is then nonzero proper, not preserved by the full supported large action. Taking differences on the other small space under the indicated maps thus forces zero projection onto the chosen summand whenever its individual linear/isometry group acts irreducibly and nontrivially: a nonzero projection generates the irreducible space already by its orbit differences. Hence mutual orthogonality outside the indicated small exceptions. For the split orthogonal plane over three with interchange, the isometries here together with the intersection label interchanging the nonsingular axes by earlier line exclusion suffice (the minus sign supported there on the sum splits the other small space along the projection, and the joint action is irreducible on the summand by the interchange together with its isometries). For a binary linear line with anti-action, use the bilinear pairing giving a fixed anti-element: each line and kernel are mutual annihilators for this pairing, with self-value 1 at the line representative. On two distinct such representatives the two cross-values agree, since left and right orthogonality to each line coincide; value 1 off-diagonal would violate the complementation of the sum. This likewise gives projector orthogonality. ◻

Proof. For \(k\ge2\), \(\Delta_{ij}=e_{F\cup i}-e_{F\cup j}\) is independent of \(F\) as in the subset argument. Disjoint-pair differences annihilate each other. Also \(e_D\) acts as identity on \(\Delta_{ij}\) on both sides when \(i,j\in D\), and as zero when \(i,j\notin D\). The latter follows using a disjoint witness for the difference, the former by adjoining a spare disjoint \(k\)-set to \(D\) (the union projector supports a witness within the union). Then \(h_i=\Delta_{ij}\Delta_{il}\) is independent of distinct auxiliary \(j,l\), including order, by disjoint annihilation. For example, with a fourth index \(a\), the triangle identities give \[\Delta_{ij}\Delta_{il} =\Delta_{ij}\Delta_{ia} =\Delta_{il}\Delta_{ia} =\Delta_{il}\Delta_{ij}.\] It meets each \(e_D\) by multiplication either as identity-action (\(i\in D\), use a partner in \(D\)) or zero. To obtain \(\Delta_{ij}=h_i-h_j\), choose \(D\) containing \(i,j\) and let \(D'\) replace them by two fresh positions \(a,b\). Then \[\Delta_{ij}=\Delta_{ij}(e_D-e_{D'}) =\Delta_{ij}(\Delta_{ia}+\Delta_{jb})=h_i-h_j.\] No division by the characteristic or by \(k\) occurs. Thus distinct \(h_i\)’s annihilate each other (separate by an \(e_D\)), and \(h_i=(h_i-h_j)(h_i-h_l)=h_i^2\). They are orthogonal idempotents, selfadjoint when relevant, and the remainder in \(e_D-\sum_{i\in D}h_i\) is constant, necessarily zero by disjointness and idempotence (each such remainder is itself an idempotent supported by \(e_D\) on both sides). For \(k=1\) use the projectors directly. ◻

Proof. In the resulting homogeneous \(k\)-set patterns choose dimensions small relative to total by orthogonality on disjoint supports. The stabilizers and disjoint-pair sum stabilizers are maximal by 32, and the sum label is \(C_{D\cup D'}\) as in the subset argument (it moves each summand). For clarity the aggregate here has the requisite irreducible collineation action: split planes over two do not arise as sums, and over three in the line test choose both norms of the same class by coloring, avoiding the split plane. Thus the aggregate projector \(e_D+e_{D'}\) depends only on the union; it is recoverable from its stabilizer, e.g. the supported large complementary special forces uniqueness among projectors with small image.

Apply 63 to write the projectors as sums of the \(h_i\). In this marker configuration, at most one \(h_i\) vanishes, and each nonzero \(h_i\) has full stabilizer label \(M_{\{i\}}\). For \(k=1\), the full stabilizers are already the singleton cuts \(C_{\{i\}}=M_{\{i\}}\). For \(k\ge2\), two vanishing \(h_i\)’s would identify distinct cut projectors by replacing one index with the other. For a nonzero \(h_i\), choose \(D,E\) meeting just at \(i\); then \(h_i=e_De_E\). Thus the paired cut-meet argument of 30 puts \(M_{\{i\}}\) in its full stabilizer. That stabilizer is proper because \(h_i\) is a nonzero proper projector, with nonzero disjoint witnesses remaining. Since \(M_{\{i\}}\) is a coatom, it is the full stabilizer.

For nonzero positions, in large fixed sizes \(l,2l\) with ample further spares, all sum-projector stabilizers on those sets \(P\) are again maximal (keep dimensions small relative to the ambient by thinning), hence are \(C_{B(P)}\) for some nonempty \(B(P)\subseteq P\), and distinct by projector uniqueness. Homogeneity/counting forces \(B(P)=P\) on a thinned reservoir (homogenize the decreasing-size test; injectivity rules out strict decrease everywhere for that size). For disjoint \(l\)-sets, if the sum dimensions differ, their meet to the combined cut is an ordinary cover, by 32 applied within the combined aggregate (both pieces now sufficiently large). Indeed projection to ring actions on that aggregate identifies the meet as full ordered splitting stabilizer, with anti-action still present in the signed linear case. These applications have the supplement hypothesis: the ancestor outer actions are supplied by the common base, and restrictions of linear/special isometries from the inner ancestor are covered modulo the aggregate special group already by the splitting stabilizer, using the two large pieces for arbitrary determinant etc. (orthogonal determinant and spinor or Dickson corrections by Witt groups) and the large outside space for compensation. The formal wreath label is strictly between the two-cut meet and the combined cut. If these product endpoints formed an ordinary cover, 55, applied before the ancestor quotient, would force normality at each original site. This is impossible on intersecting with the actual supported high special group of the combined aggregate: its projective action is simple, and the lower intersection is a proper nontrivial splitting stabilizer. The ancestor centralizer kernel is already in the lower endpoint, so this is also a valid nonnormality test in the quotient. Hence dimensions agree for disjoint \(l\)-sets, thus all individual dimensions agree (compare a pair of replacements to one common disjoint set). Align the individual summand forms themselves by finite coloring. Now on an \(l\)-set the sum-cut interval from the axis has at least the whole subgroup lattice of \(S_l\) just up to the stabilizer allowing permutations of the individual projectors (kernel the axis label, and align permutations with outside correction). The sum stabilizer exceeds this by its large full classical action, which cannot preserve that uniform direct decomposition (same orders as in the earlier direct-system pruning). Indeed the formal interval \([M_P,C_P]\) is exactly the subgroup lattice of \(S_P\), with \(|P|=l\). Its ordinary product sublattice cannot contain more labels. Yet the permutation stabilizer just described already supplies one ordinary label for every subgroup of \(S_l\), and the strictly larger full sum stabilizer supplies an additional label. This is the required contradiction. Thus these failures too are excluded. As before we get prior irreducible anchors (collineation part also in linear type). ◻

Proof. In the line-system argument of 11, the diagonal groups are still mutually included by retention, so it remains to treat its matching cases. All matchings relative to the first system would have to agree: the signs of one of the other systems already interchange first-system lines on a matched pair, simultaneously on a second pair if needed (choose two flips of equal norm class for determinant/spinor correction). There are plenty of possible second pairs by transitivity and at most two norm types (the two sign-orbit lines on a pair are of the same norm class); thus another invariant matching must pair these same two first-system lines (otherwise choose the extra flip away from a proposed different partner to keep that partner fixed). The choices on matched lines are unique as before. For the exceptional plane matchings of 12, an individual retained plane sign likewise interchanges the two old planes, forcing matching coincidence. The common intersection of all the labels is transitive on the matching edges, so for that fixed intersection group bounded choices on one edge bound the number of further systems, impossible. Here and below common intersections for any bounded family can be required previous-good by earlier rounds at sufficiently increased omission bounds. ◻

Proof. In the remaining direct-system cases the supported groups of order greater than two for compatibility are available exactly as above: use the full retained groups, or, for a missing simple-site orbit product, 54 with the previous-good intersection. (If two site orbits, can likewise use full supports from a retained one.) Centers included with these lifts are retained. In fact any missing simple orbit here forces large block dimension by the bounded-type consequence, with the action on sites of bounded pair rank by supplementation relative to \(g_{D_i}\); in particular bounded rank including the possible split four-space ambiguity cannot then occur. Block pair-orbits are bounded by the alternating actions on at most two isometry classes (one class in linear type), and the simple sites have fibers of size at most two over blocks. In applying the missing-product argument here the singly supported projections modulo centers in the intersection actually yield block-supported matrices projectively there: use representatives on that block in the supported choice of groups, since the center kernel needed when lifting already lies in the intersection. We can take all available such block matrices there, so the intersection’s semilinear collineation stabilizer of a block normalizes the corresponding supported group. The block module for the supported group again is semisimple without trivial constituents by semilinear irreducibility for that stabilizer (use the module socle and the fixed group of vectors, as before) and nontriviality of the block action. (For a missing product the singly supported subgroups are even nonsoluble.) The displacement-rank proof of 14 now applies: fixation of all the other system’s blocks forces splitting by projections using orbit differences there; otherwise movement both ways forces equal block dimensions, and forces any moving block matrix to interchange a single pair of those other blocks as an involution, identity elsewhere there and with displacement of full rank on its own support, as in 14. Its retained-case proof also applies: full retained actions force comparable refinements, whose height is absolutely bounded within the pattern by the uniform order obstruction on coarse blocks. ◻

Proof. Here the summands are direct groups of blocks of a common splitting, orthogonal in isometry cases, with the corresponding complements. Use \(W_1\) of dimension at least \(\dim W_2\). Every vector in the sum lies in some conjugate of \(W_1\) by the supported group of \(W_2\): for a nonzero \(W_2\)-projection attain it by transporting a vector of the intersection (matching norm using its high rank), and the remaining component already lies in \(W_1\). Thus we get the full symplectic or unitary transvection generation on the sum; likewise products of orthogonal symmetries/reflections of matching norm, using \(W_2\) and partners in its large intersections with these conjugates to bridge any two centers. These elements give noncentral normal generation of the special. For linear type we also have dual coverage; as in the projector argument get a transvection tensor \(v\otimes f,\ f(v)=0\), by commutators from two supported spaces of dimension \(\dim W_1\), sufficient because twice that dimension exceeds the sum dimension by a large amount. All assertions can use the natural covering matrices (the elementary generators obtained give the perfect special group), with the indicated support. ◻

Proof. Treat missing systems with two blocks first. More generally once these are excluded, two-class block systems are impossible by coarsening, and the top action on blocks, induced inside intersections by supplementation, is primitive (two blocks, or at least the alternating permutations with transitivity). Thus two disjoint-support instances give incomparable compatible systems and each block splits into at least two cells, of uniform dimension by transitivity. In the general round all those cells must in turn be mutually isometric when appropriate (else a two-class coarsening). In the two-block round there is a full \(2\times2\) grid of spaces, and for some choice of the pair of instances we can interchange two same-row or same-column cells isometrically. Indeed if this fails the form must be orthogonal in odd characteristic with odd cell dimension by similarity/transitivity. Take three failures, choosing their common intersection to be previous-good as allowed by 57. Its collineation irreducibility makes the occupied triple cells one orbit. Since its permutation action lies in \(S_2^3\), this orbit has size \(4\) or \(8\), and it projects onto all four cells for every pair. Size 8 violates odd pair-cell dimension by summing two equal triple-cell dimensions. For size \(4\), the third two-block system is the unordered diagonal partition of the four cells of the first pair (there is one third position at each occupied pair, with both third positions represented in each row and column), so the whole intersection of that pair of stabilizers preserves it, impossible by disjoint label supports.

Use a resolved pair, designated row and column, and let \(u\) be the intersection of the full row-system stabilizer with the stabilizer of the simultaneous refinement. It contains the joint intersection and is strictly smaller than the row label (uniform finer decomposition on each high-dimensional block). It permits a swap of two fine cells in a row, leaving the others in place, or a row three-cycle of cells. Indeed we have isometric choices as just described, and determinant, spinor or Dickson corrections inside the cells when of large dimension. For small cell dimension here the row block dimension is large (absolute thresholds can be made large enough to ensure at least three cells) and all cells are equivalent, so take a three-cycle via commutators of aligned permutations. Such collineations come from the ancestor socle.

We now apply 66 to the original and altered column spaces.

A column space and its altered version lose/gain one cell by the within-row permutation and have large intersection (at least half a block); their supported groups thus merge, extending the column \(X\) by a cell from another column \(Y\). If that cell is large we merge with \(Y\) immediately. Otherwise use the column label to transport within \(Y\) (alternating fine-cell permutations there while preserving \(X\)) and merge extensions along \(X\) until acquiring \(X+Y\). The resulting merged column pairs propagate to a connected graph by block primitivity, and give full generation by repeated mergers with at least a column in common. This contradicts the formal disjoint-support rule \(u\vee r_{D_{\rm col}}<M_A\), and finishes direct systems. ◻

Proof. In defining characteristic, for classical simple groups of sufficiently high rank, with natural dimension \(m\), the minimum nontrivial irreducible projective degree is at least \(m\); except for the natural module up to Frobenius/duality, it is at least \(m^2/2-O(m)\). To see this use the restricted weights on the universal cover and the Steinberg theory recalled in 9, using Steinberg (1963; Jantzen 2003). For a restricted digit in A the permutation orbit already gives the latter bound except at a single endpoint only; at that endpoint with coefficient greater than one, simple-root lowering (coefficient less than \(p\)) and the resulting orbit give it as well. In B,C,D use signed coordinate permutations (even signs in D); at least two nonzero coordinates give the larger bound in high rank (including spin weights), and a single first-coordinate support with coefficient greater than one works again by lowering. Use B only in odd characteristic. Here the single first-node coefficient one is the irreducible natural module, of degree \(m\) also in odd-dimensional B. For \(k\ge2\) nonzero coordinates out of \(e\) in a signed word the orbit has size at least \(2^k\binom e k\), allowing a loss of a factor two just at \(k=e\) with even-sign restriction; the stated quadratic bound follows (including at \(k=2\)). These arguments also give the bound by \(m^l\) for \(l\) nonzero digit places. Cross-characteristic nontrivial projective degree is at least \(P^{c m}\) for field parameter \(P\) and absolute \(c>0\), by the argument of 33; in particular the same exponential bound works for permutation degree. These bounds on nontrivial contributions work also upon restriction to perfect central lifts/quotients, pulling to the universal covers: lift projective actions there and take a nontrivial composition factor (there is one by perfection if the projective action is nontrivial). The center acts scalarly on irreducibles. In particular minimum permutation degree can be tested this way by a permutation representation in another characteristic (the high perfect central lifts themselves then have no nontrivial small permutation image). We use the classification and covering-group structure in 9. An alternating simple of high degree acting projectively nontrivially in degree \(v\) has alternating degree \(O(v)\), by the following elementary argument. Write its degree as \(n\), choose \(\ell\in\{3,5\}\) different from the representation characteristic, and take \(r=\lfloor(n-2)/\ell\rfloor\) disjoint \(\ell\)-cycles. They generate \(E\cong C_\ell^r\), leaving two spare letters. The normalizer in \(A_n\) induces every permutation of these generators: a permutation of the \(\ell\)-letter blocks has its parity corrected, when necessary, on the two spare letters. For matrix lifts of the projective action of \(E\), scalar commutators define an alternating bicharacter \(b:E\times E\to\mu_\ell\). Conjugation preserves \(b\). Interchanging any two generators therefore makes their value equal its inverse; because \(\ell\) is odd, that value is one. The lifts commute. Rescaling them makes their \(\ell\)-th powers one, and, since \(\ell\) is prime to the characteristic, they are simultaneously diagonalizable. The projective action of \(A_n\) is faithful by simplicity, so its restriction embeds \(E\) in the \(\ell\)-torsion of the diagonal torus modulo scalars. Consequently \(r\le v-1\), and \(n\le5v+1\). ◻

Proof. With a standard pinned graph/field map of height \(f_0\), coefficient \(p^j\)-twisting a defining-characteristic irreducible is equivalent on the finite cover to twisting its rational highest-weight lift by the split \(p^j\)-Frobenius of the source before restriction. Indeed in split pinned coordinates, transforming coefficients in the rational representation together with the coefficients in the argument (i.e. conjugating by coefficient Frobenius on points of target and source) still gives an equivalent rational irreducible by highest weight. The split map permutes the standard pinned finite fixed group; as in 9, digit positions thus move by \(j\) modulo \(f_0\), with diagram action upon wrapping. This reasoning up to equivalence suffices (also for testing composition factors); on a perfect covering group an equivalence of fixed projective actions lifts to linear equivalence because scalar discrepancies are characters. ◻

Proof. Let \(Y_1,Y_2\) be the supported perfect projective special groups on the two spaces, with the alternating special groups (symplectic groups) as needed on factors. They lie in \(S\). For an overgroup within \(S\) take a minimal nontrivial normal subgroup \(N\) there. Its lift algebra is semisimple by absolute irreducibility of the supporting action \(Y_1Y_2\), and has no multiple center components: the larger simple action fixes all projectors by its minimum permutation degree, then there are at most \(x\) of them (invariant spaces are modules over the larger matrix factor), so the smaller also fixes them. The center is then scalar by perfection and full spanning by \(Y_1,Y_2\).

Any proper nonscalar matrix factor fixed individually by \(Y_1,Y_2\) gives precisely the original factorization. Indeed lift the projective actions on factor spaces and tensor by perfection; all are irreducible on the universal covers jointly by tensor irreducibility. The lower bounds \(x,y\) for any respective nontrivial contributions (and \(xy\) if both in one slot) force their allocation to distinct unique slots of a nontrivial split. Their matrix spans recover the factors. Here the separate lifts commute as needed (any residual scalar commutators against a perfect factor give a trivial character); joint irreducibles of the direct product over the closure are outer tensors. The supported matrices originally span the actual factors over \(F\) by natural absolute irreducibility, and a trivial projective action on other tensor directions means these matrices belong to the indicated one by commuting with the others. Even for a permuted tensor system, \(Y_2\) on the larger space fixes all slots, and acts projectively on just one (two would cost at least \(y^2\)); \(Y_1\) preserves that one and must fix the others since their product dimension is at most \(x\). Thus a proper nonscalar lift algebra for \(N\) already forces ordered stabilization by the overgroup. If the algebra is full and \(N\) is a power of nonabelian simples, their lift algebras tensor to it by commuting as usual, so multiple components also force ordered stabilization (no swap of unequal degrees). The elementary abelian full-algebra case has the nondegenerate scalar commutator pairing, order \(v^2\) and projective linear centralizer just itself. Its normalizer inside the simple group is far too small for \(Y_1Y_2\) (at most \(v^{O(\log v)}\) by automorphisms of the elementary abelian group, while \(v\le y^2\)).

Thus otherwise \(N=B\) is a single simple with trivial projective linear centralizer and contains \(Y_1,Y_2\) by Schreier. It must be high classical in the same characteristic: bounded Lie ranks are excluded by the minimum faithful projective/linear degree of the larger simple support in any characteristic (use also bounded-dimension adjoint realizations there); alternating type and cross-characteristic classical type are excluded by the exponential bounds on embedding the larger support, together with the action of \(B\) in degree \(xy\). In the alternating alternative the alternating degree would be at most \(O(xy)\) by that action but at least exponential in \(y\) by restriction to \(Y_2\). In the other-characteristic high classical alternative the natural dimension there would already be at least exponential in \(y\) and its own cross-characteristic degree far exceeds \(xy\). Its natural dimension \(w\) is at least \(x+y\): restrict its projective natural action to the commuting supports, lift from universal covers of each (commuting linearly by perfection), and count composition factors with nontrivial contribution from each (a joint irreducible with both nontrivial is an external tensor over the closure, costing at least \(xy\)). Thus by the stronger degree bound, \(V\) is projectively the natural representation of \(B\) up to twist, \(v=w\): we have absolute irreducibility by the full matrix span, and \(w^2/2-O(w)>xy\).

In this equality case \(B\) matches the ambient root series. In ambient A, the supporting representation is not self-dual even over the algebraic closure for the finite supporting special groups; in signed types the form symmetry and dimension determine the series (and type A natural is not self-dual). In characteristic two a symplectic natural \(B\) cannot preserve the orthogonal quadratic (see also the field-transfer transitivity test in 12; over its natural finite field an invariant quadratic, even allowing coefficients in the closure, would have constant evaluation on nonzero vectors). Such finite-group preservation on perfect linear lifts would be exact. Now apply 2: either \(B=S\) or it comes from a proper root of the ambient Steinberg map, of field height strictly between zero and the ambient field height \(f\), fixing the supports pointwise projectively. The latter is impossible by restricted-weight uniqueness: on a supporting group the digit is natural at field exponent zero (parameter \(p^f\)), not its twist through such a height even up to duality. Frobenius tests on projective representations here and below agree linearly on universal covers by perfection. This proves the claim. ◻

Proof. In the field case the stabilizer includes the perfect transfer group \(Y\) over \(D\), of large natural degree \(m\). In the paired case use the paired perfect \({\rm SL}_m(F)\). Its centralizer algebra is \(Fe+F(1-e)\), by natural versus adjoint-dual representations; that for the field case is \(D\).

The full stabilizer contains a supplement to \(S\). To prove the assertion take a minimal subgroup \(N\ne1\) normal in the overgroup inside \(S\). The lift algebra is semisimple, and no center components can be permuted nontrivially by \(Y\), by its minimum permutation degree (at least exponential in an absolute positive constant times \(m f_Y\), where \(p^{f_Y}\) is its field parameter with \(m f_Y\ge v/2\); the number of center components is at most \(v\)). Thus its center lies in the displayed centralizer alternatives (perfection). Any enlargement from scalars already suffices: a larger center normalized by the overgroup then gives the indicated field (inside \(D\)) or exchanged pair. Here the high-degree centralizer facts use the transfer/paired representations as in the earlier pruning. Nor can there be a proper nonscalar central simple tensor factor, or multiple simple components of \(N\) tensoring to the full algebra: \(Y\) must individually stabilize such matrix factors in either setting (by order), acting projectively nontrivially on every factor space, as the centralizer algebras are commutative.

Indeed any such nontrivial projective representation of \(Y\) over \(F\) has degree greater than \(\sqrt v\). For the paired group this follows from \(m\); in the field case lift on its universal covering type, with some nontrivial composition factor. The factors are stable as a multiset under twist of field height \(h=\log_p |F|\). Indeed coefficient twisting the lifted representation through this height preserves the projective matrices (taken over \(F\)), hence gives the same representation by perfection on the universal cover, and in particular the same composition factors. If that constituent has orbit length \(d\), its restricted highest weight has at least \(f_Y/(d h)\) nonzero digits, since the twist circulates them modulo field height \(f_Y\) of \(Y\), where \(f_Y/h\ge n/2\). Diagram twisting does not affect the support positions. This costs dimension at least \(d m^{\lceil n/(2d)\rceil}\ge nm/2\), counting the orbit constituents with their Steinberg dimensions (use \(m^{j-1}\ge j\) for \(m\ge2,\ j\ge1\)). Thus two tensor directions are already impossible.

The full-algebra elementary abelian normalizer inside \(S\) is again too small by orders to contain \(Y\). Thus otherwise we have a full-spanning simple \(B=N\), with \(Y\le B\) as before (centralizer inside \(S\) trivial). The order/degree exclusions again force \(B\) high classical of the same characteristic, since \(Y\)’s field-height times natural degree is at least a constant times \(h v\). Let its natural dimension be \(w\ge m\), field height \(f_B\), and put \(t=f_B/\gcd(f_B,h)\). The absolutely irreducible projective representation in \(V\) costs \(v\ge w^t\) by 68. Orders compared with \(Y\) give \[h v m\le C f_B w^2\] for an absolute constant \(C\). With \(m\) large this forces \(t=1\) (otherwise \(m\le C t w^{2-t}\)), then \(w^2\ge vm/C\). Again the representation must be natural up to twist by the nonnatural lower bound of 67.

We detail series and root checks here. In ambient linear type the field support uses the natural linear type over \(D\) with its embedding-conjugates and is not self-dual. In unitary type we necessarily have odd \(n\) (involution extending the ground involution); the \(n\) embedding twists of the natural unitary type occupy distinct digit positions modulo \(n f\), so again cannot match to form a self-dual module. For a pair in unitary type the natural and conjugate-dual of \({\rm SL}_m(F)\) at distinct digits likewise do not pair to a self-dual module. Signed types match by the previous tests; in the included symplectic characteristic two field case (trivial adjoint on \(D\)), \(Y\) is symplectic over \(D\) and cannot preserve a nonsingular quadratic even with extended coefficients. For example its lift is transitive on nonzero ground vectors, impossible for such a quadratic by evaluation there (also over the binary field, using polarization or a 3-space). The square-form preservation by a perfect group would be exact. Now all supporting constituent nonzero digits lie at multiples of the ambient \(f\) with \(f_Y\) a multiple of \(f\) (also in the indicated paired representations). No proper smaller positive height could fix the action projectively, even up to duality. To see the digit-position convention explicitly, the field-transfer action over the closure uses natural copies over the embeddings of \(D\) extending the ground embedding, hence twists spaced by \(h\), with \(f_Y=hn\) in the linear or trivial-adjoint case and \(hn/2\) in the involutory case. In the latter, either \(h=2f\) in unitary type, or \(h=f\) in bilateral type and the nontrivial involution fixes \(F\), so \(n\) is even there. For a pair one uses \(f_Y=h\) with natural and adjoint-dual (thus conjugate-dual in unitary type). Wrapping may diagram-twist but does not affect the positions modulo \(f\). A fixing ambient root would intertwine the representation with a coefficient twist by its height (possibly with duality) as natural projective matrices, hence on perfect lifts as above, rotating the positions to nonmultiples of \(f\). This rules out a proper root as before and proves the assertion. In particular prime field labels and exchanged-pair labels in these uses are maximal. For a two-prime compositum the only possible maximal overgroups correspond to prime subfields. ◻

Proof. For two disjoint failures whose stabilizers respect such quadratics (transforming as the bilinear form), irreducibility of a common previous-good label forces the same quadratic by square-difference, yet the two labels join to the top, not preserving it. This applies to tensor stabilizers (factor forms then alternating), to fields with nontrivial adjoint by hermitian trace, and exchanged pairs by their canonical hyperbolic quadratic. One can exclude all such additional quadratic stabilizers as soon as previous irreducibility on the joint support is available, also for use when a later tensor label is known to force one of them (alternation of ordered factors by previous irreducible normalization; for a slot system transitivity will give alternation). ◻

Proof. The factor-form arguments of 6 apply. Take a homogeneous pattern on all \(k\)-sets with designated algebra \(A_D\) on the retained side (over \(g_D\); the supported special groups are commuting normals with soluble quotient). For disjoint sets these large supported groups normalize each other; nontrivial intersection would force equality of simple groups, hence algebras and labels. Thus they commute projectively and linearly on perfect matrix lifts, so the designated algebras commute and multiply as independent factors. By coloring one can keep all their degrees small relative to total, e.g. at most \(v^{1/10}\), by using arbitrarily many disjoint sets. Now each such factor stabilizer, and the stabilizer of the product on a disjoint pair, is maximal by 35 and supplementation of the socle by the base. Thus we have labels \(C_D\) and product label \(C_{D\cup E}\) for disjoint \(D,E\), by the three-coatom argument of 11 (the aggregate supported group moves its constituent factors). The aggregate small-side algebra \(A_D A_E\) depends only on the union: among such small sides it is unique for its full stabilizer, by the supporting-group identification of individually preserved tensor factors in the observation. By repartitioning two sets at a time, products for any larger disjoint family likewise depend only on the union.

Recovering elementary factors.

Consequently \(A_D\) tensors into elementary matrix factors \(T_i\) indexed by \(i\in D\), independent of \(D\) (scalars allowed). Here are details for \(k\ge2\). At a fixed \(B\) take external mutually disjoint \(k\)-sets \(B_i\) for \(i\in B\). Repartition using sets \(X_i\) each consisting of \(i\) and all but one entry of \(B_i\), plus the final set of leftovers. Then \(A_{X_i}\subseteq A_B A_{B_i}\); take its coefficient algebra \(L_i(B)\) in \(A_B\) relative to this tensor expression (generate from all coefficients). These coefficient algebras commute with each other since the \(A_{X_i}\) commute and the external factors are independent. They jointly generate \(A_B\), since the leftover-set factor lies entirely in the product of the \(A_{B_i}\)’s (repartitioning the auxiliary sets), and by repartition of the whole union including \(B\), \(A_B\) now lies in the product of the coefficient-generated algebra with the independent external tensor factors. Thus they are central simple tensor factors (commuting generating algebras have scalar centers and no radicals). For another choice with disjoint auxiliaries they commute across choices at different positions by the same reasoning, so coincide at matched positions by centralizers. Compare arbitrary choices through spare disjoint auxiliaries. And for a different \(B\) still containing \(i\), one can use the same \(X_i,B_i\) auxiliary choice, getting exactly the same coefficient algebra (take coefficients into the commutant of \(A_{B_i}\)). This proves the assertion; these factors are selfadjoint when needed (take coefficients also in the adjoint basis, since the algebras used are selfadjoint). They commute across arbitrary positions, hence are simultaneously tensor independent by central simplicity. At most one can be trivial by label distinctness, and each nonscalar \(T_i\) has stabilizer \(M_{\{i\}}\) as usual, since it is \(A_D\cap A_E\) for \(D\cap E=\{i\}\) (two such paired meets of cut labels, with remaining indices disjoint, join to \(M_{\{i\}}\); the stabilizer is proper by the tensor order bound). For \(k=1\) this conclusion is immediate instead.

The aggregate supplement.

Take prescribed large fixed set sizes, on nontrivial positions with degrees again kept small enough relative to total by thinning. For all such sizes product-factor stabilizers on sets \(P\) are maximal (large unequal sides), hence \(C_{B(P)}\), \(\emptyset\ne B(P)\subseteq P\); they are distinct and thus \(B(P)=P\) after homogenizing on a sufficiently large reservoir. Use sizes including \(l,2l\) for sufficiently large even \(l\). For disjoint \(D,E\) of size \(l\), \(P=D\cup E\), write \(J=C_D\wedge C_E, F_0=C_P\). On the aggregate for \(P\) we have the full supported high special projectively, say \(R\), normal in \(K_{F_0}\) and lying inside the actual ancestor simple. There is no loss from possible outer cosets: we claim \[K_J R=K_{F_0}.\] Indeed transport \(R\) as a normal product over initial sites at the product label \(F_0\), and join to \(M_P\), giving the product label of \(K_{M_P}R\) in the exact interval \([M_P,F_0]={\rm Sub}(S_{2l})\). Its meet with the alternating group there is either endpoint on \({\rm Sub}(A_{2l})\), by the earlier exact normal test. If the meet is trivial, the projected normal-multiple label in \({\rm Sub}(S_{2l})\) itself has order at most two, and thus preserves some \(l\)-set there (\(l\) even). This puts \(R\) inside the actual factor stabilizer by the cut labels, impossible on its high aggregate by the tensor order bound. So it contains the alternating group formally, giving the claim by joining with \(J\).

Degree equality and the characteristic-two exception.

In orthogonal characteristic two use instead disjoint unequal large sizes also chosen as above, with their sum. Then \([J,F_0]\) is formally just a cover (Young subgroup with unequal blocks in the combined symmetric group; any movement of the split yields a crossing transposition by conjugation). But \(R\) includes the supported symplectic group on the alternating aggregate (tensor with a further alternating space, with canonical total quadratic). Within \(F_0\) we have a proper intermediate between the split stabilizer \(J\) and the upper label by imposing the canonical tensor quadratic on the aggregate, with properness over the split by its supported high orthogonal group and tensor orders. Indeed \(K_J\) stabilizes both ordered pieces and acts by semilinear similarities on them, respecting the split’s canonical aggregate quadratic (zero on pure split tensors) with the aggregate polar multiplier. Not all of the supported symplectic group respects it. This excludes the case directly.

In the remaining types use our equal-size index sets. If the two actual side degrees differ, 35 makes \(K_J\cap R\) maximal and nonnormal in \(R\). This intersection is the full ordered stabilizer there: within \(R\), fixing one piece also fixes its commutant piece on the aggregate. Since \(K_JR=K_{F_0}\), intersections with \(R\) determine all ordinary intermediates, so \([K_J,K_{F_0}]\) would be a cover. Yet the formal interval has a proper wreath intermediate. Apply 55 at the original sites before the ancestor quotient. It would make the lower site projections normal in the upper ones, and therefore make their intersection with the actual \(R\) normal as well, contradicting the preceding nonnormality. Thus all disjoint \(l\)-products of degrees agree. Comparing replacements with spare sets makes all elementary degrees agree; finite coloring aligns their form classes.

The final permutation count.

Now under \(F_0\) the label allowing permutations of the chosen slots in \(P\) has kernel label \(M_P\) for that action. It induces all symmetric permutations. Indeed use the aligned tensor transpositions in the ambient socle, with corrections as follows. For common degree \(d\) the interchange has negative-eigenspace dimension \(\binom d2 v/d^2\) for sign purposes in odd characteristic, which is even when \(d\) is even by the many slots, and also when \(d\) is odd and the total dimension even. Linear/unitary determinant can always be corrected (by minus identity for odd total dimension). In orthogonal odd characteristic with even slots the tensor complement of the two has even dimension and square determinant by additional even slots. The negative-space form determinant modulo squares for the involution is then square by tensoring, giving also spinor membership. For odd slots use isometries on the large complement of all the chosen slots in \(P\) for determinant/spinor corrections: tensoring with identity on an odd-dimensional symmetric factor still gives both independent invariants (diagonalize that factor; determinant unchanged and spinor unchanged up to a determinant-dependent scalar class). The full aggregate group exceeds this permutation stabilizer by orders. Thus above \(M_P\) there are more elements than the entire formal interval to \(F_0\), contradiction. ◻

Proof. For remaining ordered-factor failures the missing side must still be of large dimension by the missing-site bound: the normal products of supported groups modulo centers have soluble remaining quotient and at most the two-slot ambiguity, so some missing simple orbit would otherwise have bounded type. (If there are two projective simple components on a side, use a missing transitive orbit thereof; supplementation over \(g_D\) gives the top action, here of bounded pair rank.) Thus it is the retained side that is now small. The component list and arguments of the earlier ordered-factor pruning then apply except for retained nuisance factors: a linear plane over two, an odd-characteristic bilinear orthogonal plane, or a symplectic plane over two. Indeed across a disjoint pattern all other retained components lie mutually in the stabilizer labels, with exactly the same normalization and spanning arguments giving commuting distinct component algebras (one or two per label); the cases giving forbidden direct systems there are already excluded. From arbitrarily many disjoint labels then take many distinct commuting slots. Label distinctness indeed requires many distinct slots since the designated slot set determines the designated algebra. The common intersection preserves each label’s set of at most two slots (orbit sizes at most two even on the union), so preserves an ordered aggregate of many with still many unused, impossible by 37 on the combined support. This proves the partial exclusion needed. ◻

Proof. For any prime field algebra label under consideration, the large transfer degree hypothesis of the stabilizer observation holds: the transferred special inside the ancestor simple gives, modulo its soluble center, nonabelian simple orbit product(s) with soluble remaining quotient, one of them necessarily missing by the pure-link normal test, hence of high type. To spell out this use of the standard orders/structures: if the module degree were bounded, total ground dimension large would force the extension and its parameter large; permanently soluble dimensions then give only a soluble stabilizer, and otherwise the natural perfect special modulo center gives the simple(s), with at most the split orthogonal four-space ambiguity. Semilinear field automorphisms, similitudes, extra scalars or anti-actions do not affect quotient solubility. We use the same argument for the transferred groups in simultaneous intersections below when the intersection is the full simultaneous stabilizer as indicated, over \(g_*\) there. More explicitly, any center lost on passing from such a normal transferred perfect special group to its simple site(s) is soluble normal and retained; if the simple sites are not transitive take a missing orbit (by the normal test and solubility of the rest above the supported products), with boundedly many sites and the lower pure-link endpoint supplying the orbit action by supplementation. Thus the high-type conclusion of the missing-site argument applies; we may demand the transferred natural degree exceed any needed absolute bound, by the bounded-rank test there. Thus field labels are coatoms by [fb:field-pair-overgroups,fb:canonical-quadratic-exclusion]. For two disjoint-support instances the intersection \(J\) is covered in each, with also a third coatom above it. In particular if the two fields commute with compositum a field, their intersection is the full normalizer of that field, of two-prime degree (equality would give identical labels); its large transfer and 36 rule out the third coatom. For any such application in the symplectic even case with individual trivial adjoints, the compositum also has trivial adjoint. ◻

Proof. Linear type.

In linear type the prime-field proof with retained projective tori from 6 thus works: commutation gives a field by the direct-system exclusion; the only possible noncommuting exceptional pair gives a quaternion \(M_2\) factor over three normalized by the intersection, now forbidden by 70. We thereby also exclude the remaining linear ordered factors (plane over two gives a canonical quadratic field).

Fields with nontrivial adjoint in isometry types.

In isometry types first take field patterns with nontrivial involution restriction. For two disjoint occurrences use \(R=F[D_i,D_j]\) as in the previous pruning, semisimple with either center field or two exchanged central components. We give the adaptation. If a larger center field gives an invariant central prime \(D'\), then either \(K_i\) normalizes it (forcing \(D'=D_i\) by the perfect transfer and centralizer), or the full simultaneous stabilizer of \(D_i,D'\) equals \(J\) by covering. In the latter, if \(F[D_i,D']\) is a field its large transfer inside \(J\) centralizes \(D_j\), giving commutation. If not, there must again be two exchanged components, from equal prime degrees, and the full intersection includes the paired special linear group over \(D_i\) on the components, normalized there (\(D'\) is likewise scalar over \(D_i\) on each, by the tensor of the two equal-degree fields, a product of copies of \(D_i\)). Its two spaces are dually totally isotropic and totally singular when relevant by previous irreducibility. The quotient over this paired type of group is soluble, so by the pure-link normal test and the missing-site bound its module degree on each component is again large. Indeed on a pair commuting with \(D_i\) use the transferred pairing over \(D_i\), acting by a determinant-one matrix and its inverse adjoint. In the split-algebra use this also respects \(D'\) since on each component the two prime fields act by the same field of scalars. Quadratic restrictions with zero polar on the pair spaces in characteristic two are zero by previous-good irreducibility (else the kernel sum gives an invariant proper nonzero subspace), so pairing preservation suffices there. This special lies inside the ancestor simple by perfection in the nonsoluble large-parameter/dimension range, normalized in the simultaneous stabilizer which acts by paired semilinear similarities allowing exchange (thus soluble remaining quotient; dimension one itself would just give solubility). It too centralizes \(D_j\), and its ground-field centralizer commutes with \(D_i\), as in the earlier paired situation: in the now large degree the natural and adjoint-dual constituents over the embeddings are pairwise distinguished, including across the two sides. The same argument applies directly if the center of \(R\) has two exchanged components, using their pair stabilizer which cannot contain \(K_i\) by the field centralizer; again its full intersection with \(K_i\) must be \(J\).

For center \(F\), \(R\ne\mathcal A\) since otherwise the intersection has soluble image on the fields with soluble kernel, impossible. Thus \(R\) gives an ordered tensor, necessarily with retained nuisance plane side. If \(R\) itself is that plane algebra, it cannot contain two distinct fields of the nontrivial-adjoint kind: on an odd orthogonal plane such a field has the two singular lines as eigenspaces over the closure and is uniquely recovered, and over two the quadratic field in a plane matrix algebra is unique. If its commutant is the nuisance plane instead, \(K_i\) cannot preserve the tensor (large transfer and field centralizer), so the full intersection of \(K_i\) with this ordered tensor stabilizer is \(J\). On the large factor we have the perfect transfer relative to \(D_i\), with again large degree by the missing-site argument (similarity projection on the nuisance side is soluble), supported inside \(J\). Here transfer is on that large factor space for its factor form; in the bilinear characteristic-two setting this is alternating, and the tensor with the nuisance-side identity preserves the canonical total quadratic in orthogonal type. Its centralizer in \(R\) is \(D_i\), but would include \(D_j\), impossible. In this transfer-on-a-factor use, the full simultaneous stabilizer acts on that matrix factor by semilinear similarities respecting \(D_i\), so on its factor space one again has only a soluble remaining similarity quotient over the transferred special. On the nuisance plane factor the whole similarity projection is already soluble. Likewise for the simultaneous paired transfers, after fixing field scalars and the two summands, linear action on one summand and the pairing multiplier determine the action on the other. Thus the soluble-quotient tests indeed use the corresponding supported perfect transfer images inside the simultaneous group, without needing arbitrary individual similitudes there to embed unchanged in the ancestor simple.

Consequently any remaining pair commutes. Two distinct commuting prime fields of the same degree generate a split algebra; for homogeneous adjoint type its primitive component projectors are selfadjoint (relative identifications of the fields commute with their indicated adjoints), contrary to direct-system exclusion. Distinct prime degrees give the field compositum contradiction already proved. This finishes the first prime-field case.

Exchanged pairs and trivial-adjoint fields.

Next exclude exchanged pairs \(e,e^*=1-e\) of idempotents; total singularity when necessary follows as before. Their labels likewise are coatoms by 36. For the algebra generated by two projections from disjoint instances a larger center field would give a prime \(D'\) commuting with them and normalized by the intersection. It is not normalized by the first label (large paired group centralizer). The full intersection of the first pair stabilizer with \(N_M(D')\) thus equals the pair-label intersection \(J\). Again it includes a paired \(D'\)-linear special group of large degree by the same argument. Centralizing the other projection (perfection), it forces commutation of the two projections by distinct natural and adjoint-dual constituents. But distinct commuting exchanged pairs normalized by the intersection give all four cells (zero cells paired oppositely would instead force the same pair), whose sums on adjoint orbits give a forbidden nondegenerate direct split, just as before. A center with two exchanged components likewise would force that impossibility since its pair commutes with both. For center \(F\), their generated algebra is \(M_2(F)\) by the scalar \((e-f)^2\) argument for two idempotents \(e,f\). This must have nuisance plane type by the ordered-factor exclusion. An odd orthogonal plane gives at most one exchanged pair, and the plane over two now violates the nontrivial-adjoint field exclusion. This finishes paired exclusions, hence nuisance planes too by the canonical pair or nontrivial-adjoint quadratic field there. Remaining trivial-adjoint prime fields now have the same two-label proof as above (center \(F\) immediately impossible since all such ordered factors are excluded). ◻

Proof. On compatible sets of directions call the exclusive parts \(A,C\) and common part \(B\), also denoting their aggregate spaces by these letters. Assume \(B\) has sufficiently large degree, and the two direction sets have their full tensor-supported special groups. In orthogonal ambient characteristic two we need only the canonical orthogonal specials there and on the desired union when multislotted (still no symplectic characteristic two ancestor here). Then these groups generate so as to contain the union special; nested cases need no argument, so take both exclusive parts nonempty. Work first on \(A\otimes B_0\otimes C\) where \(B_0\) is a hyperbolic summand of \(B\) of defect at most two, or the entire \(B\) in linear type. In \(AB\) take transfers \(1+N-N^*\) with \(N\) directed between signed hyperbolic \(B\)-coordinates \(s\to r\) from distinct pairs, and arbitrary \(A\)-matrix there. These are in the natural special: the adjoint arrow uses the opposite coordinates in exchanged order, so \(X=N-N^*\) has square zero and \(1+X\) is a determinant-one isometry (in odd orthogonal type unipotent of odd order gives trivial spinor norm). For the canonical quadratic in characteristic two the displacement \(X\) has totally singular image in the two \(A\)-fibers at mutually orthogonal isotropic \(B\)-coordinates, by canonical zero on pure tensors; also \(b(x,Xx)=0\) by symmetry and adjunction for its product polar form \(b\), and displacement rank is even. In \(BC\) take similarly \(t\to s\) with arbitrary \(C\)-matrix \(L\) in the transfer arrow. Use three distinct pairs. Their commutator gives the paired transfer \(t\to r\) with product arrow tensoring the two arbitrary matrices, by elementary matrix multiplication; the higher cross terms vanish. Thus we have all such arrows on \(A\otimes C\) by linear spanning. In linear type simply use single unpaired arrows with the same calculation. In individual basis coordinates now, even short paired arrows between nonopposite (distinct-pair) coordinates within one \(B\)-pair region follow by commuting through a spare pair. These standard short transfers give the split special on \(A\otimes B_0\otimes C\) supported there: ordinary elementary/root generation (in odd symplectic type commutators give also the long roots with factor two, and in hyperbolic unitary type the commutators through an intermediate index to the opposite of the initial one give all trace-zero long parameters). In orthogonal characteristic two they give the split type D generators for the indicated quadratic, which suffices. These are assertions on natural matrices (or their projective images), with identity on the complementary summand inside \(ABC\), and acting by tensor identity in all outside directions.

To see the generator step explicitly, the arbitrary \(A\otimes C\)-arrows between distinct \(B\)-pairs are additive under transfer multiplication, and the commutator used to obtain them has just the composite arrow \(N_1N_2\) and its negative adjoint when the three pairs are distinct. Choose individual hyperbolic bases on the paired \(A\otimes C\) fibers over \(B_0\), totally singular on each coordinate side (also for the indicated quadratic). Rank-one arrows between distinct big pairs give the elementary paired updates there, then any two distinct individual pairs can use an intermediate signed coordinate from a spare big pair. Thus all distinct-pair root groups are present. If a long root is needed, commutation through an intermediate index with final index opposite to the initial one yields on that last arrow the composite minus its adjoint (higher products still zero), i.e. \(2c\) in the standard odd symplectic coordinates or \(c-\bar c\) in normalized hermitian coordinates, with arbitrary composite parameter \(c\). This supplies all long parameters even in even-characteristic unitary type. Ordinary root generation of the split special here thus applies (hyperbolic orthogonal uses just type \(D\) arrows, including at characteristic two). In linear type the unpaired calculation gives all elementary matrices analogously.

If a small anisotropic part remains, take an isometric copy of it inside \(B_0\). Using also the perfect special on \(B\) (embedded in a given supported group), conjugate to the special supported instead with \(B_0\) moved onto the orthogonal complement of that copy, by Witt transitivity with corrections (the large orthogonal complement supplies determinant/spinor data as in the subspace arguments). Here hyperbolic splitting of \(B\) uses its bilateral or hermitian pairing; in bilateral characteristic two it is alternating with no remaining part. A large enough hyperbolic \(B_0\) represents an isometric subspace copy of the small anisotropic part in the other form cases (e.g. embed together with its negative in hyperbolic pairs). The supported embedding on \(B\) used for transport lies in a given special by tensoring with identity (form preservation and high-rank perfection). These two resulting large direct spaces within \(ABC\) are compatible with large nondegenerate intersection and full sum, so 66 finishes the argument. This proves the tensor merging assertion.

In applying this repeatedly in orthogonal characteristic two, use on every multislotted subset the quadratic with the product polar and zero on the fine simple tensors there. It is also zero for pure tensors across any two-part cut there (apply the canonical construction for that cut and compare on a tensor basis), so column groups or their altered versions supply the required specials. Aligned internal fine permutations carry these choices as required; when transporting whole merged sets of at least two columns the form is canonical on column tensors too, so arbitrary column similarities in those transports cause no problem. ◻

Proof. Common tensor refinement.

We now have the same full-spanning property as in 20 for a nontrivial inner subgroup normalized by a previous-good label. Thus the elementary abelian exclusion again works by scalar commutator pairing and inclusion of all retained elementary groups in the common intersection (normal there). Take then the tensor-system labels allowing permutations, with the very same reductions of exceptional small types as there. In the retained-product case we again have minimal normals in the common intersection recovering the slot systems. Otherwise take two disjoint occurrences, with common intersection \(E\), and missing transitive products with rich top action (thus high slot dimensions by the missing-site bound). The supported normal products in \(E\) supplied by 54 contain in their images the same base-core monolith (using the lower endpoint \(g_*\)). Hence the tuple-intersection commutator argument, followed by full spanning, gives a common matrix tensor refinement by intersections assigned to slot pairs. Concretely use \(N_i=\prod_a R_{i,a}\trianglelefteq E\) of singly supported subgroups in the two missing products as supplied above (the projective tensor supports here have one nonabelian simple per slot). In the quotient by the core of \(K_{g_*}\), assign each simple factor of the monolith \(W\) for each \(i\) to some \(\bar R_{i,a}\) containing it, using \([W,\bar R_{i,a}]\) since \(W\le\bar N_i\). Then \(\overline{N_1\cap N_2}\) still contains \(W\) by commutators. Just as before take intersections of the assigned \(R_{i,a_i}\) with \(N_1\cap N_2\), normal there and their images containing the assigned simple by commutators; once again taking their mutual commutators gives its containment in the image of \(R_{1,a_1}\cap R_{2,a_2}\). Thus the tuple intersections generate a nontrivial \(E\)-normalized subgroup and jointly span fully. Distinct tuples have matrix lifts commuting linearly by their distinct slot supports in some system. So their nonscalar lift algebras are commuting central simple tensor factors as before (one per occupied slot pair), recovering each coarse algebra by containment and fullness. Use the same selfadjoint/form observations as before. The fine slots have uniform degree \(h\ge2\) and forms aligned up to scalar by previous-good transitivity; each row or column contains at least two by block primitivity in \(E\) (supplementation supplies transitive top on each system including at least all alternating permutations). Indeed a singleton width would occur throughout that system by transitivity and the other system would then group it into blocks, hence coincide by primitivity and having at least two slots in both, impossible for disjoint supports. In orthogonal characteristic two the fine forms are alternating and the total quadratic is canonical, zero on fine simple tensors; this canonical quadratic is also the one for grouping by any bipartition of the fine slots (uniqueness from values on a tensor basis). Here 69 excludes symplectic characteristic two.

A local alteration of columns.

Use \(u\) the full row-system label intersected with the fine-system stabilizer, a proper decrease (order on any high row slot) above \(E\). Within \(u\) we can interchange two fine slots of a row or cycle three, leaving others in place as slots. Indeed use aligned permutations, commutators if at least three. For a transposition the determinant/spinor arguments given earlier work since there are at least four fine slots, using internal corrections on another slot for odd orthogonal degree (then \(h\) odd, at least three, repeated with odd multiplicity). In characteristic two orthogonal type the tensor swap has even displacement rank by the even \(h\) and at least four slots. Thus one column gets altered by a replacement, remaining in large overlap with its original version (common product degree at least the square root of the column degree).

Generation and the join contradiction.

By 72, adjoining \(u\) to the column label merges a column with its altered version. If the gained fine direction within another column has large degree, merge immediately with that column; otherwise transport to its other slots by aligned within-column alternating permutations (its fine-slot count then at least three) and merge extensions via their large common column. As with direct systems conjugation under the column label then gives merged pairs on a connected graph of columns by primitivity; overlapping edges (and enlarging unions) merge with at least a large column in common, giving the full special projectively, thus the ancestor simple, contradicting 53. Thus these last systems too are excluded. ◻

Proof. First extract a normal simple inside the ancestor socle, using the structural conclusions just proved. In classical type this is the minimal-normal argument of 6: elementary abelian normals have been excluded, while multiple nonabelian simple factors would give a forbidden tensor system through their commuting lift algebras. The remaining actual normal simple spans the whole matrix algebra. Its centralizer in the ancestor almost simple group therefore injects into the soluble outer quotient. In alternating type the primitive, non-affine, non-Cartesian and non-diagonal conclusions give an almost simple group, as in 7.

In either case the tested label is soluble over this simple, also after pulling back the ancestor’s soluble centralizer. The simple lifts inside the actual ancestor socle, where the quotient map is faithful; hence successive simples really are nested subgroups. For an axis descendant the pure-link normal test has lower endpoint \(H\). It puts the soluble centralizer in \(H\) and forces the simple to be missing: if it were retained, the base-core quotient would be soluble, contradicting its nonabelian monolith. Thus the axis label is supplemented by this actual simple.

For a general label above the previous-good axis, the same extraction applies when its own pure link or a contained perfect simple group supplies the nonabelian part. We need no supplementation assertion with the original base in the latter case. Solubility of the quotient over the extracted simple already forces every such contained perfect group into it. This is the conclusion used at the third coatom in the protection arguments.

Finally, at axis descendants the missing-site product test applies to their actual simple supplements inside the original \(S\), and 51 makes their types exceed any fixed threshold. Every link and intersection used here has bounded omissions; the earlier exclusion bounds were enlarged to make those intersections previous-good. ◻

Proof. Use preliminary homogeneity for a bounded number of comparisons: existence of the actual path socles and their types/continuation relative to prefixes, the structure exclusions, and whether a bad protection label for \((D,L)\) exists for a specified ordered sublist support (and of which of the two kinds), suffices. Choose extension sizes \(a<b\) from \(W\) having matching color for continuation from \(W\), with \(b\) very large compared with \((|W|+a)^4\) (use three spaced sizes). If not continuing at \(a\), use that next size with no new run protection. Otherwise take \(W\subset L\subset U\) with these increments, both continuing. For a run ancestor \(D\), any bad prime-coarsening or prime-root label for \(B_L\) (here \(B_L\) denotes the simple at \(M_L\)) occurs among those counted for \(B_U\). Indeed \(B_U\le B_L\) by actual normal simplicity and soluble centralizer as before. Individually invariant tensor slots for \(B_L\) must group atoms for \(B_U\), with just the unique prime congruence coarsening at that prime; and fixing roots still fix \(B_U\). Their stabilizer/normalizer labels for \(B_L\) are proper below \(K_{M_D}\) and contain \(K_{M_L}\). If one such bad case occurs, we can vary the entire list \(L\setminus D\) (even the positions from \(W\setminus D\)) within \(U\setminus D\), retaining by homogeneity the same test relative to \(D\) and ordered omitted support. These altered intermediate labels still have the indicated actual data, and their simple groups contain \(B_U\) by the same soluble-quotient argument (the altered \(M_L\) still contains \(M_U\), even when no longer under \(M_W\)). Indeed we are comparing throughout relative to the fixed \(D\), which still comes first in index order. If the nonempty omitted support of the bad label has size at least two this gives at least \((n/(|W|+a))^2\) different supports (vary \(L\setminus D\) over all subsets of that size in \(U\setminus D\); each specified support of size \(z\) can then occur with probability at most \((|L\setminus D|/n)^z\)), violating the combined atom-prime and root count \(O(1+n\log\log(n+3))\) against the stipulated spacing. If singleton it gives arbitrarily many coatom labels omitting distinct single positions \(x\), relative to \(D\), of the same kind, still from coarsenings or roots for this same \(B_U\). A singleton omitted support leaves no choice of side group on it. In the tensor case there is just one possible such full stabilizer per prime in the count for \(B_U\), and in the root case at most two (height and diagram uniqueness), so we still get arbitrarily many distinct primes. ◻

Proof. In the root case take distinct primes \(r,s\). Their roots fixing \(B_U\) combine to \(\Theta\) with \(\Theta^{rs}=\Phi_i\) and the two given roots as powers as in 25. Now \(Y=S_i(\Theta)=(G_i^\Theta)'\) lies in the intersection. Its presence forces \(B\) high classical in the same characteristic by 67, and the natural dimensions compared both ways force \(B\) to have dimension \(v\) and act naturally up to twist in the ancestor. The root series agrees by the same form checks using the natural \(Y\). In particular if the ancestor here is symplectic in characteristic two, its natural subfield \(Y\) from \(\Theta\) cannot preserve a polar-nondegenerate quadratic even over the closure, excluding orthogonal \(B\) acting naturally there. In natural subfield coordinates its symplectic matrices on the finite cover act transitively on nonzero finite vectors (invariance of a putative quadratic would be exact by perfection); a constant quadratic value on those vectors forces zero polar form (for the binary field use three independent vectors and polar bilinearity). In orthogonal ambient characteristic two the preservation test instead excludes a natural symplectic \(B\) itself the same way. Hence \(B=S_i(\Psi)\) with \(\Phi_i=\Psi^e\), \(e>1\). Applying 2 also to \(\Psi\) fixing \(Y\) gives \(\Psi=\Theta^h,\ eh=rs\). Lifts normalizing \(B\) commute with \(\Psi\) by uniqueness with given outer part. Thus \(K\) already normalizes the simple of one of the two given roots, impossible. ◻

Proof. Again \(B\) contains \(Y\) and is high classical in the same characteristic (\(v=m^d\) is small compared with cross-characteristic/permutation bounds for \(Y\)). Its natural dimension \(w\) is at least \(m\), by the exact minimum-degree bound in 67. Its own tensor atoms on the ancestor module are again one \(\sigma\)-orbit by tensor-system avoidance. They group those of \(Y\), so if nontrivially split, \(K\) already lies below one of the two given coarsening labels, impossible: a prime congruence coarsening here would have to come from the \(d\)-cycle of \(Y\)-atoms, necessarily at \(r\) or \(s\) and coinciding with the corresponding system used above. Thus its lift there has highest weight \(p^x\lambda\) with \(\lambda\ne0\) \(p\)-restricted. At every active node of \(\lambda\) in A the cut size or its complement has size less than \(2d\) (Weyl orbit with binomial lower bound versus \(m^d\)). In signed types there are fewer than \(2d\) nonzero coordinates of the dominant word, and no spin supports. Indeed with rank \(e\) and \(k\) nonzero coordinates the orbit has size at least \(2^{k-1}\binom e k\), and use \(e\ge(m-1)/2\), \(m\gg d^4\): on \(2d\le k\le e\) log-concavity puts the minimum of the bound at one of the endpoints, both too large. Likewise \(\binom{w}{2d}>m^d\) in A. ◻

Proof. Suppose the projective natural module of \(B\) is reducible on the finite group \(Y\) over the closure. In linear type this puts \(Y\) in a proper parabolic. In a form type an invariant subspace either gives such a parabolic, gives a proper nondegenerate splitting, or has a nonsingular radical line in the characteristic-two orthogonal case. We treat the parabolic case first, then the nondegenerate splittings with two positive ranks, and finally the line cases.

A proper parabolic preserves the filtration of every nontrivial rational highest-weight module defined by a noncentral cocharacter (on the covering group if needed). That filtration is nontrivial: the projective kernel cannot contain the cocharacter torus, so distinct cocharacter weights occur. Parabolic containment would therefore contradict irreducibility of \(L_B(p^x\lambda)\) on \(Y\). For an invariant degenerate subspace in a form type, its radical gives singular or isotropic containment; in characteristic-two orthogonal type take the kernel of the quadratic form on the radical, leaving only a nonsingular radical line if that kernel is zero. It remains to exclude the nondegenerate and line cases. We may suppress the overall weight scale \(p^x\) in the following comparisons: both the extremal restrictions and the multiplicities scale by the same factor, by Steinberg’s theorem.

Use the connected part of each indicated stabilizer on points, with covering groups: the projective finite source is perfect; component quotients for nondegenerate splittings here are soluble (independent isometries on complementary summands, each connected up to at most orthogonal component data; one can work on exact form-preserving lifts). Thus a connected covering subgroup above it would also have to act irreducibly. One can take the reduced identity component over the indicated connected stabilizer using the ambient central covering: it surjects onto that stabilizer (finite surjection with finitely many components and connected irreducible target), by a central isogeny, and projective invariance on its action already tests linear subspace invariance. Ordinary nondegenerate splittings use the product of the two connected classical groups on points (up to central quotients/covers, which do not change their Weyl actions), including an orthogonal two-space torus if needed. For a nondegenerate splitting with both ranks positive, maximal tori of the two parts use all ambient coordinates, except possibly one unused in two odd parts. Among restrictions of ambient orbit weights of \(\lambda\), those with zero on any missing coordinate have full maximal Euclidean norm (coordinate projection and highest-weight convexity), yet are not transitive under the two-part Weyl group, since the sparse nonempty support and spare zeros can be differently allocated. For instance one block has room (in torus rank) for the whole support, and then a nonzero coordinate can also be reassigned to the other. This contradicts irreducibility even on the corresponding connected group, whose simple has all maximal-norm weights in one orbit by the same convexity. In these coordinates one takes hyperbolic pairs on each part separately; if both are odd orthogonal (odd characteristic), combine the two leftover directions as one ambient pair over the closure, unused on restriction. The Euclidean norm in the surviving torus coordinates is Weyl-invariant for the product too, and pulling its torus back isogenously does not change this real coordinate calculation.

Line stabilizers.

For a line in odd orthogonal ambient type B (odd characteristic), restriction of tori to D uses the same coordinates, with even-sum root lattice. If \(k\) is the last positive position in \(\lambda\), sparsity gives \(\lambda_k=a\) with \(0<a<p\). Short root lowering gives \(\lambda-\epsilon_k\) as a weight (pairing \(2a\) nonzero in the odd characteristic), already impossible by parity (the difference after scaling is still odd-sum). For a line in type D, including the nonsingular line in characteristic two, use the standard \(B_{e-1}\) stabilizer embedding, dropping the last torus coordinate \(\epsilon_e\). Here is some detail for characteristic two. In a split hyperbolic basis \(z_i,t_i\) take the nonsingular vector \(u=z_e+t_e\). On points the connected special isometry group’s line stabilizer fixes \(u\) and injects on \(u^\perp/\langle u\rangle\) into the symplectic group: kernel elements must fix the whole hyperplane by the quadratic, and then only the nonsingular symmetry (odd displacement rank) could remain besides identity, outside the special group. A connected subgroup \(R\) fixing \(u\) has full image there: take the torus on the first \(e-1\) hyperbolic pairs, their orthogonal root groups, and for each \(i<e\) combine the two end-pair root groups of roots \(\epsilon_i-\epsilon_e,\epsilon_i+\epsilon_e\) with identical parameter \(a_0\) (and similarly in the opposite orientation). Their product sends \(z_e,t_e\) to \(z_e+a_0 z_i,t_e+a_0 z_i\) respectively and \(t_i\) to \(t_i+a_0u+a_0^2 z_i\), fixing the rest. This gives the long symplectic root groups on the quotient by squaring (surjective on points); all other required roots come from the orthogonal ones already used. Take the connected subgroup generated (with Zariski closure). Thus \(R\) has the whole stabilizer’s points and dimension of that symplectic group, and is semisimple (a connected solvable normal would have trivial image). Relative to the indicated maximal torus the generators just given have roots \(\pm\epsilon_i,\ \pm\epsilon_i\pm\epsilon_j\) on \(e-1\) coordinates (nonzero differentials also for the combined roots), exhausting the dimension. The displayed torus is maximal by the finite kernel and rank on the symplectic quotient. So this is type \(B_{e-1}\) with the claimed coordinate restriction. We can use its simply connected cover lifting into the ambient cover (central isogeny lifting as in the path proof). In odd characteristic this is the ordinary odd-dimensional special orthogonal stabilizer on the complement up to components/centers.

For the line comparisons these character coordinates on restrictions can be used on the simply connected covers: they come from the vector weights on the indicated surviving pairs. In particular for the special stabilizer just described the characters \(\epsilon_i,\ i<e\), on its torus pull back as the short-root coordinates of type \(B_{e-1}\), and \(\epsilon_e\) becomes zero even on the cover (its vector pair is fixed by this torus). Lifting from a simply connected covering source through an ambient central covering agrees projectively and hence for these linear vector matrices by absence of scalar characters.

Again let \(k\) be the last positive position and put \(a=\lambda_k\), so \(k\ll e\) and \(0<a<p\). Root lowering with pairing \(a\) gives two distinct ambient weights. On the line stabilizer they have the same restriction, since \(\epsilon_e\) restricts to zero: \[\begin{aligned} p^x(\lambda-\epsilon_k+\epsilon_e) &\longmapsto p^x(\lambda'-\epsilon_k),\\ p^x(\lambda-\epsilon_k-\epsilon_e) &\longmapsto p^x(\lambda'-\epsilon_k). \end{aligned}\] Here \(\lambda'\) is the restricted highest weight; extend a positive-chamber cocharacter by zero in the last coordinate to verify that it is highest. The two ambient weight spaces are independent, so this restricted weight has multiplicity at least two. We show that an irreducible module of highest weight \(p^x\lambda'\) has multiplicity at most one there.

After removing the scale, the relevant top Levi-face module has highest weight \(a\omega_1\) in tail type B, since the difference down by \(\epsilon_k\) uses just those tail nodes. The simple \(L(a\omega_1)\) is a composition factor of the degree-\(a\) symmetric power of the natural B Weyl module, also in characteristic two. The ordinary Weyl character of that natural module has precisely the signed coordinate pairs and one zero, independently of characteristic. Its symmetric power has a unique top weight, giving the stated constituent, and multiplicity exactly one at \((a-1)\epsilon_1\): one zero-weight term and \(a-1\) copies of the first coordinate are necessary. This bounds the multiplicity in the simple constituent by one and proves the contradiction. ◻

Proof. This natural restriction lifts as a restricted irreducible for the parameter of \(Y\), with rational highest weight \(\eta\). It gives a morphism from the simply connected algebraic group of \(Y\) into that of \(B\), as in 27. The invariant bilateral form, when relevant, is rationally invariant by self-duality and uniqueness. For the quadratic in characteristic two, if \(\eta\) has multiple digits, their self-dual factors have alternating forms and hence supply the canonical quadratic, equal to the given one by uniqueness. For a single digit, untwist to a \(p\)-restricted weight using the standard pinned choices; split Frobenius permutes the finite points. The cocycle for the finite-invariant quadratic then takes values in the square module of the dual, a nontrivial \(P_Y\)-restricted simple module because \(P_Y=p^{df}>2\). By 36 its rational cohomology class is zero. Thus the cocycle is a coboundary, and that coboundary is zero because this module has no finite-fixed vectors. This proves rational invariance.

The composed rational module is irreducible by finite irreducibility: on the finite covering points the lift into the ambient simply connected group of \(B\) differs at most centrally from lifts of the original finite projective inclusion (same projective natural matrices by construction). Thus on \(L_B(p^x\lambda)\) one still obtains projectively the embedding action just specified with its \(d\) atoms. Its highest weight, after removing the scale \(p^x\), say \(\mu\), has only single natural (or dual-natural) coefficients one at its nonzero digits. Indeed every rational digit factor restricted after twist to the finite group is one digit there and by individual tensor invariance must be an aggregate of the \(d\) twisted natural atoms, thus exactly one by restricted-weight uniqueness (an aggregate of two or more has natural/dual digits at that many distinct positions of the finite restricted tuple). Removing the scale is integral by torus restriction. ◻

Proof. Here is a cut argument to finish. Restrict natural torus weights of \(B\) to get the multiset \(\Omega\) of the module with highest \(\eta\). As in 38, one fixed Weyl translate of \(\lambda\) maximizes throughout the positive chamber for \(Y\), simultaneously on each active fundamental summand. The restriction of each such translated fundamental weight is itself dominant (its orbit maximum, also against \(Y\)-Weyl translates). Indeed Weyl actions on the image torus extend on it via ambient Weyl elements: after transport of the ambient maximal torus by an image normalizer element, conjugate it back in the centralizer of the image torus. Recall the simultaneous maximum exists since at an interior \(Y\)-cocharacter a maximizing orbit weight of the upper module must actually restrict to its unique highest on composition (weight convexity), hence maximizes on the whole positive chamber. Each active fundamental summand is then maximal there since for each cocharacter there is a common ambient Weyl element maximizing all summands. Since their positive integral sum is \(\mu\), each has support only at endpoints for A-type \(Y\), or the first node for signed \(Y\). Each corresponds to a strict upper cut in \(\Omega\) as in that result: all selected elements dominate all unselected and there are no identical vectors across the boundary, by highest multiplicity one (in signed type all active nodes here are early with spare zero coordinates). The sum of the selected weights is that fundamental restriction. Take the small side of size less than \(2d\); for a co-small A-cut this means the negatives of the unselected values, still with that same sum and strict dominance in the dual multiset instead. More explicitly an exchange across a strict coefficient boundary compares natural weights against each other throughout the \(Y\)-chamber. Equal-vector ties across it would produce two distinct ambient orbit weights mapping to the highest vector weight, contrary to multiplicity one. In signed type one similarly flips nonzero coefficient coordinates in checking the maximum (compensating with zero-coefficient coordinates in D), so the positive directed choices lie nonzero in the positive cone and one can make the same strict comparisons in the full natural multiset.

The middle-coordinate Weyl group in type A for \(Y\) (permuting positions other than first and last), or signed Weyl group away from the first coordinate, fixes this sum, thus preserves the selected multiset by strict domination across the cut. By its small size, every selected weight must individually be fixed (nontrivial orbits cost at least \(m/3\)). (A permuted selection with the same sum still attains the maximal \(Y\)-chamber interior pairing, hence is the same multiset.) This includes the highest \(\eta\) or its dual, thus supported also only at the indicated endpoints or first node as appropriate. Its active simple reflections are then outside the cut, not fixed individually there. Dominating those forces each selected weight’s difference down from highest to be supported only at the active node of any such reflection, which by individual fixedness allows no difference at all: the corresponding simple root (endpoint, or first in signed type) is not fixed by the subgroup either. Thus the cut takes just the highest line. We conclude \(\lambda\) has only endpoint coefficients in A, or just the first coefficient in signed types. ◻

Proof. It cannot have both endpoint coefficients \(a,b>0\) (ambient for \(B\) then A), giving \(\mu=a\eta+b\eta^*\). At the lowest nonzero digit exponent \(y\) of \(\eta\), if \(Y\) is type A with just one active endpoint at that digit, \(\mu\) has both active there, impossible. In fact the two coefficients at that position are nonzero modulo \(p\) by \(0<a,b<p\) and the nonzero lone endpoint digit, with no carry into that position from below. Otherwise choose torus coordinates at the full maximum with first two values \(\eta,\eta-p^y\alpha_1\), last two values \(-\eta^*, -\eta^*+p^y\alpha_1\) (last highest-coordinate minimum first in this listing). These distinct weights occur by digit activity (also for signed type with \(\eta^*=\eta\)), and middle coefficients of \(\lambda\) are constant. Here \(\eta+\eta^*\) is not proportional to \(\alpha_1\) by the support already found in high rank, so the four entries are available distinctly at the maximum, rearranging the middle coordinates. Lowering at each of the two endpoint nodes of \(\lambda\) gives two distinct weights colliding at \(\mu-p^y\alpha_1\), up to the overall scale, impossible by simple-root multiplicity one bounds in the simple composed module.

Thus up to duality \(\lambda=z\omega_1,\ \mu=z\eta\). If \(1<z<p\), a coefficient one at any active node of the lowest \(\eta\)-digit would already contradict the required digit shape of \(\mu\). For coefficient at least two there we have three distinct nonzero nonopposite weights \(\eta,\eta-p^y\alpha,\eta-2p^y\alpha\) for that simple root (again the endpoint/first-node shape in high rank ensures nonproportionality to the root). Assign these to the first three natural coordinates of \(B\) at the maximum (in signed types choose signs using also spare zero-coefficient coordinates). The upper \(A_2\)-face for \(\lambda\) gives the two symmetric-power weights lowering twice (one step to the third coordinate, or twice using the second) colliding again along just \(\alpha\). This forces coefficient one for \(\lambda\).

All the collisions just used are comparisons in the rational composed simple module. With scales displayed, the first pair of distinct upper weights restricts to the same weight \[p^x\mu-p^{x+y}\alpha_1,\] and the symmetric-power pair in the last case restricts to \[p^x\mu-2p^{x+y}\alpha.\] Each lies below the highest weight in a single simple-root direction, where the divided-power PBW bound gives multiplicity at most one, as in 38. Hence the only remaining coefficient is one. ◻

Proof. By 79, \(w=v\) and \(B\) acts naturally up to twist. Its series matches the ancestor by the previous tests: in type A the finite \(Y\) digit tuple is non-self-dual, and the symplectic characteristic-two ancestor case has been removed. The third coatom is proper and contains the supplementing base, so \(B<S_i\). By 2, a proper root of smaller height fixes \(B\), hence \(Y\).

Fixedness identifies the projective \(Y\)-action with its coefficient twist through a height \(0<g<f\), possibly with duality, using the matrix realization of the ambient point maps. These perfect lifts agree linearly. But the support of the finite restricted tuple is \[\{0,f,\ldots,(d-1)f\}\subseteq\mathbf Z/(df)\mathbf Z.\] A shift through \(0<g<f\) moves every one of these positions to a nonmultiple of \(f\). A diagram action on wrapping may exchange a natural weight with its dual, but cannot change which digit positions are nonzero. Thus no such smaller-height root fixes \(Y\), giving the required contradiction. ◻

Proof. Fix the requested number \(m\) of point labels. Choose the high-type thresholds first, and let \(D_0\) be the resulting absolute bound in 9. Reserve more than \(D_0\) path nodes. Choose the trial omission sizes successively, with the spacing required by 73; each is a finite function of the bounds already chosen. This fixes a bound on every path-prefix size. The earlier exclusion rounds can now have their further-omission bounds enlarged backwards to cover every bounded union used by a later round. All such comparisons are relative to one fixed ancestor, as in 57.

Finally choose the initial marker reservoir large enough for those finite Ramsey requirements, for \(m\) point labels whenever 30 succeeds, and for all residual pure links and missing-site tests. A terminal spare reservoir, and spaced prefixes when needed, provide the fresh indices throughout. Only the bounded case and comparison predicates already specified are colored; these choices do not depend on group dimensions or field sizes.

Start with the actual simple \(S\) in the ordinary site interval. At each subsequent bounded prefix the construction either supplies the required \(m\) point stabilizers, or continues with an actual nested simple supplement by 40. In the second case the socles satisfy the transitivity and full-spanning properties relative to every ancestor, and 51 excludes all bounded types. A continuing classical run receives both protections by [fb:protection-count,fb:singleton-roots,fb:singleton-tensors]; an alternating or otherwise noncontinuing step needs no new run protection. Thus absence of point success at the bounded prefixes would give more than \(D_0\) actual simple nodes satisfying all the group-theoretic hypotheses of 9, a contradiction. The construction must stop with point success at one of those prefixes. Its prefix bound and reservoir threshold depend only on \(m\) and the fixed choices, proving the asserted uniformity. ◻

Proof. In any overgroup the conjugates of the supported \(S_Z\) generate the full symmetric group supported on the orbit of \(Z\), by overlap (its size before expansion already exceeds half the ambient degree). That orbit absorbs entire units. Absorption in joins thus follows orbits; in particular the join of the \(x_j\)’s below will be top. We shall use the following labels: \[\begin{align*} X_D&=(D,Y_{T\setminus D}), & x_j&=X_{\{j\}}, & M_D&=X_{T\setminus D},\\ C_D&=(T\setminus D,S_D)\quad(D\ne\emptyset),& N&=C_T, & J^*&=(\emptyset,\prod_{j\in T}S_{\{j\}}). \end{align*}\] The \(C_D\)’s are precisely the coatoms (here and below test by absorption, or by the unequal pieces of each cut with the large foundation). The filter above \(X_D\) and the ideal below \(M_D\) have the templates on the remaining units (in the filter the foundation has grown); inside \(M_D\) the independent unchanged \(Y_D\) factors out. ◻

Proof. The footprint \(I\) is unambiguous in the image. In fact a collapse with changed footprint gives a comparable one (use meets), and for \(j\) gained it then identifies \(N\vee x_j=1\) with \(N\), by meeting first with \(x_j\). Indeed \(x_j\) is under the upper label, while its intersection with the lower is only residual (empty footprint). Thus the identification kills every \(x_R\) for full units \(R\) by intersection with \(N\). For a grid unit \(O\), in each direction take a label preserving a partition of \(O\cup R\) into the parallel lines of that direction plus a reference block \(R\) of the matching size. Use full independent bases, alternating top on the line blocks including \(R\), and unchanged \(Y\) elsewhere, without absorption. This contains the bottom: on parallel lines the grid groups induce even permutations (for the auxiliary grids both other lengths are even). It joins with \(x_R\) to absorb \(O\). Intersecting all these labels with \(x_O\) gives bottom: preserving each parallelism forces slotwise permutations and on the primary squares the two alternating conditions enforce both prescribed slot groups. Thus all \(x_O\) are killed as well, impossible. In particular the cut coatoms survive, distinctly with no new ones, and orbit discrepancies of residual labels can be detected by joining with absorptions. Here adjoining \(x_j\) to a label for an unabsorbed \(j\) absorbs exactly its residual orbit in addition to the old footprint. On just markers, the resulting pure lattice, in particular in the filter absorbing everything else, is exact: for a strict change of residual group (on singleton positions) intersect down to a cyclic group generated by a discrepancy element, holding the footprint fixed. A proper subgroup there has a smaller orbit on some cycle (an index divisor hits the cycle length), detected by adjoining the absorption of one singleton. The same test works inside a permutation label acting canonically on copies of a package of units modulo the prescribed internal bottom actions. ◻

Proof. Let \(f\) be a nilpotent join-preserving deflation. It kills every atom: otherwise it fixes that atom, contrary to nilpotence. In particular, \(f(x_R)=0\) for every full unit. If \(A\) is any of the parallel-line labels used in 81, then \(x_O\le A\vee x_R\), so \[f(x_O)\le f(A\vee x_R)=f(A)\le A.\] Also \(f(x_O)\le x_O\). Intersecting over the directions therefore gives \(f(x_O)=0\). Since all the \(x_j\)’s join to the top, \(f=0\). The idempotent-restriction and unary-nonunit argument of 4 applies starting with a least carrier representation among all nontrivial images of \(F\). Idempotent restriction to more than one point would still represent such an image since it gives a surjective lattice map; nonunits thus yield the nilpotent join deflation as there. Consequently this minimal representation gives an ordinary finite group interval for the dual of a nontrivial \(\bar F\), by the diagonal construction in [f:eq-diagonal-interval]. Its top order is at most \(60^m m!\) if the representing size was \(m\), taking the simple group \(A_5\). Also \(F\) itself is representable (the transitive coset action on its ordinary interval). ◻

Proof. Write \(e\) for (the label of) a base times a normal, always reasoning with labels in the forward \(\bar F\) order. Dedekind’s law makes it left-modular: in particular \(e\wedge b\le a<b\le e\vee a\) is impossible in either orientation. Projections to axis filters (by joining in the label order) again come from normals: in actual order enlarging the bottom to \(K\) projects \(HL_1\) to \(KL_1\), and shrinking the top to \(K\) projects it to \(HL_1\cap K=H(L_1\cap K)\), for \(L_1\trianglelefteq G\). Axis intervals used after deleting a bounded number of units are still nontrivial images on the corresponding templates, by interval surjectivity and footprint detection. For the normal tests below we suppose that \(e\) is a proper nonendpoint.

Here is also the intersection test used repeatedly below in the normal argument. Write \(G_b\) for the actual ordinary overgroup at label \(b\), and \(G_e=HL\) for the actual normal multiple, \(L\trianglelefteq G\) (similarly after any indicated base-core passage). If \(H<G_b<HL\), then \(I_b=L\cap G_b\) is nontrivial proper in \(L\), by \(G_b=H I_b\). It is normalized at every \(s\) for which \(\langle G_b,G_s\rangle\cap HL=G_b\), since then it is just \(L\cap\langle G_b,G_s\rangle\). The sufficient equation in parameter order is thus \(e\wedge (b\vee s)=b\) forward, \(e\vee(b\wedge s)=b\) reverse. If \(L\) was taken minimal normal nontrivial and the indicated \(G_s\)’s generate \(G\) (label join forward, label meet reverse), this normalization is impossible. ◻

Proof. Absorption and the offset comparisons. Take four disjoint equal-size unions of units in a cyclic order. Absorb the outside and allow independent full symmetric groups on the four blocks. Adjoining the square of their rotation gives a label \(a\), and adjoining the whole rotation gives \(b>a\), with strictness detected by orbits. Suppose the four counts of absorbed letters from \(I(e)\) are nonconstant, with a positive count in each parity class of positions. An odd rotation cannot preserve these counts, so \(e\wedge b\le a\). But the square rotation connects each parity pair, both of which meet the absorbed part of \(e\); hence \(e\vee a=1\). This is forbidden.

In any cohort, therefore, either every unit is absorbed or at most one is: otherwise put two absorbed copies in opposite parity positions and a nonabsorbed copy in one of the remaining positions. Each sufficiently numerous cohort consequently has a majority absorption bit. Compare a typical size-\(s\) unit together with a typical marker against a typical size-\(s+1\) offset unit. Write their majority bits as \(\epsilon,\mu,\eta\), respectively. The four-block test, using both block types and repeating a positive one in both parity classes if their counts differ, forces \[s\epsilon+\mu=(s+1)\eta.\] For bits in \(\{0,1\}\), this implies \(\epsilon=\mu=\eta\). Every required cohort and its offset therefore have the marker cohort’s bit. If that bit were one, the preceding cohort dichotomy would make every unit absorbed, contrary to \(e<1\). All majority bits are zero.

Finally, two absorbed units from different cohorts are impossible. Use four blocks each containing one copy from each of those cohorts, and put the two absorbed copies in different parity classes, choosing the other copies nonabsorbed. The same test applies. Thus \(|I(e)|\le1\). These arguments use only sizes, offsets and boundedly many copies, so they also hold with full unit actions and after the indicated bounded axis deletions.

Excluding a singleton footprint. If \(I=\{O\}\) and the residual group moves between other units, shift above \(x_j\) at the source of such a move. At least two absorptions remain (\(O\) and another unit hit from \(j\)), so the shift is top by the count just proved (the same count in a filter deleting a unit), meaning the residual group is transitive on the complement. On four copies including \(O\) the square test then gives total absorption on join anyway (we reach another unit besides \(O\), then its whole transitive complement orbit), impossible. If instead the residual group preserves units separately, use three copies including \(O\) with rotation group \(A_3<S_3\) on the blocks, outside absorbed. This increase is strict (meet with the version allowing just one transposition on blocks, then detect orbits); intersection with \(e\) cannot move the blocks while join absorbs them. So \(I=\emptyset\).

Excluding residual mixing. By shifting once at a mixed unit and applying the same conclusions, mixing units now forces transitivity on all the residual letters (there is an extra absorption in that shifted label, hence it is the filter’s top). These nested temporary shifts delete only boundedly many units before doing the counts.

For such a transitive \(e\), every marker \(x_i\) complements it. Normal complementation gives isomorphisms in actual subgroup order (when \(K\) complements \(HL\), \(L\trianglelefteq G\), project \(K\) to \(G/L\) to identify \([H,K]\) with \([HL,G]\)). Forward this forces \(e=N\) by marker atomicity (it makes \(e\) a residual coatom). Replace the base by the label \(J^*\), i.e. switch to all full unit groups, with the same transitive projected label. In the new base-core quotient take a minimal normal inside the normal giving that label; it is still missing and proper, so still transitive by what was proved thus far (nonmixing would now mean bottom), hence still maps to \(N\). For a residual marker transposition \(d\) over the new base, we have \(0<d<N,\ N\wedge(d\vee x_j)=d\) for every unit, using notation relative to the full-unit base. Thus the intersection of the minimal normal with the subgroup labeled by \(d\) is nontrivial proper and normalized by all labels \(x_j\) which generate the top, impossible. Indeed inside an actual base times that normal, all overgroups of the base factor as the base times their own intersections with the normal; and here enlarging \(d\) by any \(x_j\) leaves its intersection unchanged.

In reverse orientation the complementation identifies \([x_i,1]\) with \([0,e]\) by meeting with \(e\) in \(\bar F\). Thus \(X_{\{i,k\}}\wedge e>0\) on distinct markers, forcing every marker transposition label beneath \(e\) (the literal residual action in that intersection only has the two singleton markers available over bottom). A supported marker three-cycle group including \(i\), which lies below \(e\) by these transpositions, then cannot be in the indicated meet image: joining it with \(x_i\) absorbs all three and they admit transpositions below \(e\). These small permutation labels over bottom are exact by orbit detection. So in either orientation every proper nonendpoint normal label lies under \(J^*\). ◻

Proof. Pass now to the original base-core quotient and, if a proper nonendpoint occurs, take the normal to be minimal nontrivial therein (within the purported normal), written \(L\). This still gives a proper nonendpoint internal \(e\) (actual base times \(L\)); internal representatives can be used since \(e\le J^*\).

For a parallel direction on \(O\), write \(A<W\) for the alternating-top parallel label used earlier and its full symmetric-top version, now allowing full internal groups on other units. This increase survives: adjoin a transposition \(d\) on two spare markers to both, then intersect with a graph label coupling this swap to odd line-block action, giving different marker orbits, detectable by absorption. Suppose \(e\not\le W\). Some element of an internal representative then disrupts the partition; conjugating supported full bases and using primitive alternating top generates independent full action there (a cross-transposition in addition to the within-block symmetric groups suffices), so \(e\vee A\ge W\). Thus \(D=e\wedge W\not\le A\) by left-modularity. Forward, \(0<D<e\) in this situation, and \[e\wedge(x_k\vee D)=D\qquad(k\ne O),\] since \(x_k\), like the literal internal intersection giving \(D\), preserves \(O\) and its indicated lines; on meeting back with the internal representative one thus stays under \(W\). Hence the nontrivial proper intersection of \(L\) at \(D\) is normalized both at \(W\) (where the intersection is the same) and all these \(x_k\) (use \(x_k\vee D\)); together they absorb everything, impossible by minimal normality.

In reverse orientation in fact no internal element of a representative for \(e\) can preserve the direction’s lines on \(O\) and act oddly on them. Otherwise set \(K=e\vee d\) with \(d\) just a spare marker swap over bottom. This is strictly above \(e\) (orbit test) but residual. Take the graph label \(P'\) coupling this swap to odd line-top on \(O\), otherwise allowing internal actions preserving units and this parallelism. We have \(d\wedge P'=0\), and \(e\vee(K\wedge P')=K\) using the presumed odd element times the swap. Thus the intersection of \(L\) at \(K\), nontrivial proper by \(e<K<1\), is normalized at \(d\) (by \(K=e\vee d\)) and at \(P'\) (enlarging actually, i.e. using \(K\wedge P'\), leaves the intersection unchanged). These generate actual top by \(d\wedge P'=0\). This too is impossible; it rules out in particular \(D\not\le A\) obtained above under a disruption (an odd top element in the internal intersection with \(W\) fixes the reference block, hence acts oddly on the lines of \(O\)).

It follows in both cases that \(e\le B\), the intersection of \(J^*\) with all the \(W\)’s. This is slotwise full on the grids and differs from bottom only by primary square sign bits. Reverse has now been excluded (meet to place the representative under literal \(B\), then use the odd-line test).

In the forward case \(0<e\le B\), so the image of the literal sign-space interval is nontrivial. The subspace lattice is simple: a nontrivial congruence collapses an atom by atomisticity, and every two distinct atoms lie in a plane, where such a collapse propagates to all atoms. Hence this interval is exact. ◻

Proof. By [t:weighted-normal-internal,t:weighted-normal-parallel], only the forward sign-space case remains. Take a minimal missing normal inside any purported proper normal multiple after passing to the base-core quotient.

Forward, \([0,B]\) must now be an exact subspace lattice over \(\mathbf F_2\), since that literal interval is simple and its image is nontrivial (\(0<e\le B\)). Here the slotwise groups over literal bottom have quotient given by the sign space with two bits per primary square. Subspace lattices here are simple, e.g. by atomisticity and propagation of any atom collapse across the planes. For an active bit of the sign space of \(e\), meeting with its parallel alternating-top test \(A\) gives its kernel \(D'\). If this is nonzero, the same formula \(e\wedge(x_k\vee D')=D'\), \(k\ne O\), and normalization at \(A\) give the earlier forward contradiction. Thus \(e\) has sign dimension one with vector \(v\ne0\). Swapping the two coordinates of any primary square by \(\tau\) must fix \(v\): otherwise use as \(a\) the label adjoining \(\tau\) to bottom and as \(b\) this label extended by the sign line \(\langle v+\tau v\rangle\); \(e\wedge b\le a<b\le e\vee a\), strictness tested under \(B\). Hence at an active square both bits occur together.

Choose an active square and use its \(x\) lines in one direction, together with its size-\(x\) auxiliary unit as one additional block. Let \(a\) have independent alternating groups inside these blocks and the alternating group on their \(x+1\) positions. Match coordinates along the lines and choose any matching with the auxiliary block. Outside this union allow full internal groups, without absorption. The prescribed bottom lies in \(a\). Let \(b\) instead allow all base sign vectors of even total, with the same alternating top. Then \(a<b\): the base sign which is odd on every original line and even on the extra block lies in \(b\), and its nontriviality is detected by meeting with the exact sign interval \([0,B]\).

Write \(\alpha\) for the line-position sign and \(\beta\) for the repeated within-line sign of the chosen square. The preceding swap test gives \(\alpha(v)=\beta(v)=1\). An element of \(e\wedge b\) has \(\alpha=0\), since \(b\) has alternating top. As \(e\) is the single sign line \(\langle v\rangle\), its sign component must therefore vanish, including \(\beta\). The auxiliary bottom action is also even on its letters: its first-slot permutations repeat four times, and each binary-slot swap repeats an even number of times. Consequently \(e\wedge b\le a\).

On the other hand, a representative of the nonzero sign of \(e\) has base-sign vector \[u=(1,\ldots,1,0)\] on these \(x+1\) blocks, and may also permute the original lines. Conjugate the supported matched alternating top by this representative, then remove the top permutations using \(a\). Modulo the alternating bases this produces \(u-hu\) for every even block permutation \(h\). These differences span the augmentation space: moving the unique zero gives every pair of coordinate positions. The conjugations remain supported on this union regardless of the representative’s actions on other units. Hence \(e\vee a\ge b\), contradicting the normal-multiple test. This proves the normal endpoint property.

The lattice is nonmodular (the exact pure marker link includes the subgroup lattice of \(S_4\), already nonmodular: with a 4-cycle \(r\) and a dihedral reflection \(s\) inverting it, \(\langle r s\rangle\) has trivial meet with \(\langle s,r^2\rangle\) but joins with \(\langle s\rangle\) above it). For completeness, in the faithful base-coset action every minimal normal is transitive by the endpoint property. An abelian minimal normal would identify the interval with an invariant-subgroup interval in an abelian group, which is modular. Two distinct minimal normals commute; supplementation by one then makes the base induce all inner actions on the other. Intersections in the latter are invariant normal subproducts, again giving a modular interval. Thus there is a unique nonabelian monolith modulo the base core, and it is supplemented. The base is transitive on its simple factors, because the full group is and the monolith acts internally on each factor.

The stock bounds thus far only spent boundedly many copies for parallel tests, rotations, and nested shifts deleting at most a bounded number of units. ◻

Proof. Forward, suppose first \(P<c(P)<\mathcal W\), and put \(K_p=c(P)\). Joining a label with \(p\) to top forces subdirectness, which also suffices since \(K_p\) contains nontrivial independent support. Thus a test \(p\wedge b\le a<b\), \(p\vee a=1\), is impossible by strictness on intersections (both \(K_a,K_b\) then subdirect, intersections containing \(P>1\)). Exactly the same four-block rotation test gives \(|I(p)|\le1\). In the singleton case \(I=\{O\}\), if any other copy of \(O\) is moved in the residual action to touch a different unit, place it opposite in parity from \(O\) in the four-copy test, with destination outside the chosen four. The join again absorbs everything. If none are thus moved use the three-copy test instead. Hence \(I=\emptyset\).

Shift above \(x_j\); if its closure is not full the closure label has empty extra footprint there by the same tests (or trivially when \(x_j\) is already product). So if \(p\) mixes \(j\) with other units then \(x_j\) must project fully by monotonicity of closure, and \(x_j\vee p=1\), forcing transitivity of the residual group. In that case all marker \(x_i\)’s are subdirect and their intersections with \(p\) are bottom. Meeting also with \(X_{\{i,k\}}>x_i\) forces all marker transposition labels beneath \(p\) by strictness. The \(K_{x_i}\)’s have identical ties and identical twists restricted to the projections of \(P\), by the common intersection \(P\) with \(K_p\). Therefore the images of all the filters above \(x_i\) under meeting with \(p\) must agree (ties may be split by any invariant refinement, retaining the twists, then taking the product of the correspondingly restricted diagonals on intersection with \(K_p\)). Yet a supported marker three-cycle group \(d'\) lies beneath \(p\) by the transpositions, and occurs in the meet image for \(i\) outside its three positions, as \(p\wedge(x_i\vee d')\) (\(p\) is residual, \(i\) is singleton). It cannot occur for \(i\) inside by the same absorption contradiction as before. Thus \(p\le J^*\). Now every cut coatom is product, since subdirectness there would join to top with \(p\). The same cut conclusion holds immediately if \(P=c(P)\ne1\).

To elaborate on the common meet-image test in that argument, put \(\mathcal T=c(P)=\prod T_o\) with the index here ranging over sites. A subdirect \(K_{x_i}\) meeting \(\mathcal T\) in \(P\) restricts on each tie part to the full twisted diagonal of the corresponding \(T_o\)’s in that intersection: the projection there is exactly \(T_o\) since it comes from \(P\). These are independent across parts. Thus whether two sites are tied, and if so the restriction of their identifying isomorphism to \(T_o\), can be read from the two-site projection of \(P\) (graph of an isomorphism versus a whole product; \(T_o\ne1\)). Above \(K_{x_i}\) one may split by every operator-invariant refinement, preserving its twists within the remaining ties; this is again invariant by equivariance of the original ties, and these are all the invariant overgroups in the power here. Their intersections with \(\mathcal T\) depend only on that same restricted-twist data and the refinement. This proves the claimed agreement even if the full twists were not specified by \(P\). In marker notation with \(i\) outside the triple one can bound \(p\wedge(x_i\vee d')\) also by \(N\wedge(x_i\vee d')=d'\) calculated literally, and it contains \(d'\) in the image. ◻

Proof. Take disjoint identical nonempty packages of units and a residual label \(z=Y\rtimes S_k\) permuting \(k\) copies canonically (fix coordinated matchings of corresponding units respecting their types and actions). Foundation actions are implicit in residual notation; by a canonical subgroup label here we mean the inverse image in \(z\) of a subgroup of \(S_k\). The interval \([0,z]\) in label order is exact \({\rm Sub}(S_k)\), by the cyclic discrepancy and orbit test on these copies. We claim \(K_z\) subdirect for sufficiently large absolute \(k\), here when \(P>1\). Otherwise its closure corresponds to a nonidentity subgroup \(U\le S_k\). Labels indexed by subgroups of \(U\) generated by prime order elements are product, since the corresponding atoms (actual coatoms above the nontrivial product \(K_U\)) must be so. Hence \(U\) cannot contain a large supported \(A_v\): that subgroup is prime-generated and would give a product base \(K_{A_v}\ne1\) for an exact reverse \({\rm Sub}(A_v)\) inside this invariant interval, contrary to the reverse alternating exclusion of 10.

If \(a<c_0\) in \({\rm Sub}(S_k)\) and \(c_0\cap U=1\), their labels are subdirect (by monotonicity, the label of each closure is below \(U\) and below \(c_0\)), so \(\langle a,U\rangle<\langle c_0,U\rangle\) by intersection strictness with \(K_U\). Thus any transposition moved by conjugation by \(u\in U\), together with its conjugate, generates a group meeting \(U\) nontrivially. We get some prime-order \(h\in U\) supported on at most four letters. Match a nontrivial orbit of \(h\) to a fresh ordered set of the same size (2 or 3) and exchange them simultaneously pairwise by an involution. By the very same strict-join test its group together with its \(h\)-conjugate (dihedral, two such exchanges) meets \(U\) nontrivially. Any nonidentity element there rotates the two sets nontrivially or exchanges them in matched fashion. By finite Ramsey homogenize a large reservoir of fresh points so a constant such pattern on increasing tuples lies in \(U\). Comparing two tuples differing at just the last entry \(y,y'\) gives a supported three-cycle by taking quotients: the two permutations differ by conjugation by \((y\,y')\) and the first (say \(w\)) fixes \(y'\) and sends \(y\) to a common anchor (possibly of the original orbit) distinct from \(y\). Thus one quotient is \((w(y)\,y')(y\,y')\). This anchor is fixed as a choice of position when the prefix is fixed and the last entry varies. Varying these cycles gives a large supported alternating group, contradiction. All these Ramsey requirements are absolute (orbit lengths two or three, bounded many permutation patterns, with sufficiently many points for the absolute reverse alternating exclusion). This proves the symmetric-copy claim. Its alternating-copy subgroup also has a subdirect actual group: in reverse order that group contains the subdirect group just obtained. ◻

Proof. Suppose now \(K_p=c(P)<\mathcal W\). We have \(I(p)=\emptyset\): for any absorbed \(O\) use such a canonical permutation label on many copies including it; its alternating and symmetric versions are distinct subdirect labels, yet have the same joins with \(p\) (all copies are absorbed), impossible.

Call a label small here when supported on a bounded number of units (literally below their \(X\)-label); we need only fixed bounds, say up to ten. We show that every such label has a subdirect actual group. Otherwise an atom \(e_i\) (actual coatom) beneath its nonzero closure label in parameter order must be product, and \(e_i\le p\): the actual coatom contains the nontrivial product closure, so cannot itself be subdirect, hence is closed by maximality. It is residual and lifts to a literal label supported on the package in question at empty footprint. Take there a literal cover \(B_0<C_0\) of groups giving the images bottom and \(e_i\) (groups written on just the package). Choose many disjoint clones and write \(e_j\) for the analogous images of \(C_0\); they too are distinct atoms and the lowers collapse. Indeed image congruences within residual package intervals transfer both ways by canonical swaps: join with the swap label over \(Y\), then meet with the full residual symmetric label on the other package (only \(Y\) elsewhere). Literally this copies a supported group from one package to the other, by independent support and conjugation. Thus both collapses and noncollapses there transfer, and each supported interval maps onto its image interval. Distinctness of \(e_j\)’s follows also by their bottom meets.

Transporting product atoms. Every \(e_j\) must likewise be product. Indeed a coatom if not product is subdirect. If subdirect here for \(j\ne i\), it contains (in actual order) both intersections of \(K_{e_i}\) with the actual groups at the canonical labels of \(\langle(i\,j)\rangle\) and \(\langle(i\,j\,l)\rangle\) on copies. This is because on the parameter side, joining \(e_i\) with either permutation label contains \(e_j\) by conjugating the literal \(C_0\). Thus its ties refine both sets of permutation-label ties, since the indicated intersections with the product \(K_{e_i}\) project nontrivially and independently across each set of parts. But their common refinement is discrete: the two canonical labels are subdirect with a common subdirect lower group \(K_z\), hence consistent twists, and generate actual top (the canonical parameter subgroups meet trivially); any sites tied in both would keep that common twist on joining. This contradicts proper subdirectness of \(K_{e_j}\). Being product and above \(P\) as an actual subgroup, each is above \(c(P)\), i.e. \(e_j\le p\).

The faithful product interval. Use \(k\) large such packages. Between the products of the literal covers the interval is Boolean if \(B_0\) is not normal in \(C_0\) (any full projection conjugates the supported \(B_0\) to generate \(C_0\) there); otherwise a subspace lattice over \(\mathbf F_\ell\) for the prime order of \(C_0/B_0\). This interval maps faithfully to \([0,\bigvee e_j]\) by survival or simplicity, and all entries are product (the join of the \(e_j\)’s lies under \(p\), hence all atoms under it too must be coatom products in actual order; now use atomisticity). Take \(k\) coprime to \(\ell\) in the latter case (one of two consecutive large counts suffices). Let \(E\) be the augmentation hyperplane in that case, the whole join in the Boolean case. It is nonzero and no atom beneath it is fixed by the canonical alternating copy action (label \(a\) over \(Y\)), using the faithful literal product interval. For the hyperplane an invariant line would have a fixed nonzero vector by perfection of \(A_k\), hence constant on the copies, impossible by \(\ell\nmid k\).

The extension image. For the subdirect \(K_a\), write \(D_a\) for its extension domain with map \(\beta\) to \({\rm Aut}(S)\). By restrictions \([D_0,D_a]\) realizes exactly \({\rm Sub}(A_k)\), so the kernel of outer action must be supplemented by \(D_0\) (the normal test in 10; if retained, the base-core quotient would be soluble by Schreier, impossible there). Thus \(\beta(D_a)=(\beta(D_a)\cap S)\Lambda\), identifying \(S\) with inners. Also \[(a\vee E)\wedge p=E\] by factoring the literal join as a set product (the lift of \(E\) is normalized by the copy action) and using a representative of \(p\) above that lift, since \(a\wedge p=0\). Here \(a\wedge p=0\) comes from joining a subdirect with a nontrivial product in actual order, and literally the set product intersected with that representative of \(p\) is the lift of \(E\) times the intersection with the lift of \(a\). Hence the product closure of \(K_a\cap K_E\) is \(K_E\) (the equality gives generation with \(c(P)\), already under the closure by \(P\le K_a\cap K_E\)). Writing \(T_e={\rm proj}(K_e)\) at the site, this means \(\beta(D_a)\) normalizes \(T_E\). Indeed intersection with a product has full projection onto it precisely when its factors are matched by the diagonal twists; by transport along the tie through the site this is precisely invariance of the site’s factor under the extension image.

Here \(T_E\) is nontrivial and proper, by \(P>1\), site transitivity and the proper product \(K_E<\mathcal W\). Its normalizer in \(S\) is proper by simplicity, and is invariant under \(\Lambda\). Transporting it gives a product label \(0<f\le E\). Choose an atom \(h\le f\), and put \(B_a=\beta(D_a)\cap S\). The normalization just proved and the reversal of label order give \[B_a\le N_S(T_E)=T_f\le T_h.\] Also \(\Lambda\) normalizes \(T_h\), as it does every invariant product projection. Thus \(\beta(D_a)=B_a\Lambda\) normalizes \(T_h\). The same intersection criterion now gives \[c(K_a\cap K_h)=K_h.\] The alternating copy action moves the literal lift of \(h\) to a different atom \(h'\) under \(E\), so \(h'\le a\vee h\) in label order. Its actual group is a product and contains \(K_a\cap K_h\); it must therefore contain the displayed product closure \(K_h\). This contradicts the distinct actual coatoms \(K_h,K_{h'}\), proving the small-label claim.

In this extension-image argument, \([D_0,D_a]\) really is the whole ordinary interval: all actual overgroups of the subdirect in the invariant interval are still subdirect and correspond exactly to restrictions of its extension map. For the normalization criterion at the tie through a site \(o\), at \(t o\), \(t\in D_a\), the product factor of \(K_E\) is the transport by \(t\) of \(T_E\), whereas the entry matched by \(K_a\) to \(u\) at \(o\) is the transport of \(\beta(t)^{-1}(u)\). Thus all \(u\in T_E\) occur in the intersection exactly for stability under \(\beta(D_a)\), and similarly for the product at \(h\); other tie parts transport. Finally the “different atom” test only conjugates a literal lift within the displayed Boolean/subspace interval, where alternating copy permutations preserve the lift of \(E\) and move the given atom’s lift to a different atom there. The distinctness survives by faithfulness on this interval, and the conjugate is contained in the literal join with the copy label, so the conclusion in the image does not assume invariance of the entire quotient map under copy permutations. ◻

Proof. For small residual \(a<b\) we must have \(a\vee p<b\vee p\) by intersection strictness (actual intersections nontrivial since joins are residual). A representative for \(p\) cannot move between units. Otherwise for an element sending a letter of \(j\) into \(k\ne j\), choose a second letter outside \(k\) mapping outside \(k\) (available by sizes with copies). Use a supported full symmetric label on the one or two source units. Conjugation of the source-letter transposition in the join gives a supported transposition connecting \(k\) with somewhere else, adjoining which to the source label is a small strict increase by orbit detection at \(k\), impossible.

Preserving parallelisms. On a square both parallelisms, and on an auxiliary grid both length-two parallelisms, must be preserved. Otherwise use the parallel partition plus matching reference block with full symmetric wreath action and only \(Y\) elsewhere as \(a\), and the independent full symmetric action on that union as \(b\). Again disruption generates \(b\) in the join with \(p\); strictness is detected by meeting with the other indicated parallel label of equal length using the same reference block, then testing whether the reference unit connects to the grid (it is the only common block of the two partitions).

Uniform auxiliary flips. Thus on an auxiliary unit \(h_0\times2\times2\) a representative acts by component permutations on \(h_0\), and separate bit flips possibly per component. They must actually be uniform. If for instance the first flip varies, take \(a\) acting on the (component, second bit) positions by a full symmetric group and separately by constant first flip, with only \(Y\) elsewhere. Conjugation of its supported top permutations by the varying element produces (after removing top action) first-flip differences generating all even-sum flips on those positions: differences of the nonconstant flip vector with its translates include weight two and are supported within this unit regardless of actions on other units. Adjoining those to \(a\) strictly increases it. Indeed meet with simultaneous preservation of the two parallelisms: the first flips formerly constant induce with component permutations only even actions on the (component, first bit) lines (second-bit direction, \(h_0\) even), but flipping just one component’s first bit now occurs and acts oddly. As before distinguish by adjoining a spare marker swap then intersecting a parity graph label, which contains bottom. This violates strict joins again.

The remaining primary signs. We conclude \(p\le B\) of the earlier sign-space test, with exact \([0,B]\) since \(p>0\). On any primary slot with active sign take as \(a\) the alternating bases on those lines plus the auxiliary size-\(x\) block, with symmetric top, outside only \(Y\). In the join with \(p\), conjugation again adds independent augmentation base signs by discrepancy with the even auxiliary action, producing the strict increase \(b\) tested under \(B\) as before. This final contradiction (or triviality already if no signs) eliminates the proposed residual product. ◻

Proof. For \(P>1\) in reverse orientation, a proper closure would be residual by 86 and would contradict 87. This proves the first assertion.

If \(P=1\), apply the result just obtained in the reversed filter with parameter range \([0,M_{\{i\}}]\) for a spare marker \(i\) outside any tested small support. The actual base there is nontrivial with still the same transitive operators, so it projects fully by the reverse \(P>1\) result and the desired small labels are indeed subdirect. A proper nontrivial product group would have residual label \(p\), since containing a singleton absorption in parameter order would put independent nontrivial support inside that small proper subdirect group. The residual argument rules it out, hence all nonbottom groups are subdirect as claimed. Here \(\Lambda\) must then contain inners (otherwise its proper nontrivial inner part, or the proper-invariant-subgroup result for soluble operators in 4, yields a proper nontrivial invariant product).

Back forward, suppose \(P=1\). If some marker \(x_i\) does not project fully, its axis filter gives product success by the forward argument above (we may take the enlarged actual base as operators then). Otherwise the extension at each marker \(x_i\) has image containing inners by atomicity and the same proper-invariant-subgroup argument, now on a diagonal part: a nontrivial proper subgroup invariant there would transport and multiply to an invariant group strictly between \(P=1\) and \(K_{x_i}\). Its domain interval via restriction realizes the reverse weighted test on one less marker. If \(\Lambda\) itself does not contain inners, the kernel of that extension map is retained by the normal test (supplementation would give image just \(\Lambda\)), hence this gives an ordinary interval within the image in \({\rm Aut}(S)\), with base \(\Lambda\), top containing \(S\). Missing inners there must be supplemented, so this is a reverse invariant interval in one simple site with proper base \(\Lambda\cap S\), impossible by the reverse result just proved: a nontrivial base cannot project fully here, and for trivial base having all nonbottom actual groups project fully leaves no interior labels. Thus \(\Lambda\) contains inners and again all nonbottom labels are subdirect. ◻

Proof. Take an ordinary realization of minimum top order among both orientations of the same quotient; its order is at most the input order. If \(c(P)=\mathcal W\), the ties of \(P\) are fixed and every actual label refines them. On the tie part through a chosen site these refinements are exactly the invariant equivalences under its part stabilizer in \(H\). They give an ordinary realization of the opposite interval with top order at most \(|H|<|G|\), as in the proof of [f:case-subdirect]. This contradicts minimality. Thus absent product success we have the artificial-top extension case in the opposite orientation, with base map containing inners.