Proof. For \(D\le U\le V\le X\), the normality of \(R\) and the subgroup modular identity give \[ (DR\vee U)\wedge V =(RU)\cap V =U(R\cap V) =U\vee(DR\wedge V). \tag{1}\] If \(D<DR<X\), apply the fence to \(DR\) and its comparable complements \(U<V\). Equation (1) would give \(V=U\). Thus \(DR\) is an endpoint. In a modular lattice the same identity holds with any element in place of \(DR\), so the existence of a fence also excludes modularity. Finally, taking inverse images identifies \([D/F,X/F]\) with \([D,X]\). ◻

Proof. Factor out \(F_X\), so that the action of \(X\) on the cosets of \(D\) is faithful. Lemma 4 says that every nontrivial normal subgroup supplements \(D\), or equivalently is transitive in this action. Let \(M\) be a minimal normal subgroup. If \(M\) were abelian, (2) would identify \([D,X]\) with an interval of \(D\)-invariant subgroups of an abelian group. The latter lattice is modular: intersection is its meet and subgroup product is its join. This contradicts Lemma 4.

If \(M'\) were another minimal normal subgroup, then \([M,M']\le M\cap M'=1\). Every \(D\)-invariant subgroup of \(M\) would therefore be normalized by \(DM'=X\), and so would be either \(1\) or \(M\). The correspondence (2) would leave at most two elements in \([D,X]\), again a contradiction.

A finite minimal normal subgroup is characteristically simple, hence a direct product of isomorphic simple groups. Since \(M\) is nonabelian, these simple groups are nonabelian. Conjugation by \(X\) is transitive on the direct factors: otherwise the product of one orbit of factors would be a proper nontrivial normal subgroup of \(X\) contained in \(M\). This proves (3). Every nontrivial normal subgroup contains a minimal normal subgroup and therefore contains \(M\). Its centralizer is normal and intersects \(M\) in \(Z(M)=1\), so that centralizer is trivial. The assertion about soluble normal subgroups follows as well.

Finally \(X=DM\). Inner automorphisms of \(M\) fix each of its simple direct factors. Thus \(D\) and \(X\) have the same image on their factor set. Restoring \(F_X\) proves the statements as written. ◻

Proof. The subgroup \(R\cap B\) is normal in \(B\). Suppose first that \(R\cap B\le D\). If \(B\le DR\), then \[B=B\cap DR=D(B\cap R)=D,\] a contradiction. Comparability therefore gives \(DR\le B\), whence \(R=R\cap B\le D\). Its contrapositive, followed by Lemma 4 in \(B\), gives the asserted supplementation. The image of \(R\cap B\) modulo \(F_B\) then contains the nonabelian socle \(S_B\), by Lemma 5. This is impossible if \(R\) is soluble. For the final assertion, \(F\le F_B\) because \(F\) is normal in \(B\) and contained in \(D\). If both \(R\) and \(Q\) escaped \(D\), their intersections with \(B\) would both map over \(S_B\), while their images would commute. This would make \(S_B\) abelian. ◻

Proof. Induct on \(|I|\), separating the last coordinate. The projection to the other coordinates is, by induction, a product \(T^r\) of diagonal strips. We have a subdirect subgroup of \(T^r\times T\). The kernels of its two projections are normal in the respective factors. The kernel on the last \(T\) is either \(T\), in which case this is the full product, or \(1\), in which case the subgroup identifies a quotient of \(T^r\) with \(T\). Normal subgroups of \(T^r\) are products of its simple direct factors, so that quotient is one of the factors. The last coordinate is therefore joined to exactly one existing strip.

Two coordinates belong to the same strip precisely when the projection of \(E\) to their product is the graph of an isomorphism rather than the full product. This characterizes the partition and its links, and also proves the assertion about inclusion. ◻

Proof. The normal subgroup \(F_Y\) of \(X\) is contained in \(D\), so \(F_Y\le F_X\). Since \(Y=DN_Y\) and \(D\le X\), \[X=D(X\cap N_Y).\] The image of \(E_X\) in \(X/F_X\) is a nontrivial normal subgroup and therefore contains \(S_X\). Every simple composition factor of a subgroup of a direct product is a section of one of the factors: projection to one coordinate gives an exact sequence whose kernel lies in the product of the remaining coordinates, and induction applies. Thus \(T_X\) is a section of \(T_Y\).

If \(T_X\cong T_Y\), some coordinate projection of \(E_X\) is all of \(T_Y\); otherwise none of those smaller groups could have a composition factor of order \(|T_Y|\). The transitive action of \(D\) on \(I_Y\) makes all the projections surjective. Conversely, suppose \(E_X\) is subdirect. Lemma 7 expresses it as a product of diagonal copies of \(T_Y\). The action of \(D\) on those copies is transitive. Since \(X/F_Y=(D/F_Y)E_X\), the subgroup \(E_X\) is minimal normal in \(X/F_Y\). Its intersection with \(F_X/F_Y\) is either trivial or all of \(E_X\). The latter would imply \(X=D\), which is excluded. Its image is therefore the unique minimal normal subgroup \(S_X\), proving the equivalence and (4).

There are \(m_X\) strips on \(m_Y\) coordinates. If \(m_X=m_Y\), every strip is a singleton, so \(E_X=S_Y\) and \(X=Y\). Thus \(m_X<m_Y\). The quotient identification with \(S_X\) assigns to every strip its actual simple factor in \(I_X\) and defines (5). The subgroup \(E_X\) is normal in \(X/F_Y\), and its quotient identification with \(S_X\) commutes with conjugation by \(X\); hence this support map is \(X\)-equivariant.

For composition, factor out \(F_Z\) and let \(E_Y=Y\cap N_Z\). We have \(N_Y=F_YE_Y\) and \(F_Y\cap E_Y=1\). Since \(F_Y\le X\), \[X\cap N_Y=F_Y(X\cap E_Y)=F_Y(X\cap N_Z).\] Thus the subdirect subgroup used for \(X<Y\) is the image of the one used for \(X<Z\) under the already specified identification \(E_Y\cong S_Y\). Refinement of its strip partition is exactly composition of the support maps. ◻

Proof. Factor out \(F_Y\), and let \(C\) be the product of the coordinate projections of \(B\). It is \(X\)-invariant. The correspondence (2), with bottom \(X\) and normal subgroup \(S_Y\), identifies \([X,Y]\) with the \(X\)-invariant subgroups of \(S_Y\) containing \(B\).

Suppose \(B<C<S_Y\). If \(U<V\) are comparable complements of \(XC\), put \(E_U=U\cap S_Y\) and \(E_V=V\cap S_Y\). Their projections already contain the projections of \(C\). Hence projecting \(\langle E_U,C\rangle=S_Y\) shows that \(E_U\) is subdirect; the same holds for \(E_V\).

Every link within a strip of \(E_U\) carries the projection of \(B\) in one coordinate onto its projection in the other, because \(B\le E_U\). Consequently \(E_U\cap C\) is a product of independent diagonal copies of those nontrivial projections, one for each strip of \(E_U\). The proper inclusion \(E_U<E_V\) splits at least one strip, by Lemma 7. Intersecting with \(C\) then permits an additional independent nonidentity coordinate value, and gives \(E_U\cap C<E_V\cap C\). This contradicts the required equality of both intersections with \(B\). A fence therefore forces \(B=C\). If \(X\) is a coatom, \(B<C<S_Y\) directly contradicts maximality.

A proper subdirect subgroup has a link between two distinct coordinates. The full product \(P^{I_Y}\) contains an element which is identity in the first of these coordinates and nonidentity in the second, contradicting that link. Now let \(Z\) be a coatom above \(X\). If its projections were full, it would be such a proper subdirect subgroup. Otherwise its product closure is proper, and maximality forces it to equal that closure. Intersections of products are products.

Finally, for different labelled simple types, the projections are nontrivial because their image contains \(S_X\), and transitivity makes all of them nontrivial. If one were full, all would be full, contradicting Proposition 8. ◻

Proof. For a coatom \(M\) above \(X\), Proposition 9 writes its intersection with \(S_Y\) as \(P_M^{I_Y}\), with \(1<P_M<T_Y\). Let \(J=H_*\cap T_Y\). It normalizes \(P_M\), so \[P_MJ\le N_{T_Y}(P_M)<T_Y.\] The last inequality follows from simplicity and \(1<P_M<T_Y\). Both \(P_M\) and \(J\) are \(H_*\)-invariant, and their product therefore transports to a saturated overgroup of \(M\) which is still proper in \(Y\). Maximality gives \(J\le P_M\). Intersecting over the coatom representation of \(X\) gives \(J\le P_X\); the reverse inclusion is immediate. This proves (6).

Products above \(P_X^{I_Y}\) correspond precisely to \(H_*\)-invariant subgroups \(P_Z\le T_Y\) containing \(P_X\). By (6), these in turn correspond precisely to the subgroups \(K_Z\) of \(H_*T_Y\) containing \(H_*\): the two maps are \(P_Z\mapsto H_*P_Z\) and \(K_Z\mapsto K_Z\cap T_Y\). Since \(Z=X P_Z^{I_Y}\) and the product fixes the coordinate set, \(Z_i=X_iP_Z^{I_Y}\), proving (7). Intersections and generated subgroups of coordinate products are again coordinate products, proving the sublattice assertion. Any intermediate projected subgroup would lift to an original intermediate saturated subgroup, which proves the assertion about covers. At a strict cover Proposition 9 supplies the product hypothesis, and the bottom is itself a coatom. The ambient group \(H_*T_Y\) is a subgroup of \(\operatorname{Aut}(T_Y)\) containing \(T_Y\), hence is almost simple. ◻

Proof. Factor out \(F_Y\) throughout the proof. Put \[E=Z\cap N_Y=P_Z^{I_Y},\qquad F=F_Z, \qquad W=F\cap E.\] Let \(Q_i\) be the projection of \(W\) to the \(i\)th coordinate, and let \(\widehat Q=\prod_iQ_i\). These projections are transported into one another by \(D\), and each is invariant under its coordinate stabilizer. Thus \(\widehat Q\) is normal in \(Z\).

We claim that \(W=\widehat Q\). For \(q,q'\in Q_i\), choose \(w\in W\) whose \(i\)th coordinate is \(q\) and let \(e_i(q')\in E\) be supported only on coordinate \(i\). Since \(W\trianglelefteq E\), \[[w,e_i(q')]=e_i([q,q'])\in W.\] It follows that \([\widehat Q,\widehat Q]\le W\). The image of \(\widehat Q\) in \(Z/F\) is therefore the abelian normal subgroup \(\widehat Q/W\). Lemma 5 forces that image to be trivial. Hence \(W=\widehat Q\), proving (8). This argument also covers \(Z=X\) at a covering comparison; it does not use a projected core with bottom equal to the whole projected group.

The normal subgroup \(EF/F\) of \(Z/F\) is nontrivial, since \(Z=DE\) and \(Z>D\). It contains \(S_Z\). Therefore, if \(L=N_Z\cap E\), then \[N_Z=FL,\qquad L/W\cong S_Z.\] In particular \(L/W\) is a perfect normal subgroup of \(E/W=\prod_i(P_i/Q_i)\).

Every perfect normal subgroup \(V\) of a direct product \(\prod_iB_i\) is the product of its intersections with the factors. To see this, for \(v,w\in V\) the element \([v,e_i(w_i)]\) lies in \(V\cap B_i\) and is the \(i\)th coordinate of \([v,w]\). Hence \(V'\le\prod_i(V\cap B_i)\le V\), and \(V=V'\) gives equality. Apply this observation to \(L/W\). It splits as a product of its coordinate intersections, each a direct product of copies of \(T_Z\). Taking full inverse images gives \(L=R_Z^{I_Y}\) and (9)–(10). Conjugation by \(Z\) permutes these coordinate intersections and commutes with \(L/W\cong S_Z\), so the resulting support map is \(Z\)-equivariant. Transitivity of \(D\) on \(I_Y\) makes the multiplicity \(a_Z\) the same in every coordinate.

Any \(K_Z\)-normal subgroup strictly between \(Q_Z\) and \(R_Z\) would transport to a \(Z\)-normal subgroup strictly between \(W\) and \(L\). Its image would be a proper nontrivial normal subgroup of \(Z/F\) inside \(S_Z\). This proves the chief-factor assertion and also the transitivity of \(K_Z\) on the \(a_Z\) simple factors of \(R_Z/Q_Z\).

We have \(Z=DN_Z=DL\). Since \(L\le N_Y\) fixes \(I_Y\), for a chosen coordinate \(i\) this gives \[Z_i=D_iL,\qquad K_Z=\rho(D_i)R_Z.\] The group \(R_Z\) acts by inner automorphisms on \(R_Z/Q_Z\) and fixes its simple factors. Thus \(\rho(D_i)\) and \(K_Z\) have the same image on that factor set. The kernel of \(\rho\) centralizes the ambient coordinate and so acts trivially on this section. This identifies the action with the original fiber action and proves (11) and its assertions about stabilizers.

For the additional projected-core assertion, let \(C_Z=\operatorname{core}_{K_Z}(H_*)\). The subgroup \(Q_Z\) is normal in \(K_Z\) and lies in \(P_X\), because \(F_Z\le D\le X\). By (6) it lies in \(H_*\), and therefore \(Q_Z\le C_Z\cap P_Z\). Conversely, transport \(C_Z\cap P_Z\) over all coordinates. Its product is normal in \(Z\) and is contained in \(P_X^{I_Y}\le X<Z\). Lemma 4 for the original interval \([D,Z]\) forces this product into \(D\), hence into \(F_Z\). Equation (8) now gives the reverse inclusion.

Since \(\rho(D_i)\le H_*\le K_Z\) and \(K_Z=\rho(D_i)R_Z\), we obtain \(K_Z=H_*R_Z\). Also \(R_Z\cap C_Z=Q_Z\). The resulting image of \(R_Z\) modulo \(C_Z\) is minimal normal by the chief-factor assertion and is transitive by supplementation. Its factor action is the same for \(H_*\) and \(K_Z\), because it is already induced by \(\rho(D_i)\).

Finally suppose \(a_Z=1\). The support map (11) is then a bijection. The subgroup \(E\) fixes \(I_Y\) and consequently fixes \(I_Z\). Conjugation embeds \(EF/F\) faithfully in \(\operatorname{Aut}(S_Z)\), since \(C_{Z/F}(S_Z)=1\) by Lemma 5. Its image therefore lies in \(\operatorname{Aut}(T_Z)^{I_Y}\) and contains the inner subgroup \(S_Z\). The quotient \(E/L\cong(P_Z/R_Z)^{I_Y}\) embeds in \(\operatorname{Out}(T_Z)^{I_Y}\). Schreier’s Theorem, a consequence of the classification of finite simple groups, says that \(\operatorname{Out}(T_Z)\) is soluble; see also the explicit use in (Pálfy 2019, sec. 1). Hence \(P_Z/R_Z\) is soluble. ◻

Proof. We spell out the factor actions. A subspace or paired-flag stabilizer has at most a bounded number of Levi components. A tensor product of two spaces has two component groups. Extension-field, subfield, form, and almost simple cases have one, except for the bounded low-dimensional coincidences. A direct-sum decomposition or a tensor-induced decomposition has the full symmetric action on its blocks, with a possible alternating restriction from the determinant or spinor condition. A block contributes at most a bounded number of simple factors; for instance an orthogonal plus-type four-space contributes two. Taking a point inside a block and then the block itself accounts for this extra bounded level. In the alternating case, the intransitive and affine alternatives have bounded component actions, while the imprimitive, diagonal, and product-action alternatives have this same full block action. Exceptional types have bounded rank and hence bounded numbers of components. Central, determinant, and form quotients do not split a simple component into more factors. This proves [cls:cat-actions]; the explicit \(\mathcal C_2\) and \(\mathcal C_7\) component quotients can also be read in (Burness et al. 2017, proofs of Lemmas 5.4 and 5.9). ◻

Proof. We induct on \(r\). The assertion for one step follows from primitivity of a natural giant, or from the bounded index in the other case. For the induction put \(N=\operatorname{core}_H(H_1)\). For an arbitrary \(U\in[A,H]\), the group \(A(U\cap N)\) lies in \([A,H_1]\), and its intersection with \(N\) is \(U\cap N\). The induction hypothesis therefore controls the intersection chain and its types. A strict increase in \(U\) must increase \(U\cap N\) or \(UN/N\), since equality of both gives equality of orders. Furthermore \[[H:U]=[H/N:UN/N]\,[N:U\cap N],\] and the possible second factors are a fixed multiple of the possible indices \([H_1:A(U\cap N)]\). Thus it remains to control the images.

If \(H/N\) has bounded degree, its order is at most \(b!\) and there is nothing more to prove. Otherwise it is \(A_d\) or \(S_d\), and \(H_1/N\) is its natural point stabilizer. Project the remaining route into \(H_1/N\). A natural-giant step projects either to the same action or to a trivial action, because it is primitive; a bounded-degree step retains bounded degree. Starting with the natural \(A_{d-1}\) in \(H_1/N\), retain an alternating group normal in the current projected group and supported on all but at most \(r\) points. A natural-giant quotient either kills this alternating group, which is then retained in the next subgroup, or maps it onto the natural alternating socle of its image. In the latter case equality of orders gives the same degree, and its point stabilizer removes one supported point. A step of bounded degree retains the alternating group once its order exceeds \(b!\). After a step, take the union of all conjugates of the retained support in the new subgroup. The overlapping-support argument in the next paragraph supplies a normal alternating group on that union, without further loss of points. Apart from a possible sign on the large support, the residual quotient acts on the omitted points and has order at most \(2r!\). For \(d>2r+b+10\), each step therefore loses at most one supported point, and the terminal image contains the alternating group supported on at least \(d-r\) points. Here the identification of the next point stabilizer uses \(\operatorname{Aut}(A_e)=S_e\) for \(e\geq7\); the degree-six exception and all smaller first degrees are absorbed in the fixed finite universe. Thus we may take \(c=r\) in the large-degree argument.

An overgroup of an alternating group supported on \(d-c\) points has bounded height and boundedly many possible indices. To see this directly, assume \(d>2c+4\). The supports of any two conjugates overlap in at least three points, so their alternating groups generate the alternating group on the union of their supports. The union of all conjugate supports is an orbit of the overgroup; its conjugate alternating groups generate the alternating group on that orbit. The orbit omits at most \(c\) points. Modulo the alternating group on that orbit, the overgroup has order at most \(2c!\). Its large simple type is therefore \(A_{d-j}\) with \(0\leq j\leq c\), and its possible indices belong to the finite list of falling factorials \((d)_j\) divided or multiplied by integers bounded in terms of \(c\). The excluded small \(d\) add only bounded groups. Combining this with the intersection induction proves all three conclusions. Invariant quotient blocks correspond exactly to the overgroups \(U\) already considered. ◻

Proof. Put \(d_0=2R+b+10\) and \(M=d_0!\), and define the positive rational atoms \[\Gamma(\mathcal D)= \{u/v:1\leq u,v\leq M\} \cup\{(d)_j:d\in\mathcal D,\ d>d_0,\ 0\leq j\leq R\}.\] We claim that every index on a route of length \(r\) is a product of at most \(L_r=r(r+3)/2\) atoms or their inverses. Induct on \(r\), putting \(N=\operatorname{core}_H(H_1)\) and \(B=A(U\cap N)\). Since \(U\cap N\trianglelefteq U\), we have \(A\leq B\leq H_1\) and \(B\cap N=U\cap N\). Consequently \[ [H:U]=[H/N:UN/N]\, \frac{[H_1:B]}{[H_1:AN]}. \tag{13}\] The image factor is bounded by \(M\) when the first action is small. Otherwise the supported alternating-group calculation in Lemma 17 gives \[[H/N:UN/N]=\frac{\varepsilon(d)_j}{t},\qquad 0\leq j\leq r,\quad \varepsilon\in\{1,2\},\quad1\leq t\leq2j!.\] It therefore uses at most two atoms. The numerator \([H_1:B]\) uses at most \(L_{r-1}\) atoms by induction.

The denominator is controlled across the whole collection, not just held fixed within one route. Project the suffix route into \(H_1/N\). Its degree is exactly \[ [H_1:AN]=\prod_{j=2}^{r} [H_{j-1}N/N:H_jN/N]. \tag{14}\] Under any homomorphism with kernel \(C\), a natural step \(L_0>L_1\) projects to the same full natural action or to a trivial action: maximality gives \(L_1C=L_1\) or \(L_0\). A bounded step keeps bounded degree. Thus every factor on the right of (14) is an allowed natural degree, a bounded integer, or one, and is a single atom. In particular no arbitrary scalar-kernel divisor enters the denominator. Equation (13) gives \(L_r\leq L_{r-1}+2+(r-1)\), proving the claim.

There are at most \(M^2+K(R+1)\) atoms, so the union of all index palettes has cardinality at most \[\bigl(1+2(M^2+K(R+1))\bigr)^{L_R}.\] Only integral values of these rational recipes occur as actual indices. Invariant quotient blocks correspond to the overgroups \(U\) already allowed. If \(H'\) is any such block stabilizer, apply the argument anew to the next full route inside \(H'\); it is not necessary that \(H'\) itself admit a natural-step route from \(H\). Indices multiply through a bounded number of these operations, giving the final assertion. ◻

Proof. By Propositions 9 and 10, the starting group is a nontrivial proper coordinate product, and all designated intersections of coatoms are saturated. The shortcut \(a\prec v_{0N}\prec b\) projects to a two-step maximal chain in an almost simple group. Thus Proposition 15 gives the five-chief-factor bound for the actual projected bottom \(H_*\). The image of the original \(D\)-stabilizer need not equal \(H_*\) as a group: in general \(H_*\) is that image multiplied by the starting coordinate product. What is required is equality of the induced active permutation actions, as follows.

For every label \(z\), the equality \(z=DN_z\) implies that \(D\) and \(z\) have the same image on \(I_z\), because \(N_z\) fixes its individual simple factors. If \(I_z\to I_w\) is a product-step factor map and \(g\in z\) fixes \(i\in I_w\), write \(g=dn\) with \(d\in D\) and \(n\in N_z\). Then \(d\) fixes \(i\) and has the same action as \(g\) on the fiber over \(i\). Thus the bottom stabilizer induces the full local factor action at each level, not an arbitrary subgroup of the natural giant. This property survives the initial projection by Proposition 11.

We now account for all the shortcuts, including middles below a proper endpoint. An endpoint \(x_j\) is reached from \(b\) by the original shortcut \(b>v_{jN}>x_j\); take an empty route for \(j=N\). The first downward step is strict, because a proper subdirect group cannot contain the nontrivial full coordinate product at the bottom. Two strict covers give a concatenated route of length at most \(2r_0\). A strict cover followed by an equal-type merge instead makes the endpoint factor stabilizer an overgroup of the first route’s terminal stabilizer.

Starting at \(x_j\), the middle \(v_{ij}\) and the lower endpoint \(x_i\) are reached by at most two further original covers. An equal-type operation takes an invariant quotient block. A strict operation supplies a fresh full local route inside the actual current projected factor stabilizer \(U\leq H_*\). Indeed the image of the original \(D\)-stabilizer lies in \(U\) and already has the full local permutation image, whereas \(U\) acts through the labelled-node stabilizer because \(a\) lies below that node. The kernel of the original-coordinate projection centralizes that coordinate and acts trivially on its sections. Thus these are equal local permutation images; no equality of the stabilizer groups themselves is needed.

The factor maps and equal-type strips respect the original \(b\) coordinates. Here is the intersection identity behind this assertion. For labels \(x\leq z\) used in a route, work modulo \(F_b\) and write \[x\cap N_b=P_x^{I_b},\qquad N_z\cap N_b=R_z^{I_b},\qquad F_z\cap N_b=Q_z^{I_b}.\] These product descriptions follow from saturation and Proposition 11. Since \(F_z\leq D\leq x\) and \(N_z=F_z(N_z\cap N_b)\), intersection and the quotient map give \[\frac{x\cap N_z}{F_z} \cong \frac{x\cap N_z\cap N_b}{F_z\cap N_b} =\prod_{i\in I_b}\frac{P_x\cap R_z}{Q_z}.\] In an equal-type comparison this is the subdirect subgroup of \(S_z\) whose strips describe the merge. Because it is a full product over \(I_b\), no strip can link factors in different original coordinates: such a link would prohibit independent elements in the two factors. For a strict split followed by a merge inside \([x_i,x_j]\), repeat this identity with \(x_j\) as upper vertex. The fenced endpoint comparison makes \(x_i\) saturated in those coordinates, and the strict shortcut cover makes its middle saturated there. Thus the merge cannot cross those local starting fibers either. Altogether, every required factor, fiber, and quotient-block stabilizer uses at most four original covers, interpreted as full local routes or quotient operations. Four successive operations, each with route length at most \(2r_0\), give a conservative common bound, including all intermediate block stabilizers.

We establish the type pool before the degree and index pools. For fixed route parameters let \[\Phi(\mathcal P)=\mathcal P \mathbin{\cup}\{A_{d-j}:A_d\in\mathcal P,\ 0\leq j\leq c,\ d-j\geq5\} \mathbin{\cup}\mathcal E,\] where \(c\) and the finite exceptional set \(\mathcal E\) are supplied by Lemma 17. If the whole current projected group has nonabelian composition types in \(\mathcal P\), that lemma puts the types of every route prefix and every quotient-block overgroup in \(\Phi(\mathcal P)\). This includes any kernel of the corresponding permutation action. Retain the original types at every iteration. Starting with \(\operatorname{cf}_{\mathrm{na}}(H_*)\), of cardinality at most five, a bounded number of iterations gives one pool of absolute bounded cardinality for all these projected groups. Their induced images have no additional composition types. This proves (ii); it makes no claim about entire unprojected stabilizers in the original group.

For an equal-type stretch, fix its highest vertex. Equal-type transport identifies all earlier vertices with strip subgroups of that socle, whose support partitions strictly change along the stretch. Equal support partitions and subgroup inclusion would force equality of orders and hence equality of subgroups, including the diagonal twists. The preceding coordinate compatibility restricts these to invariant partitions of a single original \(b\)-coordinate fiber. Transitivity ensures that a strict change appears on that chosen fiber. All their block stabilizers are overgroups of a single route terminal supplied by the shortcut from \(b\) to the highest vertex. The height assertion of Lemma 17 proves (iii).

Each full giant of large degree \(d\) contributes the composition type \(A_d\) to its actual starting projected group. Every such group is a route prefix already covered by the preceding pool. Hence the set \(\mathcal D\) of giant degree values occurring anywhere in these bounded operations has absolute bounded cardinality. The values may be large. Apply Lemma 18 with this entire set \(\mathcal D\), and multiply indices through the bounded number of operations. This yields a common index palette across all the actions, including arbitrary quotient-block choices followed by fresh full routes. In particular it is a union bound, not merely a separate cardinality bound for each action.

More explicitly, put \(q_i=m_{x_i}/m_b\). The original shortcut from \(b\) to \(x_i\) realizes \(q_i\) as the degree on its active-factor fiber: a strict/strict path gives a route degree, and a strict/merge path gives a quotient-block degree. The common recipe palette bounds the cardinality of \(\{q_i\}\). Since \(m_{x_i}/m_{x_j}=q_i/q_j\), its quotient palette also has bounded cardinality. This proves (iv). ◻

Proof. The exact product/core identity in Proposition 11 gives, before projection, \[(F_i\cap N_b)/F_b=Q_i^{m_b},\qquad (N_i\cap N_b)/F_b=R_i^{m_b},\qquad R_i/Q_i\cong T_i^{m_i/m_b}.\] Since \(F_j\leq D\leq x_i\), we have \(Q_j\leq P_i\). Intersecting with the lower group and then with its core gives the displayed formulas for \(J_{ij}\) and \(B_{ij}\). Also \(C_j\leq C_i\) and \(R_j\leq P_j\), whence \[C_j\cap R_j=Q_j.\] Thus \(uQ_j\mapsto uC_j\) embeds \(B_{ij}\) in \(C_i/C_j\).

The \(C_i\) form a decreasing chain of normal subgroups of the one group \(H_*\). Each nonsoluble successive quotient consumes at least one nonabelian chief factor of \(H_*\). There are at most five, by Lemma 19. Cutting at these transitions leaves at most six stretches; one is arbitrarily long. The core of a shortcut middle lies between the cores of its two endpoints, so its killed layers are controlled by the same soluble quotient. No separate bound on a collection of unrelated shortcut cores is being assumed.

For the final assertion work with the original labelled vertices \(x_i<x_j\) and put \[E=x_i\cap N_j,\qquad W=F_i\cap N_j,\qquad L=N_i\cap N_j.\] These subgroups are normal in \(x_i\), with \(F_j\leq W\leq L\). The kernel of the conjugation action of \(E\) on \(S_i=N_i/F_i\) is exactly \(W\), because \(C_{x_i/F_i}(S_i)=1\). When \(m_i=m_j\), the product support map \(I_i\to I_j\) is a bijection. Since \(E\leq N_j\) fixes \(I_j\), it fixes every factor of \(S_i\). Its socle preimage is \(L\), and therefore \[E/L\hookrightarrow\operatorname{Out}(T_i)^{m_j},\] which is soluble by Schreier’s Theorem. In the \(x_j\) coordinates write \[E/F_j=P_{ij}^{m_j},\qquad W/F_j=Q_{ij}^{m_j},\qquad L/F_j=R_{ij}^{m_j}.\] Proposition 11 gives \(R_{ij}/Q_{ij}\cong T_i\), and the preceding embedding makes \(P_{ij}/R_{ij}\) soluble.

The killed kernel is exactly the one controlled in the initial \(b\) projection. Indeed \(N_j=F_j(N_j\cap N_b)\) and \(F_j\leq F_i\) give \[\begin{align*} F_i\cap N_j&=F_j(F_i\cap N_j\cap N_b),\tag{15}\\ W/F_j&\cong \bigl((Q_i\cap R_j)/Q_j\bigr)^{m_b}. \tag{16}\end{align*}\] It is soluble on the retained stretch, so \(Q_{ij}\) is soluble as well. Thus \(P_{ij}\) has exactly its one active simple factor as a nonabelian composition factor. The same intersection calculation identifies the killed kernels for either half of a shortcut. This argument takes place in the original labelled \(x_i\); it uses neither projected quasiprimitivity nor a supplementation dichotomy at an unlabelled bottleneck. A shortcut middle which still has several active factors can retain a nonabelian factor-permutation quotient; this is not asserted to be soluble. Its degrees are controlled by the common type pool, and it occurs only in the split–merge case considered separately below. ◻

Proof. Project the strict comparison from \(a\) to \(u\) using [cls:test-alternating]. The bottom is a full product and an intersection of coatoms. The coatoms and their intersection \(c_u\) are saturated, so Proposition 10 carries the entire interval \([c_u,u]\) to an \(M_{16}\) interval ending at an almost simple group with socle \(T_u\). For an alternating socle of degree at least seven that group is alternating or symmetric, contradicting Proposition 15. The exceptional automorphisms in degree six concern only bounded-order types. ◻

Proof. Apply (Larsen and Pink 2011, Theorem 0.5). Outside (i), \(P\) lies in a proper algebraic subgroup \(B<X\) from a fixed constructible family. Apply (Larsen and Pink 2011, Theorem 12.3) to that family. It supplies an envelope \(H\leq B\), with \(P\leq H\), whose connected reductive quotient has simple adjoint factors \(X_1,\ldots,X_t\) such that the image \(L_i\) of \(P\cap H^\circ\) satisfies \[E_i=(X_i^{F_i})'\leq L_i\leq X_i^{F_i},\] with \(E_i\) simple. Each \(\dim X_i\leq\dim H^\circ<\dim X\). The kernel of the map from \(H^\circ\) to the product of its simple adjoint quotients is soluble: it is an extension of the unipotent radical by the full center of the connected reductive quotient. That center can be disconnected, but is abelian. If \(L\) is the image of \(P\cap H^\circ\) in \(\prod_i L_i\), then \(L'\) is a subdirect subgroup of \(\prod_iE_i\), because each \(L_i/E_i\) is abelian. The subdirect product Lemma makes \(L'\) a product of full diagonals. Its simple factors are therefore among the \(E_i\), counted at most once per position. There are at most \(\dim X/3\) positions.

The component bound in (Larsen and Pink 2011, Proposition 1.4) bounds \(|H/H^\circ|\) throughout this constructible family. Its contribution has bounded order and bounded composition length. This proves (ii). Applying the same argument to every subgroup of a finite group of rank at most \(R\) proves the last assertion: the ambient algebraic dimensions range over a finite set, and the finitely many small orders and characteristics in the exceptional alternative are absorbed in a function of \(R\). ◻

Proof. First, the pointwise centralizer of \(E=(X^{F_0})'\) in \(\operatorname{Aut}_{\rm alg}(X)\) is trivial when the field parameter is sufficiently large. Indeed the fixed groups \(X^a\), for nonidentity algebraic automorphisms \(a\), form a constructible family of proper closed subgroups: take the equalizer of the action and projection maps over the finite-type algebraic automorphism group, remove the identity section, and stratify. Now use (Larsen and Pink 2011, Proposition 3.5).

Here is the isogeny calculation needed to include field and exceptional graph operations. Let \(\mathcal I(X)\) be the group of bijections of \(X(\overline{\mathbb F}_p)\) generated by self-isogenies and their inverses. The isogeny Theorem (Conrad 2014, Theorem 6.1.16(1) and Proposition 6.1.13(2)) gives a homomorphism \[\nu:\mathcal I(X)\longrightarrow\mathbb Z, \qquad\ker\nu=\operatorname{Aut}_{\rm alg}(X).\] For completeness, after fixing a Borel pair, a self-isogeny permutes the simple roots by \(\pi\) and raises their parameters to powers \(p^{e_i}\). Use the row-coroot convention \(C_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle\) for the Cartan matrix. The compatibility condition is \[p^{e_j}C_{ij}=p^{e_i}C_{\pi(i)\pi(j)}.\] The labeled undirected Coxeter diagram is preserved. Unless a multiple edge is reversed, connectedness forces all \(e_i\) equal. An edge reversal is possible only for \(B_2,F_4\) in characteristic two, or \(G_2\) in characteristic three. The exponents on the two root lengths then differ by one. Thus every self-isogeny is \(a\theta^m\), where \(a\) is algebraic, \(m\geq0\), and \(\theta=F_p\) except in those three cases, when \(\theta^2=F_p\); see also (Conrad 2014, Remark 6.2.10). Root-group constants are absorbed by the algebraic automorphism, using the pinned uniqueness in the isogeny Theorem. Conjugation by \(\theta\) preserves algebraic automorphisms, so the same normal form holds with \(m\in\mathbb Z\) in \(\mathcal I(X)\). Inseparability degree proves uniqueness of \(m\), and \(\nu(a\theta^m)=m\) is the asserted homomorphism. Positive \(\nu\) gives a Frobenius map: Lang–Steinberg removes its inner factor, leaving a positive diagram–field map (Liebeck and Seitz 1998, Proposition 2.7(i), p. 435). Its field parameter is \(p^{\nu}\), or \(p^{\nu/2}\) in the three exceptional cases.

Let \(C_E\leq\mathcal I(X)\) fix \(E\) pointwise. The preceding centralizer observation makes \(\nu|_{C_E}\) injective. Its image is a nonzero subgroup of \(\mathbb Z\), since it contains \(\nu(F_0)\). Thus \(C_E=\langle F_1\rangle\) for a Frobenius map \(F_1\), and \(F_0=F_1^b\) for some \(b\geq1\). Put \(d=\dim X\) and let \(q_0\) be the parameter of \(F_0\). The fixed-point order bounds and bounded diagonal index in (Larsen and Pink 2011, Theorem 3.4(b),(d)) give \[\frac{(q_0-1)^d}{z_\Phi} \leq |E|\leq |X^{F_1}| \leq c_\Phi q_0^{d/b}.\] Enlarging \(c_\Phi\) includes the finitely many possibilities for the parameter of \(F_1\) below four. For sufficiently large \(q_0\), this forces \(b=1\). Consequently \(C_E=\langle F_0\rangle\). Since \(F\) fixes \(E\), it follows that \(F=F_0^r\) with \(r>0\).

The quotient \(P/E\) is abelian and \(E\) is perfect, so \(P'=E\). Every \(m\in M\) therefore normalizes \(E\). The finite automorphism Theorem (Steinberg 1967, Theorems 30 and 36) lifts \(m\) to an element \(\widetilde m\in\mathcal I(X)\). The commutator \(\widetilde mF_0\widetilde m^{-1}F_0^{-1}\) has \(\nu=0\) and fixes \(E\) pointwise. It is therefore the identity. Thus \(m\) normalizes \(E_e=(X^{F_0^e})'\) for every divisor \(e\) of \(r\).

We have \(r>1\), since \(r=1\) would give \(T\leq M\). If \(r\) were composite, choose \(1<e<r\) dividing \(r\). Fixed-point order estimates, after increasing the original field threshold, give \(E_e\not\leq P\) and \(1<E_e<T\). Hence \[M<ME_e\leq N_A(E_e)<A.\] The last inequality follows because \(T\) is simple and cannot normalize a nontrivial proper subgroup of itself. This contradicts maximality and proves that \(r\) is prime. ◻

Proof. Consider a subsequence of distinct endpoint types of ranks at most \(R\). A lower simple type is a section of each later one by Proposition 8; in particular their orders increase strictly. Remove the boundedly many types below the order thresholds of Lemma 22, for the finitely many relevant algebraic types. Every two-edge shortcut has intermediate types of rank bounded in terms of \(R\), by the last assertion of that lemma. Increase the threshold accordingly. On each strict cover read downwards, the characteristic is then unchanged and the algebraic dimension does not increase. The only case of equal algebraic dimension is the sufficiently general alternative. An equal-type cover changes neither characteristic nor dimension.

There are finitely many possible dimensions and algebraic families. If the subsequence were arbitrarily long, it would have an arbitrarily long further subsequence with one dimension and family. On every prescribed two-edge shortcut between its endpoints the dimension cannot drop. Each strict cover is therefore a normalized prime subfield step by Lemma 23; a possible equal-type cover contributes no field change. With one fixed family at the endpoints, the conversion between its usual exponent \(f\) and the Frobenius parameter exponent cancels. Thus, for every \(i<j\), \[f_j/f_i\in\mathbb Z_{>1},\qquad \Omega(f_j/f_i)\leq2,\] where \(\Omega\) counts prime factors with multiplicity. Four successive endpoints would give \[\Omega(f_4/f_1)=\sum_{i=1}^3\Omega(f_{i+1}/f_i)\geq3,\] a contradiction. All discarded sets and pigeonhole choices have bounds depending only on \(R\), proving the first assertion.

For any desired rank threshold, remove the boundedly many endpoints below a sufficiently larger threshold. If a remaining shortcut middle had bounded rank, its lower endpoint, being a simple section of it, would have rank bounded by Lemma 22. The larger threshold was chosen to exclude this. Arbitrarily long chains remain, proving the second assertion. ◻

Proof. Take counterexamples of lengths tending to infinity, retaining the original endpoints initially. A lower simple type is a section of an upper one, so equality of the endpoint types of a segment forces equality throughout that segment. Lemma 19 bounds the lengths of these equal-type stretches. Choose one vertex from each stretch and discard the original initial stretch. There are arbitrarily many remaining vertices, and none has the type of the original \(a\). Their shortcut middles are above a retained lower vertex, so also cannot have type \(T_a\). Lemma 21 now excludes all large alternating types at both endpoints and middles. Sporadic groups and all other bounded-order coincidences form a finite list. Bounded-rank Lie types can be removed using Lemma 24. This gives (i), with arbitrarily large minimum rank.

All later coordinate projections still use the original tests. For a retained upper endpoint or middle \(u\), condition [cls:test-alternating] applies to \(a<u\), with \(T_a\not\cong T_u\). Proposition 10, with original lower vertex \(a\), gives \(H_a\cap T_u=P_a\). A retained vertex \(z\) saturated in these coordinates then has \[K_z=H_aP_z,\qquad K_z\cap T_u=P_z,\] because \(P_a\leq P_z\). Saturation for a distinct-type endpoint comparison comes from its fence, and saturation at a strict shortcut cover comes from its coatom status. Thus the required inner-intersection identity never relies on inheriting a local coatom-meet representation from the global one below \(b\), or on a new test at a retained bottom.

Use the common cardinality bound on ratios in Lemma 19 to color pairs by \(m_i/m_j\). Finite Ramsey gives an arbitrarily long homogeneous subchain. On a homogeneous triple the common ratio \(t\) satisfies \(t^2=t\); since \(t>0\), it is one. This proves (ii). Next apply Lemma 20 in the original fixed projection: deleting the at most five nonsoluble transitions leaves an arbitrarily long stretch. All shortcut cores between its endpoints lie in the corresponding soluble gap. Passing to any later active coordinate takes subgroups and images of these soluble killed kernels, and hence preserves solubility. Lemma 20 and Proposition 11 give (iii).

The common type pool and index bounds are precisely Lemmas 19 and 18; none of these bounds counts abelian factors. They remain valid under all the preceding thinning, giving (iv).

Read a shortcut downwards from its upper endpoint. Its first cover cannot be a proper subdirect merge, because the lower endpoint is a nontrivial full product. If both covers are strict, their positive integral splitting multiplicities multiply to the endpoint ratio one. Both are therefore one. Their killed kernels are soluble by (iii), and their tails are soluble by Proposition 11. If the second cover is an equal-type merge, the first must split; the only unbounded-rank maximal-subgroup cases with such an active split are the geometric block and tensor cases described after Proposition 16. The merge stays inside each original upper coordinate. Its possible indices are already covered by (iv).

The inactive nonabelian labels in a shortcut belong to the pool in (iv). To see this, write its local inner group as \(P\trianglelefteq K\), with soluble killed kernel \(Q\) and active chief factor \(R/Q\cong T^a\). The faithful conjugation argument in Proposition 11 gives \(P/Q\hookrightarrow\operatorname{Aut}(T^a)\). The kernel of the factor-permutation map on \(P/R\) lies in \(\operatorname{Out}(T)^a\) and is soluble. Since \(P\) is normal in \(K\), its permutation image is normal in the full local permutation image. Lemma 19 identifies that full image with an image of one of the projected stabilizers counted in (iv). Its normal subgroup therefore has its nonabelian composition types in the same pool. Thus all nonabelian composition types outside the active chief factor \(R/Q\) lie in this pool. The active type \(T\) can still vary along the chain.

Apply now the structural observations following Proposition 16. There are boundedly many choices of inactive simple labels by (iv), and the repetition indices also have boundedly many values. The resulting omitted dimensions therefore have boundedly many possible values. When all inactive components are soluble, their individual natural dimensions are absolutely bounded. This proves (v) and all the assertions passed to the next section. ◻

Proof. For the dimension and field statements, take the full stabilizers of the indicated structures. On a nondegenerate direct summand the active group has its natural module; on a totally singular summand the dual summand has the contragredient module. A tensor stabilizer acts as the tensor product of the component natural modules. Restriction of scalars multiplies dimension by its degree \(b\) and gives \(\epsilon_s f_s=b\epsilon_t f_t\) for the natural matrix fields. For example, the mixed \(\operatorname{GU}_m(q)\) row in \(\operatorname{Sp}_{2m}(q)\) has \(b=2\) and equal usual exponents, since \(\epsilon_s=2\) and \(\epsilon_t=1\). It is a restriction of scalars, with a direct sum after scalar extension, and is not a twisted-tensor row. A subfield inclusion changes only the field. If an ordinary field degree were composite, its proper intermediate field structure would be preserved by the normalizer, contradicting maximality. The classical stabilizer descriptions with prime degree are given explicitly in (Burness et al. 2017, proofs of Lemmas 5.5 and 5.7). Unitary conjugation and the diagram maps have bounded order, which accounts for the permitted bounded ratios in the remaining field operations. The twisted-tensor assertion is proved separately in Lemma 28 below.

The only soluble classical component groups have bounded natural dimension: a linear group in growing dimension contains a growing special linear simple section, and the corresponding assertions for the form groups follow by their natural hyperbolic subspaces. Thus the omitted dimensions in a one-active-component subspace or tensor stabilizer are bounded. The explicit block description of a parabolic gives a radical of order \(q^e\), with \(e\leq Cn_t^2\) and \(e\geq c n_t\) whenever that radical is nontrivial in the retained large-rank cases. For example, the linear radical attached to a \(k\)-space has exponent \(k(n_t-k)\); the symplectic and orthogonal exponents count the off-diagonal blocks subject to one symmetric or alternating relation. The smallest nonzero retained exponent is linear in \(n_t\). The other factors are the displayed classical groups and scalar groups. Consequently their orders are determined by the dimensions, field, form signs, and the bounded choices of interchanged flags. Intersecting the full stabilizer with the ambient socle adds only its prescribed determinant or spinor condition; it does not introduce a freely chosen subgroup of a scalar group. This proves [cls:cat-inactive].

For a symplectic-type normalizer put \(N=n_t=r^d\), and let \(W\) be the projective image of its normal \(r\)-group, of order \(r^{2d}\). The full inner-normalizer rows, including their minimal-field conditions, are given in (Bray et al. 2013, sec. 2.2.6, pp. 70–71, Table 2.9). Their small exceptions disappear on retaining \(d\geq4\). Ordinary linear conjugation fixes scalars pointwise, so it preserves the nondegenerate scalar commutator pairing \(W\times W\to\mu_r\). Moreover the projective linear centralizer of \(W\) is \(W\): a centralizing operator twists the representing matrices by a character of \(W\), that character is implemented by an element of \(W\) through the pairing, and the remaining centralizer is scalar by irreducibility.

In the linear and unitary rows, the normal group is an exponent-\(r\) extraspecial group for odd \(r\), and a central product \(C_4\circ2^{1+2d}\) for \(r=2\). In both cases the ordinary projective normalizer has quotient \(\operatorname{Sp}_{2d}(r)\) on \(W\). Indeed every symplectic change of generators lifts to a center-fixing automorphism and is implemented by the irreducible representation with its specified central character. For \(r=2\), multiplying a generator by the central element of order four restores its required square; thus a square form attached to a pure extraspecial subgroup is not an invariant of this scalar-extended normalizer. In the unitary case an implementing operator preserves the Hermitian form up to a multiplier, which is removed by a scalar using the surjective finite-field norm. The standard tensor generators of \(W\) have determinant one: for odd \(r\) this holds for the cyclic shift and diagonal root-of-unity generators, and for \(r=2\) their determinants are \((-1)^{2^{d-1}}=1\). The map from this normalizer to the abelian quotient of the projective linear or unitary group by its simple socle therefore factors through \(\operatorname{Sp}_{2d}(r)\). That group is perfect in the retained range, so the map is trivial. Hence the full ordinary projective normalizer is already the inner intersection \(P\). Quotienting by the entire ambient scalar center leaves kernel exactly \(W\), and \[P/W\cong\operatorname{Sp}_{2d}(r),\qquad |P|/|T_s|=(2,r-1)N^2.\] This uses the quotient, without requiring the extension to split.

The pure symplectic and orthogonal form rows have \(r=2\). Here the extraspecial square form is invariant. The cited inner rows give \(P/W=O_{2d}^{\varepsilon}(2)\) or \(\Omega_{2d}^{\varepsilon}(2)\), with active simple group \(T_s=\Omega_{2d}^{\varepsilon}(2)\) and index at most two. These are the inner rows after the determinant and spinor conditions; the scalar kernel of the natural cover is removed in \(W\). Consequently \(N^2\leq |P|/|T_s|\leq2N^2\) in every retained row. There is no extra factor from \(r-1\) or the ambient field exponent.

The target-field bound uses the minimal-field condition of the maximal row. For odd \(r\), the degree \(e=\operatorname{ord}_r(p)\) divides \(r-1\); the usual target exponent is \(e\) in the linear row and \(e/2\) in the unitary row. The binary and pure form rows have usual exponent one. Thus \(f_t\leq\max(1,r-1)\leq |T_s|\). The representation characteristic is different from \(r\), and the active source is over the prime field of order \(r\). This proves [cls:cat-symplectic].

For an absolutely irreducible representation of a central cover \(\widetilde T_s\), the dimension is at most \(|\widetilde T_s|^{1/2}\): the square of each simple-module dimension is bounded by the dimension of the group algebra. The field-of-definition bound can be obtained directly, including in modular characteristic. For a finite group \(K\) in characteristic \(p\), there are at most \(|K|\) absolute simple module types, by the semisimple quotient of its group algebra. Frobenius permutes these types, so an absolute simple representation has a Frobenius orbit of length \(a\leq |K|\). An isomorphism with its \(p^a\)-Frobenius twist descends the representation to \(\mathbb F_{p^a}\): choose a finite field containing its matrices and an intertwiner, scale the intertwiner to have norm one, and apply matrix Hilbert’s Theorem 90. The scalar norm adjustment is possible because finite-field norms are surjective. Apply this with \(K=\widetilde T_s\). The Schur multipliers of the large classical groups have order at most a polynomial in \(|T_s|\); the finitely many exceptional multipliers do not change the bound. Forms cost an extension of degree at most two. Maximality outside the subfield collection makes these bounds applicable to the target field itself. For the cross-characteristic lower bound, the classical tables in (Häsä 2014, sec. 3, Tables 3–6) give degree at least \(q_s^r/3\) once the source rank \(r\geq6\), including the smaller Weil degrees. Their excluded families have bounded rank. As \(n_s\leq2r+1\), this is at least \(q_s^{n_s/4}\) after increasing the rank threshold. This proves the numerical assertions of [cls:cat-cross].

For the defining-characteristic quadratic bound, use (Lübeck 2001, Theorem 5.1 and Table 2) in rank \(r>11\). Modules outside that table have dimension greater than \(r^3/8\) in type \(A\), and greater than \(r^3\) in types \(B,C,D\). Every listed nontrivial, nonnatural, nondual degree is at least \(r^2/2\). Consequently the retained restricted module has dimension at least \(r^2/2\geq n_s^2/10\). Type \(B\) in characteristic two is read through its type-\(C\) natural simple quotient, as in our parameter convention; the natural-size special-isogeny cases are already assigned to geometry.

A single occupied ordinary Frobenius digit has full source period: any shift strictly between zero and \(f_s\) moves it to a different position in the unique finite restricted label. Returning through \(f_s\) applies at most the bounded diagram operation. An internal tensor decomposition of that digit changes neither its occupied position nor this uniqueness argument. Thus the minimal scalar field degree is \(f_s\) times one of the bounded diagram periods. The matrix degree is separately \(\epsilon_t f_t\). The form-compatible minimal-field argument in Lemma 27 bounds the ratio of this matrix degree to the minimal degree outside the geometric collections. This gives the fixed finite rational set of usual exponent ratios and proves [cls:cat-nonlinear]. Lemma 27 below supplies the final normalizer assertion. ◻

Proof. Lift the irreducible projective representation to the perfect central cover of \(Y\). A projective linear element centralizing its image commutes with that representation up to a linear character of the perfect cover, hence commutes exactly. Schur’s Lemma makes it scalar, so it disappears projectively.

If a semilinear element in this kernel has scalar-field action \(\sigma\), the representations \(\rho^\sigma\) and \(\rho\) are isomorphic. If \(\sigma\) has order \(a\) on the scalar field \(E\), its intertwiner \(A\) satisfies \[A\sigma(A)\cdots\sigma^{a-1}(A)=cI, \qquad c\in E^{\sigma,*}.\] Surjectivity of the scalar norm adjusts \(A\) so that the product is one; matrix Hilbert’s Theorem 90 then gives a model over \(E^\sigma\). An invariant bilinear form is a line defined by linear equations over the minimal field, and invariant quadratic-form coefficients satisfy linear equations there as well. In characteristic two, two invariant quadratic forms with the same polarization differ by the square of an invariant linear form. Absolute irreducibility makes that difference zero, so the prescribed quadratic form itself descends up to scalar. Matching finite-field form classes costs at most a quadratic extension. For a unitary group the involution on the minimal field is either nontrivial, giving Hermitian descent and surjective norms, or trivial, giving a preserved bilinear structure and hence a geometric form case. A larger field period would therefore provide a proper subfield structure normalized by the whole irreducible normalizer, contrary to this step being outside the subfield collection.

To justify the assertion about the whole normalizer, put an absolutely irreducible model over its minimal field \(E_0\). A normalizing matrix intertwines this model with the model obtained by a source automorphism and a field automorphism; both have entries in \(E_0\). The intertwining equations therefore have coefficients in \(E_0\). Their solution space over the algebraic closure is one-dimensional by absolute irreducibility. It consequently has a nonzero \(E_0\)-rational solution, necessarily invertible. Up to a scalar, the normalizer thus preserves the projective \(E_0\)-structure. The same argument applies after dualizing in a graph component, and the uniqueness of the invariant form gives the already allowed quadratic extension. A graph operation contributes at most another factor of two in unbounded rank. This bounds the kernel independently of the ambient field degree.

Finally the diagonal, field, and graph description of automorphisms gives respectively factors at most \(n_Y\), \(f_Y\), and an absolute constant in the outer group. ◻

Proof. Let \(S\) be the simple source, \(M\) the almost simple maximal subgroup, and \(T\) the ambient simple classical group. Work over an algebraic closure on the cover supplied by Lemma 44, and write the irreducible module as \[V=\bigotimes_{i\in I}V_i, \qquad \mathcal A_i=\operatorname{End}(V_i)\otimes1 \ \leq\operatorname{End}(V),\] where \(V_i\) are the nonzero ordinary restricted Frobenius digits, with their indicated twists. The dimensions of these nontrivial digits tend to infinity with the source rank.

We first verify preservation of the actual tensor structure. A source field automorphism translates ordinary digit positions, applying the diagram operation when a position passes the source period; inner, diagonal, and diagram automorphisms act on the restricted modules at those positions. Ambient semilinear operations give the same shifts, and a graph operation dualizes the modules. Uniqueness of the finite restricted highest-weight parametrization matches the resulting whole digit modules. Tensor their individual intertwiners, with the matching permutation, to obtain an intertwiner of \(V\). The discrepancy between a projective and linear intertwining identity is a linear character of the perfect cover and is trivial. Absolute irreducibility makes the space of whole-module intertwiners one-dimensional. Every actual normalizing operator therefore differs from this tensor intertwiner by a scalar and permutes the embedded algebras \(\mathcal A_i\). For a graph operation the corresponding map on endomorphisms is an anti-isomorphism, which has the same conclusion. This argument preserves a whole ordinary digit even if that restricted module has an internal tensor decomposition in a very special characteristic.

Replace each digit by its full connected special linear group, or by its full connected bilinear-form group when required. Bilinear self-duality makes each ordinary digit self-dual separately. Unitary descent acts on the special linear factors, permitting duals. In characteristic two every nontrivial irreducible self-dual digit has an alternating form: a nonalternating invariant form would give an invariant linear functional by taking square roots of its diagonal. The tensor product of at least two nondegenerate alternating spaces has a quadratic form with the product polarization and value zero on pure tensors (Liebeck and Seitz 1998, Proposition 2.2(iv)). It agrees with the prescribed finite quadratic form, after a scalar normalization of the polarization: their difference is the square of an invariant linear form, which vanishes by nontrivial absolute irreducibility. These constructions give a connected semisimple tensor subgroup \(K\) in the ambient form group.

The groups are intrinsic to the algebras \(\mathcal A_i\) and their restricted adjoint involutions. Hence the actual ambient Frobenius \(F\) and the full finite normalizer preserve \(K\). In growing classical rank, finite ambient automorphisms have lifts commuting with \(F\); diagonal lifts satisfy this projectively because \(F(x)x^{-1}\) is central. Graph and field lifts also commute, with only the bounded-rank triality and exceptional graph cases removed. This is the setup of (Liebeck and Seitz 1998, Theorem 2 and Section 4). Henceforth view \(K\), and later \(K'\), in the ambient adjoint group. Thus \(M\) normalizes the terminal derived subgroup \(H=(K^F)^{(\infty)}\). Since \(H\) is perfect it lies in \(T\), and \(S\leq H\) because \(S\) is a perfect subgroup of \(K^F\).

For each orbit of \(F\) on the factors, fixed points are obtained by choosing one coordinate fixed by the corresponding power of \(F\); see (Liebeck and Seitz 1998, Proposition 2.8(ii)). Consequently \(H\) is a central product of classical components, one per orbit. It is proper in \(T\) also as a finite group. To check this uniformly, put \(q=p^{f_t}\), let the individual digit dimensions be \(d_1,\ldots,d_k\), put \(d_{\min}=\min_i d_i\), and write \(n_t=\prod_i d_i\). The earlier bound for nontrivial digits gives \(d_{\min}\to\infty\) with the source rank. Elementary classical order bounds give \[|H|\leq q^{2\sum_i d_i^2},\qquad |T|\geq q^{n_t^2/8}.\] For \(k\geq2\), \(\sum_i d_i^2/n_t^2\leq k/d_{\min}^{2(k-1)}\leq2/d_{\min}^2\). Thus \(H<T\) for all sufficiently large source ranks. If \(MH\) were the entire almost simple ambient group, its normal subgroup \(H\leq T\) would equal \(T\). Hence \(MH\) is proper, and maximality gives \(H\leq M\). But the perfect group \(H/S\) is a subgroup of the soluble group \(M/S\), so \(H=S\). There is therefore one Frobenius orbit, and its full natural classical component is the source. If its length is \(b\), the digit dimension is \(n_s\) and \(n_t=n_s^b\) exactly.

The field relation is read from this actual natural-factor cycle. Put \(t=f_t\). For a linear target, \(F^b\) on one factor is the standard field map with exponent \(bt\), up to an inner conjugacy. For a unitary target, write \(F=F_{p^t}\tau\) with \(\tau\) the inverse-transpose operation. On one factor the return map is \(F_{p^{bt}}\tau^b\), again up to an inner conjugacy. Its source is unitary when \(b\) is odd and linear when \(b\) is even, in both cases with usual exponent \(bt\). An orthogonal graph choice changes the form sign, not this exponent. The characteristic-two quadratic construction likewise changes no field exponent. Thus \[ f_s=b f_t \tag{19}\] exactly, in the convention of this paper. The natural factors have positions \(0,t,\ldots,(b-1)t\) modulo \(bt\), which are distinct. For the odd unitary convention using positions \(2jt\), multiplication by two permutes the same residues modulo \(b\); the prime two-cycle in a unitary target has a linear source. These are precisely the usual-field alternatives in (Liebeck and Seitz 1998, sec. 4(5)).

Finally suppose \(b=uv\), with \(u,v>1\). Number the factors by \(\mathbb Z/b\mathbb Z\), and group the cosets of its unique subgroup of order \(u\). The normalizer’s permutation normalizes the regular cyclic descent action and preserves this block system. The algebra for a block \(B\) is the algebra generated by its \(\mathcal A_i\), namely \(\operatorname{End}(\bigotimes_{i\in B}V_i) \otimes1\). Replace each block by its full special linear or bilinear-form group. For a characteristic-two quadratic target use the full symplectic group on each grouped alternating space. The resulting quadratic tensor form is unchanged: it has the same polarization and vanishes on every fully pure original tensor, and such tensors span \(V\). Hence this grouped subgroup \(K'\) still preserves the prescribed ambient form and is invariant under \(F\) and \(M\).

Its terminal derived fixed-point subgroup \(H'\) has one classical component on dimension \(n_s^u\) over the field with usual exponent \(v f_t\), with the unitary parity or orthogonal sign prescribed by the return map. For \(n_s\geq16\), the elementary bounds \(Q^{d^2/8}\leq|\operatorname{Cl}_d(Q)|\leq Q^{2d^2}\) give \[|S|\leq q^{2uv n_s^2} <q^{v n_s^{2u}/8}\leq|H'|, \qquad |H'|\leq q^{2v n_s^{2u}} <q^{n_s^{2uv}/8}\leq|T|.\] The strict inequalities follow from \(n_s^{2u-2}>16u\) and \(n_s^{2u(v-1)}>16v\). As before \(M\) normalizes \(H'\) and \(S\leq H'\). Its properness in \(T\) forces \(H'\leq M\), and perfectness then forces \(H'=S\), contradicting the first inequality. Therefore \(b\) is prime. ◻

Proof. Write the actual shortcut as \(U=z_i\prec V\prec W=z_j\), and work modulo \(F_W\). In one \(W\)-coordinate the last cover has \(Y\trianglelefteq P_A\le\operatorname{Aut}(Y)\). Its killed kernel from the middle is soluble by Proposition 25 and normal in \(P_A\), hence is trivial. The active subgroup in that coordinate is therefore the actual simple group \(Y\). Proposition 11 gives a \(V\)-equivariant bijection \(\beta:I_V\longrightarrow I_W\) because the multiplicities agree. For \(\alpha\in I_V\) and \(t=\beta(\alpha)\), the actual stabilizers in \(U\) coincide: \(U_\alpha=U_t\). Let \(\rho_V,\rho_W\) be their middle and upper coordinate actions, and let \(\phi:N_{\operatorname{Aut}(T_j)}(Y)\longrightarrow\operatorname{Aut}(Y)\) be conjugation. The quotient-induced actions satisfy \[\phi\circ\rho_W|_{U_t}=\rho_V|_{U_\alpha}, \qquad \phi(M_{ij})=M_B,\] where \(M_B\) is the actual first-cover coordinate stabilizer image.

The inner-intersection assertions use the original lower endpoint \(a\) from Definition 13. All retained comparisons start strictly above this \(a\), so none uses an excluded middle \(v_{0j}\). Condition [cls:test-alternating] makes \(a\) an intersection of coatoms in \([a,V]\) and \([a,W]\) for every retained middle and upper endpoint. Proposition 10 in these original-\(a\) projections identifies the inner part of a saturated image with its transported inner projection. For \(U\prec V\) the coatom case of that Proposition also applies directly. In particular, \[M_B\cap\operatorname{Inn}(Y)=P_B.\] There is no assumption that a meet of coatoms in the original global interval is a meet of coatoms in every smaller interval. The bounded kernel from Lemma 27 now gives \[|M_{ij}|\le C|M_B| \le C|P_B|\,|\operatorname{Out}(Y)| \le C|P_B|n'f',\] which implies (21). This includes diagonal, field, and graph operations. All higher-core losses are already kernels of the displayed actual coordinate actions; no uncontrolled factor from the field of the final target is inserted. ◻

Proof. Let \(S\) contain the primes occurring in the numerators or denominators of members of \(\mathcal C\). For \(\ell\in S\), the integer \(v_\ell(x_j/x_i)\) lies in a fixed bounded interval: the contribution of \(c_{ij}\) is bounded, and that of \(u_{ij}\) has absolute value at most \(h\). Outside \(S\) all valuations have sign \(\varepsilon\), and the sum of their absolute values is at most \(h\). Color \((i,j)\) by the valuation vector on \(S\) and this last sum. This is a finite coloring with the asserted uniformity.

For \(i<j<k\) we have \[v_\ell(x_k/x_i)=v_\ell(x_j/x_i)+v_\ell(x_k/x_j).\] On a homogeneous triangle each coordinate of the common valuation vector therefore equals twice itself, so it is zero. Outside \(S\) there is no cancellation because all valuations have the same sign. Their total absolute value likewise equals twice itself, hence is zero. Every valuation of every pair ratio is zero, so all ratios equal one. Finite Ramsey gives the last assertion. ◻

Proof. Color first by the order and kind of the two steps. By [grw:input-fields], their combined dimension formula is either \[n_j=b_{ij}n_i\quad\hbox{for every }i<j, \qquad\hbox{or}\qquad n_j=n_i^{b_{ij}}\quad\hbox{for every }i<j,\] where \(b_{ij}>1\) is prime or belongs to a fixed bounded set of integers. In the first case put \(x_i=n_i\), and in the second put \(x_i=\log n_i\). In either case \(x_j/x_i=b_{ij}\) has uniformly bounded prime complexity. Lemma 30, with \(\mathcal C=\{1\}\), gives a homogeneous triangle with \(b_{ij}=1\), a contradiction. Notice that field ratios alone would not suffice: ratios \(r_j/r_i\) of distinct primes are compatible on arbitrarily many vertices. It is the exact dimension formula that excludes this possibility. ◻

Proof. Color by the two cover categories, the endpoint family, and whether a split and merge occur. There are finitely many colors. The endpoint characteristic is common because every step is in defining characteristic. If both an increasing and a decreasing exceptional field move occur, Lemma 31 applies. Otherwise all exceptional primes in a pair ratio have the same sign. The product of the two bounded rational factors belongs to a fixed finite set, and at most two exceptional degrees occur. A bounded exceptional degree has uniformly bounded prime complexity as well. Apply Lemma 30 to \(x_i=f_i\). We may therefore assume that all endpoint fields and families agree.

The exceptional degrees at the individual covers are now bounded integers. Indeed, with no opposite field moves their product, or its reciprocal, is a member of the fixed finite set of bounded twisting ratios, because \(f_j/f_i=1\). Each positive degree divides the numerator or denominator of one of these finitely many rationals.

Fix a lower endpoint \(i\). We show that its upper natural dimension has only boundedly many possible values. For block and parabolic shapes, the ambient dimension is the active dimension, or twice that dimension when its dual is present, plus the dimensions of the omitted blocks. The dimensions of nonabelian omitted blocks are determined, with bounded ambiguity, by the bounded-cardinality pool of inactive types. Soluble omitted blocks have bounded dimensions. A tensor shape multiplies the active dimension by a product of the omitted dimensions. A replicated base multiplies it by a replication degree from its bounded-cardinality pool. A form change has the same natural dimension, with the bounded natural-module exceptions included in the family color. Field restriction multiplies dimension by one of the bounded degrees just obtained, and a twisted-tensor step raises it to one of those degrees. Thus one geometric cover gives a bounded list of possible outputs from a fixed input. Two covers give a list whose size is at most the product of the two bounds. For a split followed by a merge the second step has the same simple label, so gives no new natural dimension at all.

All labels now have the same family and field. Their strict growth in order makes their natural dimensions distinct. There cannot be more upper endpoints than the bounded number of possible outputs from the fixed lower endpoint. This contradicts arbitrary length. ◻

Proof. The active factor \(T_i\) is a composition factor of \(P_{ij}\), so its order divides \(|P_{ij}|\). Saturation and supplementation give \[[z_j:z_i]=[T_j^m:P_{ij}^m] =\left(\frac{|T_j|}{|P_{ij}|}\right)^m.\] Multiplying group indices through \(z_j\) and taking the positive \(m\)th root gives \[\frac{|T_k|}{|P_{ik}|} =\frac{|T_k|}{|P_{jk}|}\frac{|T_j|}{|P_{ij}|}.\] Rearranging and dividing by \(|T_i|\) proves the formula. The same calculation applies at a shortcut because its three multiplicities agree. No identification of its intermediate label with a chain label is required. ◻

Proof. For \(U>z_i\), the original test [cls:test-normal] provides a fenced interval or a labelled bottleneck at \(U\). If \(U=z_i\), the endpoint itself is labelled. Since \(F_j\trianglelefteq z_j\) and \(F_j\le D\), apply Lemma 5 or Corollary 6 in the quotient by \(F_j\), with bottom \(D/F_j\). The soluble-normal conclusion gives (22). For ([grw:normal-commuting]), the full inverse image of \(\overline K\) is normal in this same \(U\) and contained in \(D\), so the commuting-normal conclusion kills one product. Since \(D\le z_i\), a killed product has its coordinate subgroup in \(P_{ij}\). ◻

Proof. Use the natural-space field \(E=\mathbb F_{p^e}\); thus \(e=f\), or \(e=2f\) in the unitary case. Apart from the bounded-rank graph exceptions already excluded, automorphisms in the form cases are represented by projective semilinear similarities. In the linear case there may also be correlations, which interchange the natural space and its dual. The ordinary orthogonal graph operation is a similarity in this description; characteristic-two type \(B\) uses its symplectic natural quotient.

First suppose that \(M\) acts on the natural-space side. Choose one semilinear representative \(s_g\) for each projective element, with \(s_1=1\). The products satisfy \(s_gs_h=c(g,h)s_{gh}\) for nonzero scalars \(c(g,h)\in E\). For an unknown vector \(v\in E^n\) impose \[B(s_gv,s_hv)=0\quad(g,h\in M)\] in a bilinear or Hermitian form case, and impose also \(Q(s_gv)=0\) in a quadratic form case. In a plain linear space no equations are necessary. Field automorphisms and semilinear maps are \(\mathbb F_p\)-linear. These are therefore at most \(e(m^2+m)\) scalar polynomial equations of degree at most two in \(en\) prime-field variables. Their sum of degrees is less than \(en\).

For completeness, the elementary polynomial count giving a nonzero solution is as follows. If \(F_1,\ldots,F_t\) have total degree sum less than the number \(N\) of prime-field variables, the number of their common zeros, reduced modulo \(p\), is \[\sum_{x\in\mathbb F_p^N} \prod_{a=1}^t(1-F_a(x)^{p-1}).\] Every expanded monomial has degree less than \(N(p-1)\), so some variable has exponent less than \(p-1\). The sum over that variable is zero in \(\mathbb F_p\), including exponent zero. The displayed sum is zero. All our equations vanish at zero, so their common zero set has another element. This is the needed Chevalley–Warning argument (Chevalley 1935; Warning 1935).

Choose such a nonzero \(v\) and put \[U=\operatorname{span}_E\{s_gv:g\in M\}.\] Then \(1\le\dim U\le m\). Scalar discrepancies between representatives do not change this span, so \(U\) is \(M\)-stable. The imposed equations make \(U\) totally singular in the form cases. In particular, in characteristic two we used the quadratic equations as well as the polar equations. Thus the stabilizer of \(U\) is a proper parabolic normalized by \(M\).

Now let \(T\) be linear and suppose \(M\) contains correlations. Put \(H=M\cap\operatorname{P\Gamma L}(V)\) and choose \(d\in M\setminus H\). On subspaces the correlation has the form \[d(S)=\operatorname{ann}(F(S))\] for an invertible semilinear map \(F:V\longrightarrow V^*\). Choose representatives \(s_h\), \(h\in H\), and impose \[F(s_hv)(s_kv)=0\qquad(h,k\in H).\] Again there are at most \(e|H|^2\) quadratic scalar equations. The same count gives \(v\ne0\). Let \(U=\operatorname{span}_E\{s_hv:h\in H\}\) and \(W=d(U)\). The equations say \(U\le W\). Since \(H\) is normal in \(M\), both subspaces are \(H\)-stable. Moreover \(d(W)=d^2(U)=U\), because \(d^2\in H\). Thus the flag \(U<W\) is preserved by all of \(M\), as an unordered pair under correlations. Its dimensions are \(r,n-r\), where \(1\le r\le |H|<n/2\), so it is a proper flag. Its parabolic stabilizer is normalized by \(M\). No involution assumption on \(d\) has been used.

In these constructions the small subspace has dimension \(1\le r\le m<n/4\) once \(n\) is sufficiently large. The upper triangular block matrices in the radical supply at least \(n/4\) independent prime-parameter root coordinates, each of order at least \(q\). For example the linear case has \(r(n-r)\) coordinates, and the flag case has at least \(r(n-2r)\); in the polar cases the block from the nondegenerate middle space to the singular \(r\)-space supplies at least \(r(n-2r-2)\) coordinates. These lower bounds are at least \(n/4\) after enlarging the fixed dimension threshold. We may take a smaller absolute \(c_0\) to cover all the families. Projectivization does not remove unipotent coordinates: the scalar center has order prime to \(p\).

Let \(J\) be this proper parabolic and \(R\) its radical. The group \(R\) is soluble and normalized by \(M\). In the coordinate application, take the actual enlargement (24). Its inner part is \[\overline U\cap S_j=(P_{ij}R)^{I_j},\qquad P_{ij}R\le N_{T_j}(J)<T_j.\] The last inequality holds because \(J\) is a nontrivial proper subgroup of the simple group \(T_j\). Thus \(U<z_j\): properness only in \(\operatorname{Aut}(T_j)\) would not suffice. The soluble coordinate normal test gives \(\widehat R\le D/F_j\), hence \(R\le P_{ij}\le M_{ij}\). Its order contradicts \(q_j^{c_0n_j}>|M_{ij}|\). ◻

Proof. Every ascending same-characteristic geometric step preserves both natural dimension and \(n\log q\) up to fixed positive lower factors, as established in [grw:input-degrees]. The same holds for a same-characteristic nonlinear step by its quadratic degree bound and fixed positive field ratio. If the last cover crosses, a preceding same-characteristic cover leaves \(n'\log q'\) at least a fixed multiple of \(n_i\log q_i\). A preceding crossing has the same consequence from \(n'\ge\exp(c n_i\log q_i)\) and \(\log q'\ge\log2\) at sufficiently large rank. The last crossing then gives the endpoint lower bound. If only the first cover crosses, it gives \(n'\ge\exp(c n_i\log q_i)\), and the last cover preserves natural dimension up to a fixed factor. Decreasing \(c\) and increasing the fixed minimum-rank threshold, we obtain in every case \[n_{i+1}\ge\exp(c n_i\log q_i).\] Write \(u=n_i\log q_i\). Classical order formulas give \[\log a_i\le C_1 n_i^2\log q_i\le C_2u^2.\] After the first step every additional comparison gives at least the map \(x\mapsto F(x)=\exp(c\log(2)x)\). For fixed \(C_3\), \[\frac{F(\exp(cu))}{\exp(C_3u^2)}\longrightarrow\infty \qquad(u\longrightarrow\infty).\] If \(L(u)/R(u)\longrightarrow\infty\) and \(R(u)\longrightarrow\infty\), then \(F(L(u))/\exp(R(u))\longrightarrow\infty\) as well. Induction therefore shows that \(h+2\) comparisons dominate \(\mathcal E_h(\exp(C_3u^2))\), and hence \(\mathcal E_h(a_i^C)\) when \(C_3\ge CC_2\). This also proves the last assertion. The target prime characteristic need not be bounded in this argument: its only use in the iteration was \(\log q\ge\log2\). ◻

Proof. Suppose otherwise, and color by the two step categories. Every pair in the resulting arbitrarily long chain has a crossing. There are three possibilities for its last step \(A\).

The last step is geometric in the same characteristic. The first step \(B\) crosses. Hence \(n',f'\le a_i^C\) by (20). An ordinary geometric last step changes dimension by bounded factors, except for an extension degree. By (17) that degree is \(b=\epsilon'f'/(\epsilon_jf_j)\le2f'\). A last subfield degree can be arbitrarily large but leaves dimension unchanged. A twisted-tensor last step has exponent \(b=f'/f_j\le f'\) by (18). Therefore \[n_j\le (C n')^{C(1+f')}+C(1+f')n'+C \le\exp(a_i^{C'})\] for a fixed \(C'\). Lemma 36 contradicts this bound on a sufficiently distant pair.

The last step is symplectic type. Its source \(Y\) is a symplectic or orthogonal group over a prime field, say of order \(r\), and its target dimension is \(r^{d}\) with \(2d+O(1)=n'\). First suppose \(B\) is in the same characteristic. Then \(r=p_i\) and \(f'=1\). In a subfield first step the formula \(1=\rho f_i b\), with \(\rho\) in a fixed finite set of positive rationals, bounds both \(f_i\) and \(b\); in the untwisted formula such a proper step is impossible. An ordinary extension degree in \(B\) satisfies \(b=\epsilon_i f_i\le2f_i\), because the target here has \(f'=1\) and \(\epsilon'=1\); a twisted-tensor exponent satisfies \(b=f_i\). The ordinary geometric formulas give a fixed-height tower bound on \(n'\) in terms of \(a_i\). If \(B\) is nonlinear, its irreducible degree is at most \(a_i^C\), giving the same conclusion. Since \(p_i\le a_i\), the formula \(n_j=p_i^{d}\) gives \(n_j\le\mathcal E_h(a_i^{C'})\) for fixed \(h,C'\). This contradicts Lemma 36.

It remains in this case that \(B\) also crosses. Let \(b_{ij},s_{ij}\) be the inner multipliers of \(B,A\). By [grw:input-multipliers], \[1\le b_{ij}\le a_i^C, \qquad c n_j^2\le s_{ij}\le C n_j^2, \qquad e_{ij}=b_{ij}s_{ij}.\] The bound for \(b_{ij}\) is the outer-order bound when \(B\) is almost simple; in the symplectic-type case it follows from \(b_{ij}\le C(n')^2\le a_i^{C'}\). For \(i<j<k\), Lemma 33 now gives \[c n_j^2\le e_{ij} =\frac{b_{ik}s_{ik}}{b_{jk}s_{jk}} \le\frac{C}{c}a_i^C.\] The factors involving the common upper dimension \(n_k\) have canceled. This contradicts Lemma 36 on a sufficiently distant pair \(i<j\).

The last step is almost simple irreducible. Let its source be \(Y\). If \(B\) crosses, then \(n',f'\le a_i^C\) and \(|P_B|\le a_i^C\): for an almost simple \(B\) this uses \(|P_B|\le a_i|\operatorname{Out}(T_i)|\), and for a symplectic-type \(B\) it uses the preceding bound on its multiplier. Equation (21) gives \[ |M_{ij}|\le a_i^{C'}. \tag{25}\] It is irrelevant here whether \(A\) itself is a crossing.

Suppose next that \(B\) is in the same characteristic, so that \(A\) crosses. If \(B\) is not a subfield step, its geometric dimension formulas, or its irreducible degree bound when nonlinear, bound \(n'\) by a fixed-height tower in \(a_i\). Its field satisfies \(\log q'\le C\log q_i\): restriction of scalars contributes the bounded factor \(\epsilon_i/\epsilon'\), and the other non-subfield rows have their fixed field factors. Thus \(|Y|\) is bounded by a fixed-height tower in \(a_i\), and \(n_j\le |Y|^C\) contradicts Lemma 36.

Finally suppose \(B\) is subfield. Its rank equals the lower rank and its characteristic equals \(p_i\). Its inner projection has order at most \(a_i^C\), but \(f'\) can be unbounded. The last crossing gives \[\log n_j\ge c n' f'\log p_i, \qquad f'\le C\log n_j.\] Equation (21), with \(n'\) comparable to \(n_i\), therefore gives \[ |M_{ij}|\le a_i^{C'}(1+\log n_j)^{C'}. \tag{26}\] The same upper bound includes (25). By Lemma 36, after a fixed separation we have \(a_i^{C'}\le n_j^{1/16}\); and, for sufficiently large rank, \((1+\log n_j)^{C'}\le n_j^{1/16}\). Hence \(|M_{ij}|\le n_j^{1/8}\). This satisfies the dimension inequality of Lemma 35, and \(q_j^{c_0n_j}>n_j^{1/8}\ge |M_{ij}|\). That Lemma gives the final contradiction.

The three possibilities exhaust the last cover in a crossing path. ◻

Proof. Propositions 32 and 37 leave a defining-characteristic nonlinear cover on every comparison, with the common characteristic of the endpoints. Color by endpoint family and cover categories. A nonlinear cover has a bounded rational field ratio; only the other cover can carry an exceptional prime degree. Lemma 30 applied to \(f_i\) therefore gives equal endpoint exponents. The exceptional degree at the remaining cover must cancel a member of a fixed finite set of rational twisting ratios, so it is bounded. The intermediate ratios consequently belong to a fixed finite set.

The nonlinear cover increases natural dimension at least quadratically. All remaining geometric steps increase it by at least a fixed positive factor. Thus its position as first or last cover makes no difference to (27), after decreasing \(c\). ◻

Proof. Use Lemma 38, and color by the position of the parabolic cover. The other cover is nonlinear.

If the parabolic cover is first, its active dimension is \(n_i\) and its ambient dimension is comparable to \(n_i\). Its inner quotient is at most \(q^{C n_i^2}\). The multiplier of the last nonlinear cover is at most \(|\operatorname{Out}(Y)|\), which is absorbed by this bound, since \(n'\) is comparable to \(n_i\) and \(f'/f\) is bounded. Hence \[ e_{ij}\le q^{C_1 n_i^2}\qquad\hbox{for every }i<j. \tag{28}\] For \(j<k\) the radical in its first cover also gives \(e_{jk}\ge q^{c_1 n_j}\), with the bounded field ratio absorbed in \(c_1\). By Lemma 33, \[q^{c_1 n_j}\le e_{jk}\le e_{ik}\le q^{C_1 n_i^2}.\] Iterating (27) through a fixed number of intermediate chain vertices makes \(n_j/n_i^2\) tend to infinity as the minimum rank grows. The displayed inequalities are incompatible.

If the parabolic cover is last, write \(e_{ij}=b_{ij}s_{ij}\), where \(b_{ij}\le|\operatorname{Out}(T_i)|\) is the nonlinear multiplier and \(s_{ij}\) is the parabolic multiplier. For a fixed upper endpoint \(k\), the values \(s_{ik}\) have a uniformly bounded number of possibilities by [grw:input-multipliers]. List those values in increasing order and color \((i,k)\) by its position in that list. This gives a uniform finite coloring even though the lists for different \(k\) may be different. On a homogeneous triple, \(s_{ik}=s_{jk}\), and therefore \[e_{ij}=\frac{e_{ik}}{e_{jk}} =\frac{b_{ik}}{b_{jk}} \le |\operatorname{Out}(T_i)|.\] But the parabolic radical on the pair \((i,j)\) gives \(e_{ij}\ge q^{c_2n_j}\). The classical outer-order bound is \[|\operatorname{Out}(T_i)|\le C n_i f \le q^{C_2(1+\log n_i)},\] so even (27) contradicts these inequalities at sufficiently large rank. ◻

Proof. There is at least one nonlinear cover. For a remaining geometric cover with active label \(T\) and natural dimension \(n\), the other classical factors have bounded natural dimension and are soluble. Their orders, scalar factors, and bounded component-permutation factors are bounded by \[q^{C_0}(1+n)^{C_0}(1+f)^{C_0}.\] Field and twisted-tensor degrees are bounded by Lemma 38; their semilinear factors satisfy the same bound. These are precisely the block, form, tensor, and field structures left after Lemma 39; no unipotent radical or extra nonabelian factor is being omitted from the estimate.

If the geometric cover is first, its intermediate natural dimension is at most \(C_1(1+n_i)^{b_1}\) for fixed constants. Its multiplier has the displayed bound in the lower parameters, and the last nonlinear multiplier is at most \(|\operatorname{Out}(Y)|\le Cn'f'\). Since \(|T_i|\le q^{C n_i^2}\) and \(f\le q\), their product satisfies (29).

If the nonlinear cover is first, its intermediate degree is at most \(a_i^{C_2}\) and \(f'/f\) is bounded. The geometric multiplier is bounded by \(q^{C_3}(1+n')^{C_3}(1+f')^{C_3}\), which is at most \(q^{C_4 n_i^2}\). Multiplying by \(a_i\) and the first nonlinear outer multiplier proves the same bound. With two nonlinear covers only the two outer multipliers occur, and the latter argument applies without the geometric factor. ◻

Proof. Parts (i)–(iv) collect Propositions 32 and 37 and Lemmas 38, 39, and 40. For the spacing assertion put \(x_i=c n_i\). Then (27) says \(x_j\ge x_i^2\) on every consecutive comparison. After \(s\) comparisons, \(n_j\ge c^{-1}(c n_i)^{2^s}\). Choose \(2^s>d\) and then increase the minimum rank to obtain \(n_j\ge n_i^d\).

The coordinate normal test is inherited from the original tested intervals and is not altered by deleting chain vertices. Every proper parabolic of a large-rank classical group has unipotent radical of order at least \(q^{c_3 n_j}\) for an absolute \(c_3>0\). One can see the uniform bound by choosing a maximal parabolic containing it: the omitted simple root has at least the relative rank many positive-root coordinates outside the Levi subsystem. Their product contributes at least \(q^{c_3 n_j}\); unitary root orbits can only increase the corresponding root-group orders. The radical of the smaller parabolic contains that of the maximal one. Fixed-spacing thinning makes \(c_3n_j>Cn_i^2\) uniformly. If \(M_{ij}\) normalized such a parabolic \(J\) with radical \(R\), use (24). Since \(P_{ij}R\le N_{T_j}(J)<T_j\), its full inverse image is proper in \(z_j\). The soluble product \(\widehat R\trianglelefteq\overline U\) then satisfies (22). In particular \(R\le P_{ij}\), contrary to \(|R|\ge q^{c_3 n_j}>q^{C n_i^2}\ge|P_{ij}|\). This proves (v). ◻

Proof. Set \(Q=P_0\cap E_1\) and \(A=\pi(Q)\). The group \(\overline P_0/A\) is a quotient of \(P_0/Q\), which embeds in \(P_1/E_1\), so it is soluble. A perfect group has trivial image in a soluble group. Consequently \(\overline E_0\leq A\). Let \(B\) be its full inverse image in \(Q\). Then \(B\lhd P_0\) and \[B/Z_0\cong\overline E_0, \qquad Z_0=B\cap\ker\pi=B\cap Z(E_1)\leq Z(B).\] Because \(B/Z_0\) is perfect, \(B=B'Z_0\), and therefore \(B'=[B'Z_0,B'Z_0]=B''\). Thus \(B'\) is perfect. The inverse image in \(B\) of \(Z(\overline E_0)\) has central commutators, and its intersection with \(B'\) is central in \(B'\): for an element of this inverse image, commutation with \(B'\) is a homomorphism from the perfect group \(B'\) to the abelian group \(Z_0\). It is trivial. It follows that \(B'\) is a perfect central extension of the nonabelian simple group \(\overline E_0/Z(\overline E_0)\), hence is quasisimple.

The quotients \(P_0/Q\), \(Q/B\cong A/\overline E_0\), and \(B/B'\) are soluble. Hence \(P_0/B'\) is soluble. Since \(B'\) is perfect, it equals \(P_0^{(\infty)}\). Finally \(B'\leq Q\leq E_1\), and the terminal derived subgroup is characteristic. ◻

Proof. Steinberg’s multiplier Theorem has only a finite list of exceptions for the split groups, and the accompanying twisted statement has another finite list (Steinberg 1981, Theorem 1.1 and the opening paragraph of p. 528). Their ranks are bounded, independently of the field size. We supply perfectness separately: the root-presentation statement (Steinberg 1967, Lemma \(32'\), original pp. 51–53) applies to the universal split groups, and (Steinberg 1967, Theorem 34 and its proof, original pp. 187–188) gives the twisted case outside its finite list of exceptions. The root-generated groups used there are the full fixed-point groups for simply connected groups over finite fields: see (Steinberg 1967, Corollary 3 to Theorem 7, original p. 65, and the Corollary after Lemma 64, original p. 186). Increase the discarded rank bound beyond these finite exception lists. Thus \(H_1(U,\mathbb Z)=H_2(U,\mathbb Z)=0\) and \(u\) is the usual central simple quotient. This argument retains every field size, including \(q=2,3,4\).

Let \(A=\ker a\leq Z(L)\). Pull back \(a\) along \(u\) to obtain the central extension \[1\longrightarrow A\longrightarrow U\times_S L \longrightarrow U\longrightarrow1.\] The action on \(A\) is trivial. The universal coefficient sequence \[0\longrightarrow \operatorname{Ext}^1_{\mathbb Z}(H_1(U,\mathbb Z),A) \longrightarrow H^2(U,A) \longrightarrow\operatorname{Hom}(H_2(U,\mathbb Z),A) \longrightarrow0\] gives \(H^2(U,A)=0\). The extension splits, and projection of a splitting to \(L\) supplies \(\pi\). Two such maps differ by a homomorphism \(U\to A\), which is zero because \(U\) is perfect. If \(H=\pi(U)\), then \(L=HA\) and \(A\) is central. Therefore \[L=[L,L]=[HA,HA]=[H,H]\leq H,\] so \(\pi\) is onto. Its kernel is contained in \(\ker u\leq Z(U)\).

For a projective representation, compose with \(\pi\) and pull back \(\operatorname{GL}(V)\to\operatorname{PGL}(V)\). This is a central extension of \(U\) by \(k^\times\). The same cohomology argument splits it; perfectness gives uniqueness of the resulting linear lift. ◻

Proof. Put \(I=\operatorname{ind}_{\mathbf G^\sigma}^{\mathbf G}k\). The induction filtration of (Bendel et al. 2015, arXiv version, Proposition 3.1.2), obtained from the coordinate-ring filtration and the Lang map, gives \(I\) a filtration with sections \[\nabla(\lambda)\otimes\nabla(\lambda^*)^\sigma, \qquad \lambda\in X^+,\] each once. Write such a section as \(\nabla(\lambda)\otimes\nabla(\mu)^{[f]}\), where \(\mu\) is obtained from \(\lambda^*\) by the diagram automorphism. Choose the regular height \[h(\xi)=\left\langle\xi,\sum_{\alpha>0}\alpha^\vee\right\rangle.\] It is invariant under diagram automorphisms and duality, is positive on every nonzero dominant weight and every positive root, and satisfies \(h(\mu)=h(\lambda)\).

We first justify removing the actual constants from this filtration. A nonconstant section cannot contain a trivial submodule. Indeed a map from \(k\) into it would transpose to a nonzero map \(\Delta(\lambda^*)\to\nabla(\mu)^{[f]}\). The image contains the simple socle \(L(q\mu)\), so its highest weight must occur in the source. This gives \(qh(\lambda)\leq h(\lambda)\), impossible when \(\lambda\neq0\). Let \(I_j\) be the first filtration stage containing the actual one-dimensional space of constants. That space injects into \(I_j/I_{j-1}\), so this section is the unique section for \(\lambda=0\). The constants split it and \(I_j=I_{j-1}\oplus k\). The images of the earlier stages in \(I/k\), followed by \(I_i/k\) for \(i\geq j\), give a filtration of \(I/k\); omitting the repeated stage removes exactly the constant section. Thus its sections are precisely the nonconstant ones.

We prove that a nonzero \(q\)-restricted simple \(L(\nu)\) cannot map to a nonconstant section. Suppose otherwise. Transposing the second tensor factor gives a nonzero homomorphism \[L(\nu)\otimes\Delta(\mu^*)^{[f]} \longrightarrow\nabla(\lambda).\] Its image contains the simple socle \(L(\lambda)\). Filter the Weyl module by simple composition factors and apply Steinberg’s tensor product Theorem (Steinberg 1963, Theorems 5.1 and 6.1). Since \(\nu\) is \(q\)-restricted, each simple composition factor of the source has highest weight \(\nu+q\eta\) for some dominant \(\eta\). Consequently \[ \lambda=\nu+q\eta,\qquad h(\lambda)\geq h(\nu). \tag{30}\] Transposing the first tensor factor instead gives a nonzero homomorphism \[ \Delta(\lambda^*)\otimes L(\nu) \longrightarrow\nabla(\mu)^{[f]}. \tag{31}\] Its image contains \(L(q\mu)\), so the weight \(q\mu\) occurs in its source. The maximum height of a weight of that source is \(h(\lambda^*)+h(\nu)=h(\lambda)+h(\nu)\). Thus \[ qh(\lambda)\leq h(\lambda)+h(\nu). \tag{32}\] Equations (30) and (32) are inconsistent when \(q>2\).

When \(q=2\), both inequalities are equalities. Hence \(\eta=0\) and \(\lambda=\nu\). The only weight of maximum height in the source of (31) is \(\lambda^*+\nu\), and its weight space is the one-dimensional span of the tensor \(a\otimes b\) of highest-weight vectors. If \(2\mu\neq\lambda^*+\nu\), the required weight does not occur. Otherwise the homomorphism must be nonzero on \(a\otimes b\). Choose a simple root \(\alpha\) with \(\langle\nu,\alpha^\vee\rangle=1\), which is possible because \(\nu\) is nonzero and \(2\)-restricted. For the Chevalley raising and lowering operators, \[e_\alpha(a\otimes f_\alpha b)=a\otimes b.\] The target of (31) is Frobenius twisted, so its Lie algebra action is zero. Every module homomorphism to it kills the displayed vector, a contradiction.

A homomorphism from a finite-dimensional simple module to \(I/k\) has image in a finite stage of the filtration. Taking the first stage with nonzero intersection with that simple image gives a nonzero map into one of its sections. We have proved \[\operatorname{Hom}_{\mathbf G}(L(\nu),I/k)=0.\] Apply this also to the dual of the coefficient module. Tensoring \(0\to k\to I\to I/k\to0\) with \(L(\nu)\) shows that the map from \(H^1(\mathbf G,L(\nu))\) to \(H^1(\mathbf G,L(\nu)\otimes I)\) is injective. The tensor identity for induction and derived Frobenius reciprocity identify this with the asserted restriction map. For split groups this is also the special case \(U=k\) of (Cline et al. 1977, Theorem 7.4); the argument above includes the diagram twist and the equality case \(q=2\). ◻

Proof. The polar form \(B\) of \(Q\) is nonzero: otherwise \(Q\) is the square of a linear form, and that linear form would be invariant under the nontrivial irreducible finite action. Its radical is invariant, so \(B\) is nondegenerate. Thus the finite representation is self-dual. The injective parametrization by \(q\)-restricted highest weights in Steinberg’s finite-group restriction Theorem (Steinberg 1967, Theorem 43) implies that \(V\cong V^*\) rationally. Schur’s Lemma, first for \(\mathbf G\) and then for \(\mathbf G^\sigma\), shows that their spaces of invariant bilinear forms are both one-dimensional and agree. In particular \(B\) itself is \(\mathbf G\)-invariant.

Polarization gives an exact sequence of rational modules \[ 0\longrightarrow(V^*)^{[1]} \longrightarrow\operatorname{Sym}^2(V^*) \longrightarrow\bigwedge^2(V^*)\longrightarrow0. \tag{33}\] The first map sends a twisted linear form to its square. The obstruction to refining \(B\) by a rationally invariant quadratic form is its connecting class \[c_B\in H^1(\mathbf G,(V^*)^{[1]}).\] This class restricts to zero on \(\mathbf G^\sigma\), because \(Q\) is a refinement there. This description is also (Garibaldi and Nakano 2016, Proposition 3.6 and Corollary 3.9).

Let \(\mathbf G_1\) be the first Frobenius kernel. Its action on \((V^*)^{[1]}\) is trivial. The inflation–restriction sequence therefore begins \[0\longrightarrow H^1(\mathbf G/\mathbf G_1,(V^*)^{[1]}) \longrightarrow H^1(\mathbf G,(V^*)^{[1]}) \longrightarrow \bigl(H^1(\mathbf G_1,k)\otimes(V^*)^{[1]}\bigr) ^{\mathbf G/\mathbf G_1}.\] The required first-kernel calculation is \[H^1(\mathbf G_1,k)^{[-1]}\cong \begin{cases} N,&\mathbf G\cong\operatorname{Sp}_{2n},\ n\geq1, \quad N\text{ its natural module},\\ 0,&\text{otherwise}. \end{cases}\] It is reproduced in the proof of (Garibaldi and Nakano 2016, Proposition 7.1, p. 411), citing (Bendel et al. 2004, Theorem 3.1(C)(a),(f)). Here the symplectic case includes \(A_1=C_1\) and \(B_2=C_2\). The exceptional module is \(L(\omega_1)\) in \(C_n\) numbering, including \(n=1\), and \(L(\omega_2)\) in \(B_2\) numbering. It is simple and self-dual. After untwisting, the last invariant space is therefore zero under our hypothesis. Inflation, followed by untwisting the quotient by \(\mathbf G_1\), gives an isomorphism \[H^1(\mathbf G,V^*)\xrightarrow{\ \sim\ } H^1(\mathbf G,(V^*)^{[1]}).\]

Write \(c_B=F^*c\), where \(F\) is ordinary Frobenius. Put \(\Gamma=\mathbf G^\sigma\) and \(M=V^*\). Since \(F\) commutes with \(\sigma\), its restriction \(F_\Gamma\) is an automorphism of \(\Gamma\). Naturality gives the commutative square \[\begin{array}{ccc} H^1(\mathbf G,M)&\xrightarrow{\ F^*\ }&H^1(\mathbf G,M^{[1]})\\ \operatorname{res}_M\downarrow&&\downarrow\operatorname{res}_{M^{[1]}}\\ H^1(\Gamma,M)&\xrightarrow{\ F_\Gamma^*\ }& H^1(\Gamma,F_\Gamma^*M). \end{array}\] The lower arrow is an isomorphism, with inverse obtained by pulling back along \(F_\Gamma^{-1}\) with the transported coefficients. This does not identify \(M\) with its Frobenius twist. Hence the vanishing of \(\operatorname{res}(c_B)\) implies that of \(\operatorname{res}(c)\). This and Lemma 45, applied to \(V^*\), imply \(c=0\) and \(c_B=0\). Sequence (33) supplies a \(\mathbf G\)-invariant quadratic refinement \(Q_0\) of \(B\).

Finally \(Q-Q_0\) has zero polarization, hence is the square of a linear form \(\ell\). Its finite invariance implies that \(\ell\) is invariant under \(\mathbf G^\sigma\). Such a linear form is zero on this nontrivial irreducible module. Therefore \(Q_0=Q\). ◻

Proof. A line stabilizer fixes \(v\), because a scalar preserving its nonzero quadratic value has square one, and the field has characteristic two. Choose a complement \(U\) to \(\langle v\rangle\) in \(v^\perp\) and choose \(t\) with \(B(v,t)=1\) and \(B(t,U)=0\). The space \(U\) is symplectic. An isometry inducing the identity on \(\overline W\) sends \(u\in U\) to \(u+\ell(u)v\). Preservation of \(Q\) forces \(\ell(u)^2Q(v)=0\), so it fixes \(U\) pointwise. Preservation of \(B\) then forces its image of \(t\) to be \(t+av\). Preservation of \(Q\) says \(a^2Q(v)+a=0\), hence \(a=0\) or \(a=Q(v)^{-1}\). These are exactly the identity and \(r_v\).

For completeness, let \(s\in\operatorname{Sp}(U)\). The difference \(Q(u)+Q(su)\) has zero polarization, so, because the field is perfect, there is a unique linear form \(\ell\) satisfying \[\ell(u)^2Q(v)=Q(u)+Q(su).\] The map \(v\mapsto v\), \(u\mapsto su+\ell(u)v\) is an isometry of \(v^\perp\). Witt’s extension theorem (Taylor 1992, Theorem 7.4, pp. 57–58) extends this isometry to \(W\): the ambient polar radical is zero, so its radical-intersection criterion is automatic even though \(v^\perp\) has degenerate polar form. This proves surjectivity. The formula for \(r_v\) also shows that it commutes with every isometry fixing \(v\).

The Dickson invariant of an isometry in even characteristic is \(\operatorname{rank}(g-1)\) modulo two; in particular it is one on \(r_v\). Its zero kernel is \(\Omega(W,Q)\) over a finite field of even order (Taylor 1992, Theorems 11.43 and 11.51, pp. 160, 164–165); the exceptional group \(\Omega^+(4,2)\) is outside the dimension range here. Each fibre of the displayed surjection has two elements differing by \(r_v\), so exactly one lies in this kernel. The last finite assertion follows.

We also justify the algebraic assertion, including its scheme convention. Put \(W_0=v^\perp\). Restriction defines \[H=\operatorname{SO}(W,Q)_v\longrightarrow \operatorname{SO}(W_0,Q|_{W_0}).\] The kernel of restriction from the full orthogonal vector stabilizer is the constant group of order two computed above: over an arbitrary \(k\)-algebra its remaining parameter satisfies \(a+a^2Q(v)=0\), an étale equation. Its nonidentity element has Dickson invariant one, so the displayed map has trivial scheme-theoretic kernel. We give its inverse over an arbitrary \(k\)-algebra \(A\). Put \(c=Q(v)\) and retain the decomposition \(W=kv\oplus U\oplus kt\) used above. An element \(h\in\operatorname{SO}(W_0)(A)\) has symplectic quotient action, of determinant one; the odd special orthogonal determinant condition therefore makes it fix \(v\). Write \(h(u)=su+\ell(u)v\) for \(u\in U_A\). There is a unique \(w\in U_A\) such that \(B(su,w)=\ell(u)\) for all \(u\). The extension fixing \(v\) and restricting to \(h\) is obtained by setting \[g(t)=t+w+av,\qquad ca^2+a+Q(w)=0.\] The bilinear condition is \(B(su,w)+\ell(u)=0\), and the quadratic condition is exactly the displayed equation. Its derivative is one and its leading coefficient is a unit, so adjoining a solution gives an étale double cover. The two extensions differ by \(r_v\) and have opposite Dickson invariants. The unique Dickson-zero extension on this cover agrees on overlaps and descends to \(A\). This is a functorial inverse, proving the asserted scheme isomorphism. The odd special orthogonal group is smooth, connected, and adjoint of type \(B_r\) by (Conrad 2014, author version, Theorem C.2.11 and Proposition C.3.10). This restriction calculation is also the one in the proof of (Conrad 2014, author version, Proposition C.3.1). Its action on \(W_0/\langle v\rangle\) is the purely inseparable isogeny to the symplectic group described in (Conrad 2014, author version, Remark C.3.6); consequently this quotient is the simple module \(N_{B_r}\). Both outer factors of the flag are trivial. We used the vector stabilizer: the line-stabilizer scheme \(\operatorname{SO}(W,Q)_{\langle v\rangle}\) is \(H\times\mu_2\). Indeed its scalar on \(v\) has square one, and multiplying by the inverse central scalar makes it fix \(v\); the two factors have trivial intersection. Its reduced subgroup and its finite-field points are the ones just described. ◻

Proof. Finite components and their normalization. We first specify the components in the structures of Proposition 26. A nondegenerate block or form subgroup contains the derived classical group on each active block. A tensor subgroup contains the central product of the derived groups on its tensor factors. An extension-field or subfield subgroup contains the derived classical group over its indicated field. In the retained twisted tensor construction, the single Frobenius orbit of the tensor factors has fixed-point group equal to the classical group over the corresponding extension field; its derived group is the active quasisimple group. The almost-simple case contains its quasisimple socle preimage.

The reductions leave one nonabelian active factor. Hence every permitted normalizer preserves it. All other base factors commute with it modulo scalars, and their surviving images are soluble. The retained diagonal, field, graph, and factor-permutation quotients are soluble as well; the exclusion of other nonabelian factors here includes factors in the permutation quotient, not just base factors. Thus the explicitly displayed active derived group is normal and the quotient is soluble. Scalar preimages introduce only central kernels: if \(C/Z\) is the displayed perfect active factor, with \(Z\) central, then \(C=C'Z\) and \(C'=C''\), giving its quasisimple derived preimage. For the characteristic-two nonsingular-line case, Lemma 48 gives the asserted component directly. For a two-edge projection, we verify the precise hypothesis of Lemma 42. Use the common upper-coordinate notation \(P_0\leq P_1\) and \(Q_1\leq R_1\leq P_1\) from Lemma 20. Here \(Q_1\) is soluble, \(R_1/Q_1\) is the middle simple active factor, and \(P_1/R_1\) is soluble. The upper retained edge supplies the quasisimple component \(E_1=P_1^{(\infty)}\). Perfectness gives \(E_1\leq R_1\). Its image in \(R_1/Q_1\) is nontrivial, since \(Q_1\) is soluble, and is normal, hence is the full simple group. Therefore \[E_1Q_1=R_1,\qquad E_1\cap Q_1=Z(E_1).\] Both \(E_1\) and \(Q_1\) are normal in \(P_1\). Thus, for \(x\in Q_1\), commutation with \(x\) maps \(E_1\) into \(E_1\cap Q_1=Z(E_1)\). It is a homomorphism to an abelian group, so perfectness makes it trivial. Consequently \(Q_1\) centralizes \(E_1\). Conjugation on \(E_1\) agrees with conjugation on \(E_1/Z(E_1)\cong R_1/Q_1\): an automorphism of the perfect group \(E_1\) acting identically modulo its centre differs from the identity by a homomorphism \(E_1\to Z(E_1)\) and is therefore the identity.

The original-core product identity in Proposition 11 gives \(Q_1\leq P_0\). Indeed its transported product comes from the middle original core, which is contained in \(D\) and hence in the lower labelled group. It follows that \[J=(P_0\cap R_1)/Q_1\] is the actual first-edge inner projection of Lemma 20. Its retained structure supplies the normal quasisimple component \(J^{(\infty)}\) with soluble quotient. This is a structural assertion about this particular \(J\); the excluded parabolic or module semidirect extensions have not been inferred to split.

Let \(\pi:P_1\to\operatorname{Aut}(E_1)\) be conjugation. By the centralization and simple-quotient identification just proved, \(\pi(P_0\cap R_1)\) is the inner image of \(J\), isomorphic to \(J\). It is normal in \(\pi(P_0)\), and \(\pi(P_0)/\pi(P_0\cap R_1)\) is soluble because it is a quotient of \(P_0/(P_0\cap R_1)\leq P_1/R_1\). Hence the image of \(J^{(\infty)}\) is normal quasisimple in \(\pi(P_0)\), with soluble quotient. These are exactly the hypotheses of Lemma 42, which produces the actual lower component inside \(E_1\).

Finally, if \(P\leq P'\) and \(L'\) is the component of \(P'\), the image of the perfect group \(L\) in the soluble group \(P'/L'\) is trivial. Hence \(L\leq L'\). Characteristicity gives normalization by the common bottom. Each intermediate inner projection is normal in its own coordinate stabilizer, which contains the lower coordinate stabilizer; its characteristic component is therefore normalized by that lower stabilizer too.

To check that the active component is not killed, use the notation \(Q\leq R\leq P\) of Proposition 11. In our multiplicity-one coordinate, \(R/Q\) is the nonabelian simple active factor and \(P/R\) is soluble. Hence \(L\leq R\). Since \(P/L\) is soluble, the simple group \(R/Q\) has trivial image modulo \(LQ\), and \(LQ=R\). Thus \(L\) maps onto \(R/Q\). Transporting \(L\) over all upper coordinates gives a normal subgroup of the lower labelled group, mapping onto its socle modulo its original bottom core. If this product were contained in the common bottom, normality would put it inside that core, a contradiction.

For each actual finite component \(L_i\), fix its standard cover \(U_i=\mathbf G_i^{\sigma_i}\) and the surjection \(\pi_i:U_i\twoheadrightarrow L_i\) of Lemma 44, over the specified simple quotient.

Rational maps for the geometric edges. We have obtained the finite components, their nesting, and their survival under the original bottom-core quotient. We now lift their inclusions. For the geometric cases we describe the maps on natural modules over \(k\), which also proves the assertion about their composition factors. Write \(N\) for the source natural simple module.

1. A block embedding acts on its active block by \(g\), on a dual block, when present, by \(g^{-\mathsf T}\), and trivially on omitted spaces. Its constituents are \(N\), \(N^*\), and trivial modules. A form embedding is the natural action with its prescribed invariant bilinear or quadratic form. These are rational matrix maps.

2. For a tensor factor with an omitted space \(U\), the map is \(g\mapsto g\otimes1_U\). On the active factor its natural module is the direct sum of \(\dim U\) copies of \(N\), or of a prescribed dual or Frobenius twist of \(N\). The commuting omitted factor changes the multiplicity space, not this formula.

3. Restriction of scalars from \(\mathbb F_{p^{ab}}\) to \(\mathbb F_{p^a}\), followed by extension of scalars to \(k\), has the explicit decomposition \[k\otimes_{\mathbb F_{p^a}}\mathbb F_{p^{ab}} \cong\bigoplus_{j=0}^{b-1}k, \qquad x\longmapsto(x,x^{p^a},\ldots,x^{p^{a(b-1)}}).\] Thus its map is the direct sum of the rational representations \(N^{[aj]}\). If a form pairs two summands, the corresponding summand is expressed as the required dual. A subfield inclusion is the same natural matrix map, possibly precomposed with its specified field or diagram automorphism. On natural modules in the ranks in use, a diagram automorphism gives \(N\) or \(N^*\); triality has already been discarded with bounded rank.

4. In a twisted tensor orbit the map is \[g\longmapsto \rho_0(F^{a_0}g)\otimes\cdots\otimes\rho_{b-1}(F^{a_{b-1}}g),\] where each \(\rho_j\) is natural or dual. This is a rational representation and it is the defining matrix construction of the finite subgroup. Its natural module is \(\bigotimes_j N_j^{[a_j]}\), \(N_j\in\{N,N^*\}\). The orbit lengths, field degrees, and omitted tensor dimensions are bounded by Proposition 41.

In characteristic two, two further classical maps must be interpreted literally. The nonsingular-vector stabilizer in a group of type \(D_{r+1}\) has connected reduced subgroup of type \(B_r\); its action on the \(2r+2\)-dimensional natural space has, by Lemma 48, the invariant filtration \[0<\langle v\rangle<v^\perp<W\] with factors \(k,N_{B_r},k\). Its simply connected cover therefore supplies the required \(B_r\)-to-\(D_{r+1}\) map. The middle symplectic action is an isogeny from this adjoint \(B_r\) stabilizer to \(\operatorname{Sp}_{2r}\). It has no nontrivial kernel on \(k\)-points, consistently with Lemma 48; it is not a finite unipotent radical. In the following display, instead, \(\mathbf B_r=\operatorname{Spin}_{2r+1}\) and \(\mathbf C_r=\operatorname{Sp}_{2r}\) denote the simply connected groups. The preceding vector stabilizer is the adjoint quotient of \(\mathbf B_r\); the kernels of its symplectic action and of the simply connected maps are not being identified. For the two special isogenies \[s:\mathbf B_r\longrightarrow\mathbf C_r, \qquad t:\mathbf C_r\longrightarrow\mathbf B_r, \qquad st=F_p,\quad ts=F_p,\] one has \[s^*N_C=N_B,\qquad t^*N_B=N_C^{[1]}\] by (Dowd and Sin 1996, author preprint, §I.3.2). If the inverse of \(s\) on \(\mathbb F_q\)-points is needed, it is induced by the rational map \(tF_p^{f-1}\), since \(q=p^f\). No inverse inseparable morphism is asserted. These formulas preserve the claimed description of simple constituents, including the trivial factors of the full odd orthogonal module.

Rational maps for the nonlinear edges. For a defining-characteristic almost-simple edge, inflate its actual finite irreducible representation along \(\pi_i:U_i\twoheadrightarrow L_i\), using the unique linear lift of Lemma 44 if the natural action was given projectively. Surjectivity preserves every invariant subspace and every prescribed form. A form similitude multiplier is a character of the perfect group \(U_i\), hence is trivial. Only now apply Steinberg’s restriction Theorem (Steinberg 1967, Theorem 43): the resulting \(U_i\)-module is the restriction of its \(q_i\)-restricted rational simple module, with the specified field and diagram twist. Here \(\sigma_i=F_{p^{f_i}}\gamma_i\) uses \(f_i\) ordinary Frobenius digits, including in the unitary case; the unitary natural scalar field \(\mathbb F_{p^{2f_i}}\) does not change this restriction parameter. This yields a rational map to \(\operatorname{GL}(V)\) whose determinant is trivial: a connected semisimple group has no nontrivial rational character. Thus it maps to a linear target, including the algebraic group underlying a finite unitary target. In a symplectic target or an orthogonal target in odd characteristic, finite self-duality and the injective highest-weight parametrization identify the invariant rational bilinear form with the finite form, by Schur’s Lemma. Its symmetry or alternating property is consequently the prescribed one. For an orthogonal target in characteristic two the remaining nonlinear module satisfies Proposition 46, which lifts its actual quadratic form. The natural and special-isogeny cases are the geometric maps already constructed. The image is connected and hence lies in the identity component of the isometry group.

Compatibility through the fixed central covers. These constructions initially give maps into whichever central form of the target is specified by its natural action. A homomorphism from a simply connected semisimple source lifts through a central isogeny of semisimple algebraic groups; see (Conrad 2014, author version, Exercise 6.5.2(iii), p. 236). Thus they give maps between simply connected covers as claimed. Let \(\phi_i:\mathbf G_i\to\mathbf G_{i+1}\) be such a map and let \(\iota_i:L_i\hookrightarrow L_{i+1}\) be the actual finite inclusion. By construction their induced maps to the target adjoint group agree on \(U_i\). For \(u\in U_i\), the target adjoint image of \(\phi_i(u)\) is fixed by \(\sigma_{i+1}\), so \[\delta(u)=\sigma_{i+1}(\phi_i(u))\phi_i(u)^{-1} \in Z(\mathbf G_{i+1})(k).\] Centrality gives \(\delta(uv)=\delta(u)\delta(v)\). Perfectness of \(U_i\) forces \(\delta=1\), so \(\phi_i(U_i)\leq U_{i+1}\), including when the algebraic centre is nonreduced. The two homomorphisms \(\pi_{i+1}\phi_i|_{U_i}\) and \(\iota_i\pi_i\) to \(L_{i+1}\) agree modulo its centre. Their discrepancy is consequently a homomorphism \(U_i\to Z(L_{i+1})\), and again is trivial. Thus we have the exact commutative diagram \[ \begin{array}{ccc} U_i&\xrightarrow{\ \phi_i|_{U_i}\ }&U_{i+1}\\ \pi_i\downarrow&&\downarrow\pi_{i+1}\\ L_i&\xrightarrow{\ \iota_i\ }&L_{i+1}. \end{array} \tag{34}\] This proves compatibility with the actual finite inclusions through the fixed central-cover maps. It gives no uniqueness assertion for rational maps agreeing on finite points. ◻

Proof. The finite compatibility statements for the edges compose, so the first assertion follows by induction on \(j-i\). Every subspace invariant under the algebraic image is invariant under the contained finite image. Finite irreducibility therefore implies algebraic irreducibility, without a density assertion or a bound on the highest weight of the composite representation. The kernel of a nontrivial map from a connected simple algebraic group is finite as a group scheme: its connected reduced normal subgroup cannot be the whole source. Its image is closed and connected, and is simple up to the allowed central form. Inseparability can interchange the types \(B\) and \(C\) in characteristic two, but does not change rank. The last assertion follows from the rank inequality. ◻

Proof. Work over the algebraic closure. Any proper submodule or nontrivial composition filtration for the \(L\)-action would be an \(H\)-invariant one. Thus the target module is a single nontrivial composition factor for \(L\). Proposition 49 expresses it as a tensor product of twists of the source natural module or its dual. If one factor had a nonzero proper \(H\)-invariant subspace, tensoring that subspace with all the other factors would give a nonzero proper invariant subspace of the product. Hence each factor is irreducible on \(H\). Taking a dual or applying a field Frobenius to all matrix entries preserves irreducibility, so the source natural module itself is irreducible on \(H\). ◻

Proof. This is Proposition 41(v), with its fixed-spacing thinning already incorporated in the retained chain. ◻

Proof. Work first over the finite natural scalar field. Put \[S=\operatorname{soc}_E(V),\quad R_0=\operatorname{rad}_E(V),\quad U=S\cap R_0,\quad W=S+R_0.\] If \(V\) is not semisimple, \(R_0\ne0\), and a simple submodule of \(R_0\) shows that \(U\ne0\). Also \(W<V\): the proper submodule \(S\) is contained in a maximal submodule, which necessarily contains \(R_0\). Socles and radicals are intrinsic, so every semilinear normalizer preserves these constructions.

For a linear group without graph operations, \(U\) gives an \(M\)-normalized parabolic, contrary to Lemma 53. Suppose that \(M\) also contains duality operations. The side-preserving subgroup has index two. A duality acts on subspaces by a semilinear correlation, and its transport of a module to the dual exchanges the socle with the annihilator of the radical, and the radical with the annihilator of the socle. Consequently it exchanges \(U\) and \(W\) as members of a flag. Thus the flag \(U\le W\) is preserved by all of \(M\); if \(U=W\) it is a one-member flag. Both members are nonzero and proper. This again contradicts Lemma 53. This argument uses correlations on subspaces and does not treat a graph automorphism as a linear operator on \(V\).

In a bilinear or Hermitian space, identify the module with its appropriate dual using the preserved form. Duality of socles and radicals gives \[R_0^\perp=S,\qquad S^\perp=R_0.\] Hence \(U\) is totally isotropic. This supplies a forbidden parabolic in the symplectic, unitary, and odd-characteristic orthogonal cases.

It remains to consider a quadratic form \(Q_V\) in characteristic two. On a totally polar-isotropic space \(U\) its restriction has the form \(\ell^2\), for a linear functional \(\ell\), since the finite field is perfect. Its zero space is an \(M\)-invariant singular subspace. Unless \(U\) is a nonsingular line, this zero space is nonzero and contradicts Lemma 53. We are therefore left with the nonsingular line \(R=U\).

The space \(R^\perp/R\) has a nondegenerate alternating form. If its \(E\)-module were not semisimple, the argument with socle and radical would give a nonzero \(M\)-invariant totally isotropic subspace \(A\) of that quotient. Its inverse image \(B\) in \(R^\perp\) is totally polar-isotropic and contains \(R\). On \(B\) the quadratic form is the square of a linear functional nonzero on \(R\). Its zero hyperplane is therefore a singular complement to \(R\), projecting isomorphically onto \(A\). This complement is \(M\)-invariant and again gives a forbidden parabolic in the original quadratic space. The quotient is semisimple.

The flag \(0<R<R^\perp<V\) adds precisely two one-dimensional composition factors to that quotient. Both are trivial for \(E\) because \(E\) is perfect. By Lemma 48, the inner stabilizer has the canonical finite-group isomorphism \[\theta:\Omega(V,Q_V)_R\xrightarrow{\sim} \operatorname{Sp}(V^\circ).\] Every \(m\in M\) induces a semilinear similarity \(m^\circ\) on \(V^\circ\). Evaluation on quotient vectors gives \[\theta(mhm^{-1})= m^\circ\theta(h)(m^\circ)^{-1} \qquad(h\in\Omega(V,Q_V)_R).\] Consequently inverse images under \(\theta\) preserve normalization by the actual \(M\), inclusions, and exact commutation of inner subgroups. Since \(P=M\cap T_j\) preserves \(R\), it lies in this inner stabilizer. Write \(P^\circ=\theta(P)\) and \(E^\circ=\theta(E)\); then \(P^\circ/E^\circ\cong P/E\). Every candidate section subgroup is pulled back through \(\theta\) before applying a normal test. Every lifted enlargement lies in the proper original line stabilizer, even if its symplectic image is the whole symplectic group. Transport over the original coordinates uses the same normal higher core \(F_j\) in ([grw:normal-soluble]) and ([grw:normal-commuting]). There is no additional finite kernel in this inner-group isomorphism.

Finally, extending the finite scalar field to its algebraic closure preserves semisimplicity here. Indeed the semisimple image algebra of the finite group is a product of matrix algebras over finite fields; those fields are separable, and scalar extension preserves semisimplicity of that algebra. ◻

Proof. Write \(C=C_{T_j}(E)\) in the original natural realization. The group \(MC\) normalizes \(E\) and cannot contain \(T_j\): the proper nontrivial \(E\) cannot be normal in the simple ambient group. Hence the full preimage of \(\overline U=\langle z_i/F_j,\widehat E,\widehat C\rangle\) is proper in \(z_j\). The two products \(\widehat E,\widehat C\) are normal in \(\overline U\) and commute exactly. Their full preimages commute modulo \(F_j\trianglelefteq U\), with \(F_j\le D\). The active product \(\widehat E\) is not killed, so ([grw:normal-commuting]) kills \(\widehat C\) and forces \(C\le P\). Since \[C\cap E\le Z(E),\qquad C/(C\cap E)\hookrightarrow P/E,\] \(C\) is soluble.

For the symplectic section of Lemma 54, put \(E^\circ=\theta(E)\) and take the inverse image under \(\theta\) of \(C_{\operatorname{Sp}(V^\circ)}(E^\circ)\). The \(M\)-equivariance and injectivity of \(\theta\) show that this inner inverse image and \(E\) commute exactly and are normal in their enlargement with \(M\). Their transported products again commute in \(N_j/F_j\), and their original preimages commute modulo the normal subgroup \(F_j\le D\). The enlargement lies in the proper original nonsingular-line stabilizer even if its symplectic image is the full symplectic group. Condition ([grw:normal-commuting]) and the same quotient calculation prove that the symplectic centralizer is soluble. ◻

Proof. We give the linear-algebra details needed for a bound independent of the fields and constituent dimensions. Let \(U\) be a simple module over the finite natural field \(F\). Its endomorphism division ring is a finite field \(F'\). On an isotypic summand \(U^m\) the commuting endomorphism algebra is \(\operatorname{Mat}_m(F')\). After scalar extension, \(U\) splits into distinct absolute simples, each once; thus \(m\) is exactly their common absolute multiplicity.

In the linear case the centralizer contains \(\operatorname{GL}_m(F')\). Its subgroup \(\operatorname{SL}_m(F')\) has determinant one on the whole \(F\)-space: its determinant there is a power of the norm of its \(F'\)-determinant. Thus its derived subgroup survives the ambient special-linear condition.

For a form group, pair the simple isotypic summands by duality. A pair of distinct dual summands again supplies \(\operatorname{GL}_m(F')\), acting on the second summand by the inverse adjoint. On a self-paired summand the adjoint operation on \(\operatorname{Mat}_m(F')\) is an involution. If it acts nontrivially on \(F'\), the commuting isometries are a unitary group on an \(m\)-dimensional multiplicity space. If it fixes \(F'\), they preserve a nonsingular symmetric or alternating bilinear form on that multiplicity space. These descriptions follow by writing the invariant pairing as a matrix of pairings between the \(m\) copies of \(U\); Schur’s Lemma makes each matrix entry an element of \(F'\), and the symmetry of the original form is precisely the adjoint condition on that matrix.

There is one useful even-characteristic qualification. A nonsingular symmetric multiplicity form need not be alternating. Over a perfect finite field its diagonal function is the square of a linear functional. The vector representing that functional allows one to split off a nonsingular subspace of dimension one or two; its perpendicular is alternating and nonsingular. Thus a multiplicity space of dimension \(m\) still contains a nonsingular alternating subspace of dimension at least \(m-2\).

If \(m\ge16\), every group just described contains a subgroup isomorphic to \(\operatorname{SL}_3(F'')\) for a suitable finite field \(F''\). In a general linear group use a three-dimensional block. In a symplectic, orthogonal, or unitary multiplicity space choose paired totally isotropic three-spaces and let \(g\in\operatorname{SL}_3(F'')\) act as \(g\) and its inverse adjoint, fixing their perpendicular complement. Finite-field forms of dimension at least \(16\) have such paired spaces; the anisotropic part has dimension at most two. The preceding paragraph handles the nonalternating symmetric form in characteristic two in the same way.

These embedded groups are perfect and have nonabelian simple projective quotients. Being perfect, their determinants, similitude multipliers, spinor norms where applicable, and Dickson invariants in even-characteristic orthogonal groups are trivial. Therefore they lie in the actual inner classical group, rather than merely its full similitude group. A scalar quotient cannot make their images soluble.

In an orthogonal target of characteristic two, preserving the polar form on a nontrivial isotypic summand also preserves the quadratic form. Indeed two \(E\)-invariant quadratic refinements of that polar form differ by \(\ell^2\) for an \(E\)-invariant linear functional \(\ell\). On a sum of nontrivial simple \(E\)-modules there is no such nonzero functional. A commuting polar isometry consequently preserves the given refinement. On the trivial isotypic space use its actual quadratic isometry group; the same paired-three-space construction applies. The trivial space is nondegenerate, since it is orthogonal to every nontrivial isotypic summand.

We have proved that \(m\ge16\) would put a nonsoluble subgroup in the centralizer. Lemma 55 excludes this. ◻

Proof. Let \(V_1,V_2\) be two such unions, and put all remaining parts in \(V_3\). Let \(H_1,H_2\) be the derived groups of the full linear or isometry groups on \(V_1,V_2\), acting trivially on the other parts. They commute, are normalized by \(M\), and are normal in the enlargement generated by them and \(M\). Its inner part preserves the proper decomposition and cannot contain \(T_j\). Thus \(\overline U=\langle z_i/F_j,\widehat H_1,\widehat H_2\rangle\) has proper full preimage in \(z_j\). Both \(H_1,H_2\) are nonsoluble: the explicit \(\operatorname{SL}_3\) construction in the proof of Lemma 56 lies in each derived group and in the ambient inner group.

The two transported products are normal in this \(\overline U\) and commute exactly; their original preimages commute modulo \(F_j\trianglelefteq U\), with \(F_j\le D\). Condition ([grw:normal-commuting]) kills one product and forces, say, \(H_1\le P\). The kernel of the nontrivial action of \(E\) on \(V_2\) is central in \(E\), so \(H_1\cap E\le Z(E)\). Hence \[H_1/(H_1\cap E)\hookrightarrow P/E\] would make \(H_1\) soluble, a contradiction. On the symplectic section, take inverse images under the finite \(M\)-equivariant isomorphism \(\theta\) from Lemma 54. They commute exactly and their enlargement lies in the proper original line stabilizer, so the same application uses the original higher core \(F_j\), as in Lemma 55.

Suppose that a nontrivial packet has dimension at least \(16\). If another nontrivial packet occurs, apply the first assertion to the large packet and its entire complement, including the trivial packet. The complement has dimension at most \(15\). If no other nontrivial packet occurs, the complement is the trivial isotypic space, whose dimension is at most \(15\) by Lemma 56. If no packet has dimension \(16\), but their total dimension were at least \(46\), accumulate packets until their dimension first reaches \(16\). The accumulated dimension is at most \(30\), leaving at least \(16\) dimensions in the other union. This contradicts what was just proved. Thus in this case the total is at most \(45\). ◻

Proof. We work on simply connected covers and use Frobenius positions modulo the group parameter, rather than the degree of its natural scalar field. Write a classical finite group with parameter \(p^r\) as the fixed points of \(\tau F_p^r\), where \(\tau\) is a diagram operation. Steinberg’s Restriction and Tensor Product Theorems give the unique parametrization \[\lambda=\lambda_0+p\lambda_1+\cdots+p^{r-1}\lambda_{r-1}, \qquad\lambda_i\text{ $p$-restricted},\] of its absolute simple modules; see (Steinberg 1967, Theorems 41 and 43) and (Steinberg 1963). For a nontrivial simple module define \[\operatorname{supp}_r(\lambda) =\{i\in\mathbb Z/r\mathbb Z:\lambda_i\ne0\}.\] For an arbitrary module take the union of these supports over its nontrivial composition factors. A field twist by \(F_p^b\) translates support by \(b\). Diagram operations and contragredience preserve positions. When a position wraps modulo \(r\), its restricted weight is transformed by \(\tau^{-1}\), which does not change the position.

Consider one edge, with source and target exponents \(r,t\), and let \(d=\gcd(r,t)\). Suppose that the nonempty union \(\mathcal E\) of source digit supports in the target natural module lies in one residue class modulo \(d\). If a normalizer has target scalar exponent \(b\) and source field exponent \(a_0\), its transport of a composition factor has type \[(L(\lambda)\circ\gamma^{-1})^{[b-a_0]},\] where \(\gamma\) records the source diagram operation. This formula follows directly from \(x(mw)=m(\alpha_m^{-1}(x)w)\) and the \(p^b\)-semilinearity of \(m\). It applies to composition factors by transporting a composition series; semisimplicity on an individual edge is not required. An ambient correlation adds contragredience. Inner-diagonal automorphisms do not affect the absolute module type, because they are algebraic inner conjugations on the cover. Normalization therefore implies \[\mathcal E+(b-a_0)=\mathcal E.\] Since \(\mathcal E\) is nonempty and contained in a single residue class modulo \(d\), we obtain \[ b-a_0\equiv0\pmod d. \tag{36}\]

Assume for the moment that this support condition holds for both edges of the shortcut. Their source and target exponents are \(f,f'\) in opposite orders, so their common divisor \(d\) is the same integer \(h\). For an element of \(M\), denote its induced intermediate field exponent by \(a_1\). Its existence on the same intermediate component is the normalization conclusion of Proposition 49. The natural realization of that automorphism has scalar exponent \(a_1\), with any graph part recorded separately. Applying (36) to the two edges gives \[a_1-a\equiv0\pmod h,\qquad s-a_1\equiv0\pmod h.\] Their sum proves (35).

To bound a packet, let \(\Phi\) denote the operation of field twisting on absolute source types, and let \(\Gamma\) be the diagram group. The rank assumption gives \(|\Gamma|\le2\). These operations commute, and \(\Phi^f=\tau^{-1}\). Every stabilizer action lies in \(\langle\Phi^h,\Gamma\rangle\). If \(\tau=1\), that group has order at most \(2f/h\). If \(\tau\ne1\), the diagram involution already belongs to \(\langle\Phi^h\rangle\), whose order is at most \(2f/h\). Thus in either case the order is at most \(2v\). Descent over the upper natural field has exponent \(f\) or \(2f\) and belongs to the same group. Duality is the opposition diagram operation, also already included. The packet bound follows.

It remains to verify the support hypothesis for every remaining edge of Proposition 41 and Proposition 49.

1. In a nonlinear edge the natural module is a Frobenius twist of a nonzero restricted module. Its support is a singleton. The starting Frobenius position may be arbitrary; it cancels in (36).

2. A block, form, or subfield edge has natural or dual-natural nontrivial composition factors in one Frobenius position, allowing repetitions and trivial factors. A tensor edge with one active component has the same property: its other tensor factors are trivial for that perfect component, so they supply repetitions. The characteristic-two nonsingular-line representation has these composition factors as well, with the additional trivial factors already described.

3. For restriction of scalars, the source constituents are a direct sum of the Galois conjugates of the natural module. To distinguish group parameters from natural scalar fields, put \(\epsilon_s,\epsilon_t=2\) for a unitary source or target, respectively, and \(1\) otherwise. A scalar restriction of degree \(b_{\rm sc}\) satisfies \[\epsilon_s r=b_{\rm sc}\epsilon_t t.\] Its constituent positions are \(t_0+j\epsilon_t t\), so they all lie in one residue class modulo \(d=\gcd(r,t)\). For example, restricting the natural matrix field of a unitary component with parameter \(p^r\) to a symplectic target with the same parameter has scalar degree two. Over the algebraic closure its module is \[N\oplus N^{[r]} \cong N\oplus N^*.\] Both positions are zero modulo \(r\); this repetition causes no tensor carry because the operation is a direct sum.

4. For a retained natural twisted-tensor edge, let \(b_{\rm cyc}\) be the prime tensor-cycle length. Lemma 28 gives the exact usual-field relation \[r=b_{\rm cyc}t.\] In the finite realization supplied by Proposition 49, the cyclic fixed-point construction gives the following actual finite-source modules, up to a common initial Frobenius twist (Liebeck and Seitz 1998, Proposition 2.8(ii) and Section 4(5), p. 449). Here \(W\) is the source natural module in each case.

   Linear, symplectic, and orthogonal targets. The module is \[\bigotimes_{j=0}^{b_{\rm cyc}-1}W_j^{[jt]},\] where each \(W_j\) is \(W\) or \(W^*\) as prescribed by the diagram operation. Its positions are \(jt\) for \(0\le j<b_{\rm cyc}\); diagram changes affect the weights rather than these positions.

   Unitary target, odd cycle. The source is \(\operatorname{SU}_m(p^r)\) and the module is \[\bigotimes_{j=0}^{b_{\rm cyc}-1}W^{[2jt]}.\] Indeed, the cycle initially gives \(W^{[jt]}\) for even \(j\) and \((W^*)^{[jt]}\) for odd \(j\). On this finite source, \(W^*\cong W^{[r]}\). The integers \(j\) for even \(j\) and \(j+b_{\rm cyc}\) for odd \(j\) are precisely \(0,2,\ldots,2b_{\rm cyc}-2\), giving the displayed module. Its positions \(2jt\) are distinct modulo \(r\) because \(b_{\rm cyc}\) is odd.

   Unitary target, two-cycle. Since the cycle length is prime, this is the only even case. The source is \(\operatorname{SL}_m(p^{2t})\) and its module is \[W\otimes(W^*)^{[t]},\] with distinct positions \(0,t\) modulo \(r=2t\).

   Thus in every case the positions are distinct modulo \(r=b_{\rm cyc}t\) and lie in one residue class modulo \(d=\gcd(r,t)=t\). A common initial twist adds the same \(t_0\) to every position and preserves both assertions. Steinberg’s Tensor Product Theorem therefore gives the required support condition with no coincident-digit carries. This distinctness comes from the exact finite cyclic construction.

Changes between the type \(B\) and type \(C\) conventions in characteristic two do not alter the numerical field exponent. The special isogenies are defined over \(\mathbb F_2\), commute with Frobenius, and have Frobenius as their composites (Dowd and Sin 1996, sec. I.3.1). Thus each edge can use the convention in which its displayed module has the indicated support, while the intermediate field action has the same exponent in either convention. ◻

Proof. Apply Lemma 54. On its semisimple natural section, Lemma 56 bounds each absolute multiplicity by \(15\), and Lemma 58 bounds each packet by \(2B\) absolute types.

If there is a large nontrivial packet, it contributes at most \(30B\) factors. All other nontrivial packets have total dimension at most \(15\) by Lemma 57. If no packet is large, their total dimension is at most \(45\). In either case the number of nontrivial composition factors is at most \(30B+45\). The trivial isotypic space has dimension at most \(15\), again by Lemma 56. Thus the semisimple section has length at most \(30B+60\). Adding the possible two factors from the nonsingular-line case gives the displayed bound.

All constants came from the remaining-edge bounds, explicit finite-dimensional linear algebra, and the common cyclic congruence (35). They are independent of field exponent, rank, and chain length. ◻

Proof. Apply the corrected irreducible-triple Theorem to \(V\) and use Table 1. Only \(\mathrm S_6\) and \(\mathrm {MR}_5\) require an additional argument. Each of their \(t\) blocks contributes a nontrivial constituent of the natural restriction, of dimension \(2r\). In \(\mathrm {MR}_5\), passing from the odd orthogonal natural module to its simple quotient removes one trivial constituent and leaves all \(t\) nontrivial constituents. In \(\mathrm S_6\), there are also two trivial constituents, which do not affect the lower bound \(t\) on length. Different Frobenius twists may change the isomorphism types of these constituents, but not their dimensions or their number counted with multiplicity. Thus \(t\leq c\). The image conventions and (38) justify the same calculation on the original groups. No complete-reducibility assertion is used. ◻

Proof. Use the same representation \(\varphi:Y\longrightarrow\operatorname{SL}(V)\) throughout the proof. For \((Y,H,V)\), every row of Table 1 outside \(\mathrm S_6\) and \(\mathrm {MR}_5\) gives the conclusion. Suppose one of these two rows occurs. Then \(p=2\), the image of \(H\) is \(B_r\), the image of \(Y\) is \(D_m\) or \(B_m\), and \(V\) is a half-spin or spin module, respectively. Let \(W\) be the orthogonal natural module of this fixed ambient image, on a cover when necessary. Every nontrivial composition factor of \(W|_H\) has dimension \(2r\).

Since \(H<X\), irreducibility on \(H\) implies irreducibility on \(X\). Apply the irreducible-triple Theorem a second time to \((Y,X,V)\), keeping the ambient group and representation unchanged. The theorem requires restrictedness and tensor-indecomposability on \(Y\), so it makes no such assumption about the \(X\)-module \(V|_X\). If the ambient image is \(B_m\), the only row with a simple subgroup of rank at least ten is \(\mathrm {MR}_5\). If it is \(D_m\) and \(V\) is half-spin, the only such rows are \(\mathrm {IV}_1\), \(\mathrm {IV}'_1\), \(\mathrm {IV}'_2\), and \(\mathrm S_6\). In all these cases the image of \(X\) is \(B_s\) and every nontrivial composition factor of \(W|_X\) is a Frobenius twist of \(N_X\), of dimension \(2s\).

By hypothesis and (38), each such constituent remains irreducible on \(H\). Restricting a composition series for \(W|_X\) and refining it over \(H\) therefore exhibits an \(H\)-composition factor of dimension \(2s\). This contradicts the preceding dimension \(2r\), since \(s>r\). The argument works for nonsplit natural restrictions as well as for direct sums.

There is no extra case from \(\mathrm {MR}_4\): its ambient image is \(C_m\), whereas the image of the fixed representation here is \(B_m\) or \(D_m\). One cannot change that image between the two applications. Equations (37)–(38) preserve the natural irreducibility hypothesis on passing to these image groups, including an inseparable isogeny. They do not replace the fixed spin representation \(V\) by a natural representation.

Finally, if \(X=Y\), the irreducibility of \(N_Y|_H\) directly excludes \(\mathrm S_6\) and \(\mathrm {MR}_5\), since their natural restrictions have at least two nontrivial composition factors. ◻

Proof. Let \(U_1,\ldots,U_a\) be the nontrivial \(L_j\)-constituents, counted with multiplicity. If \(U_t|_{L_i}\) had only trivial composition factors, the image of the perfect group \(L_i\) would be conjugate into an upper unitriangular group and hence would be both perfect and solvable. It would be trivial, contrary to the nontriviality hypothesis. Thus, writing \(a_t,b_t\) for its nontrivial and trivial composition counts, we have \(a_t\geq1\) and \(b_t\geq0\). Additivity of composition multiplicities along restriction gives \[a=\sum_{t=1}^a a_t,\qquad b=b+\sum_{t=1}^a b_t.\] It follows that \(a_t=1\) and \(b_t=0\) for every \(t\), as required. ◻

Proof. By Theorem 59, fix an absolute integer \(c\) bounding \(\ell_{L_i}(N_j)\) for every retained comparison \(i<j\). The pair of nontrivial and trivial composition counts has at most \((c+1)^2\) possible values. Finite Ramsey applied to these pair colors gives arbitrarily long subchains with constant counts \((a,b)\). These restrictions are nontrivial, and \(a\geq1\) follows because a perfect group cannot act nontrivially with only trivial composition factors. Moreover, a nontrivial simple representation of a quasisimple group has central kernel. Its restriction along the noncentral earlier component is therefore nontrivial. The hypotheses of Lemma 62 hold.

For each \(j<k\), realize every nontrivial constituent \(U\) of \(N_k|_{L_j}\) as the restriction of the corresponding restricted finite-group highest weight module of \(\mathbf G_j\). Steinberg’s Tensor Product Theorem decomposes that rational simple module into Frobenius twists of \(p\)-restricted simple modules; see (Steinberg 1967, Theorem 41). If a restricted factor is tensor decomposable, continue factoring it until its nontrivial factors are tensor-indecomposable. This terminates since every factor has smaller positive dimension. Every resulting factor is \(p\)-restricted: its dominant highest weight is a summand of a \(p\)-restricted dominant weight.

Color a triple \(i<j<k\) red if some such \(U\) has a nontrivial terminal factor \(V\), not of natural type, and blue otherwise. In a red triple, Lemma 62 says that \(U|_{L_i}\) is irreducible. Hence each tensor factor \(V^{[d]}\) is irreducible on \(L_i\). Use a path lift from \(\mathbf G_i\) to \(\mathbf G_j\) and replace its image by its \(p^d\)-Frobenius image. The module \(V\) is irreducible on this closed connected simple subgroup: any proper rational invariant subspace would also be invariant under the corresponding finite component. The subgroup has rank \(r_i\) and is proper because the endpoint ranks increase.

The natural restriction on this subgroup has algebraic composition length at most \(c\). Indeed, restriction from an algebraic group to a finite subgroup is exact and cannot reduce composition length, and Frobenius transport preserves length. The finite restriction here is a twist of \(N_j|_{L_i}\). Lemma 60 consequently bounds the number of possible middle ranks \(r_j\) by \(2c+7\), for fixed \(i\).

A red homogeneous chain with more than \(2c+9\) vertices is impossible: fix its first and last vertices \(i,k\) and vary its middle vertex \(j\). The distinct ranks \(r_j\) would exceed that bound. A second application of finite Ramsey therefore gives arbitrarily long blue homogeneous subchains.

Fix \(i<j<k\) in a blue subchain and choose a nontrivial constituent \(U\) of \(N_k|_{L_j}\). There is at least one nontrivial terminal tensor factor, and every such factor is now of natural type. Since \(U|_{L_i}\) is irreducible, a natural or dual natural Frobenius factor restricts irreducibly to \(L_i\). Undoing its twist, dual, and the isogenies (37) shows that \(N_j|_{L_i}\) is irreducible. Delete the last vertex of the blue subchain, so that a suitable \(k\) exists for every remaining pair \(i<j\). All modules and restrictions were taken over \(k\), so the resulting finite restrictions are absolutely irreducible. ◻

Proof. A tensor factor of a simple module is simple, since any proper nonzero submodule of the factor would tensor to one of the whole module. Highest weights add in an irreducible tensor product. The highest weights of the natural-type factors have the form \(p^a\omega_1\), or \(p^a\omega_r\) in type \(A_r\) for the dual. Equation (37) shows explicitly that the special isogenies add no further natural weights. Since their sum is \(p\)-restricted and all fundamental-weight coefficients are nonnegative, every exponent \(a\) is zero. The terminal factors are therefore \(N_J\) or \(N_J^*\).

Any two of these factors have reducible tensor product. For two identical factors, the kernel of the flip minus the identity is a nonzero proper invariant subspace: it contains \(v\otimes v\), while \(v\otimes w-w\otimes v\ne0\) for independent \(v,w\). This also proves properness in characteristic two. For opposite factors, \(N_J\otimes N_J^*\cong\operatorname{End}(N_J)\) contains the proper invariant line of scalar endomorphisms. The same arguments apply to dual identical factors. Tensoring such a proper submodule with all remaining factors contradicts simplicity of \(M\). There is thus exactly one terminal factor. ◻

Proof. Irreducibility on \(L\) implies irreducibility on the source component, so the target restriction has a single absolute composition factor. It is nontrivial because the target natural dimension exceeds one. By hypothesis this factor is a tensor product of natural-type modules. Irreducibility of its further restriction forces each factor to restrict irreducibly. Undoing a twist, a dual, and the natural isogenies yields the assertion for \(N_J\). For retained geometric edges this also follows from Corollary 52. For the stated one-digit edges it follows from Lemma 64, after restoring the common twist. ◻

Completion of the proof of Theorem 14. Suppose tested chains with different endpoint labels have unbounded length. Apply Proposition 25 and Proposition 41. We obtain arbitrarily long configurations with strictly increasing endpoint ranks, with their minimum rank tending to infinity, and with two-edge shortcuts on all pairs. There are absolute constants \(c_0>0\), \(C\geq1\), and \(B\geq1\) such that the natural endpoint dimensions satisfy \[ n_j\geq c_0 n_i^2\qquad(i<j), \tag{40}\] and every retained geometric edge satisfies \[ n_{\mathrm{target}}\leq C n_{\mathrm{source}}^B. \tag{41}\] Every other edge is a defining-characteristic irreducible representation with one restricted digit after removing a common twist. At least one such nonlinear edge occurs in each shortcut. Proposition 49 and Lemma 50 supply the actual component homomorphisms and their algebraic path lifts. By Proposition 63, pass to arbitrarily long subchains with all natural endpoint restrictions absolutely irreducible.

Color a pair of endpoints red if its chosen shortcut has a one-digit edge with a nontrivial restricted tensor-indecomposable factor not of natural type. Color the pair blue otherwise. Finite Ramsey leaves arbitrarily long homogeneous configurations of one of these two kinds. We exclude both.

First consider a blue configuration. On every one-digit edge, Lemma 64 bounds the target natural dimension by the source natural dimension; restoring a Frobenius twist does not change dimension. Thus every edge of every shortcut satisfies (41), enlarging \(C,B\) if necessary. Composing at most two edges gives the uniform bound \[ n_j\leq A n_i^E\qquad(i<j),\qquad A=C^{B+1},\quad E=B^2. \tag{42}\] Choose a fixed integer \(d\) with \(2^d>E\). Along \(d\) consecutive comparisons of a retained chain, (40) gives \[n_d\geq c_0^{\,2^d-1}n_0^{2^d},\] whereas the shortcut from its first to its last endpoint gives \(n_d\leq A n_0^E\). These inequalities contradict each other as the minimum rank, and hence \(n_0\), tends to infinity. This proves the bounded-natural-factor case without imposing a bound on the number of Steinberg digits of an arbitrary composite endpoint representation.

Now consider a red homogeneous configuration. Fix an early endpoint \(H\) of rank \(r\geq9\). Choose eight later pair comparisons with disjoint ordered index intervals, leaving a gap between successive intervals. For each such comparison, let \(X\) be its lower endpoint, and choose the last edge in its shortcut that has a restricted factor not of natural type. Let \(Y\) be the algebraic source of that edge and \(Z\) its target. Every edge after it has a natural tail in the sense of Lemma 65: it is either geometric or a one-digit edge with only natural restricted terminal factors. The upper endpoint natural module is irreducible on the finite component of \(H\). Applying that lemma backwards along the tail shows that \(N_Z\) is irreducible on the same component.

Remove the common Frobenius twist of the \(Y\)-representation on \(N_Z\), and choose a nontrivial restricted tensor-indecomposable factor \(V\) not of natural type. Irreducibility of the whole restriction implies irreducibility of the corresponding twisted factor on the finite \(H\). Compose the actual algebraic maps along the path from \(H\) through \(X\) to \(Y\), and apply the removed Frobenius morphism to both lower images. Then \(V\) is irreducible on the resulting connected simple image of \(H\). Its rank is \(r\), and the resulting image of \(X\) has its original rank \(s>r\). Its natural simple module remains irreducible on the image of \(H\), by the endpoint conclusion and (38). Finite irreducibility implies each of these rational irreducibility statements, since a rational invariant subspace would also be invariant under the actual finite image.

If the algebraic image of \(X\) is proper in \(Y\), apply Lemma 61. If that image is all of \(Y\), the map from \(X\) to \(Y\) is an isogeny. This can occur even when the finite groups are strictly nested, for example on a subfield edge. The pullback of \(N_Y\) is a natural simple module or its dual, up to Frobenius twist and the isogenies in (38), by the root-datum classification of isogenies (Conrad 2014, Definition 6.1.8, Example 6.1.9 and Theorem 6.1.16(1)). Indeed, for compatible simple roots an isogeny has a root bijection \(\delta\) and powers \(q_i\) of the characteristic. Write the Cartan matrices in the row-coroot convention \(A^X_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle\) and \(A^Y_{ab}=\langle\beta_b,\beta_a^\vee\rangle\). The compatibility equation is \(q_j A^X_{ij}=q_i A^Y_{\delta(i)\delta(j)}\). Pairing the pullback of the natural fundamental weight with the source simple coroots gives \(q_i\) at the node mapped to the natural node and zero elsewhere. In the retained ranks this node is the natural node, or its type-\(A\) dual; the only change of root lengths is the characteristic-two \(B/C\) case in (38). Since \(N_X\) is irreducible on the lower \(H\), it follows that \(N_Y\) is irreducible on the corresponding image of \(H\). The \(X=Y\) case of Lemma 61 therefore applies. Thus every one of the eight chosen source ranks belongs to \(\mathcal R(r)\). They are nevertheless distinct. Along an algebraic edge the rank cannot decrease, because a maximal torus maps with finite kernel into a torus of the image. Hence each chosen source rank lies between the ranks of the two endpoints of its comparison. The gaps between the ordered comparison intervals, and strict increase of endpoint ranks, make these eight intervals disjoint in rank. This contradicts \(|\mathcal R(r)|\leq7\).

Both homogeneous alternatives are impossible. All Ramsey bounds and all path lengths used here depend only on the absolute constants in the preceding lemmas. In particular only fixed finite paths are needed before taking the large-rank limit. The assumed unbounded family of tested chains cannot exist, proving the asserted absolute chain bound and completing Theorem 14. ◻

Proof. The prescribed order is transitive: every route from an old vertex into the fresh interior passes through \(a\), and every route back passes through \(b\). This proves (48) and antisymmetry. For an old vertex \(v\) incomparable with a fresh interior vertex \(w\), the cross operations are \[v\wedge w=v\wedge a, \qquad v\vee w=v\vee b,\] where the operations on the right are those of \(P\). In the comparable cases, the smaller and larger vertices give the meet and join. For two fresh vertices, the operations in \(Q\) remain their operations: an old common lower bound lies below \(a\), and an old common upper bound lies above \(b\). An analogous argument shows that an old meet or join cannot be changed by a new bound. This proves the lattice assertion.

If a fresh vertex \(w\) lies strictly between the old cover \(c\prec d\), then (48) gives \(c\leq a<b\leq d\). The cover condition forces \(a=c\) and \(b=d\). Conversely, an insertion with those endpoints and a fresh interior vertex destroys the cover. ◻

Proof. Vertices in different components of the proper comparability graph have meet \(a\) and join \(b\). Indeed, a common lower bound strictly above \(a\), or a common upper bound strictly below \(b\), would connect them. For a given interior vertex, choose the two comparable vertices in one of the two specified components different from its own. They are its required complements.

Consider an insertion into \([c,d]\). A fresh vertex can belong to \([a,b]\) only if \(a\leq c<d\leq b\), by Lemma 67. If both \(c,d\) are proper vertices of \([a,b]\), they lie in the same comparability component. All comparisons from the new interior to the old proper part then enter that component. The same is true if just one of \(c,d\) is proper. If \(c=a\) and \(d=b\), the new interior has no comparisons with the old proper part. Thus no insertion merges two old components. The two reserved comparable pairs survive, proving persistence. A fencing insertion supplies two such separate components. ◻

Proof. Each shortcut edge is a cover immediately after its one-vertex insertion. By the cover criterion in Lemma 67, a later shape insertion could destroy it only by using that exact edge as its endpoint pair. The listed pairs never do so. The only apparent possibilities would be \([a,v_{0j}]\), which is excluded from (49), and \([v_{iN},b]\), which is excluded from (50). This proves (i).

All pair fences are present in the third batch and persist by Lemma 68. The chain targets \(x_j\) already have the fence on \([a,x_j]\) from that batch; the other upper targets are listed there expressly. Each reserved coatom witness is fresh when inserted and is not an endpoint of a subsequent shape insertion. Its edge to the upper endpoint therefore survives. Its meet with its companion remains the lower endpoint, by Lemma 67.

When an \(M_{16}\) branch is inserted, its interval \([m_u,u]\) has no other vertices: an old vertex above \(m_u\) must be above \(u\). A subsequent shape insertion could add a vertex inside this interval only if both its endpoints lay in \([m_u,u]\). None of \(m_u,t_{u,k}\) is used subsequently as a shape endpoint. The remaining possible endpoint is \(u\), which cannot by itself form a nondegenerate interval inside \([m_u,u]\). Thus this interval remains exactly \(M_{16}\). These arguments give (ii) and (iii).

For completeness, the numbers of new vertices in the first four batches are respectively \[N-1,\qquad \frac{N(N+1)}2,\qquad 4N^2, \qquad \frac{17N(N+1)}2.\] The fifth batch adds at most \(2N(N+1)\) vertices. Their sum is \(15N^2+12N-1\leq27N^2\). ◻

Proof. We first record how the two shape constructions are separated. Every fresh vertex of a forward shape construction lies below no proper original vertex. This follows inductively from (48): its upper insertion endpoint is either \(1\) or an earlier fresh forward vertex with the same property. In contrast, every vertex of a reverse shape construction lies in a forward interval \([v,u]\) with \(0<v<u<1\) original. Thus every vertex of \(\mathcal R_C\) lies below a proper original upper bound, and no fresh forward vertex belongs to \(\mathcal R_C\).

These observations also show that the shape constructions do not spoil each other’s designated intervals. Different forward tests have disjoint fresh interiors; their only shared shape endpoints are original vertices and \(1\). Hence neither uses a fresh endpoint of the other’s designated cover. No reverse shape insertion can use such a forward endpoint as its lower endpoint, since it lies below a proper original upper bound. For reverse tests, two different tests again have disjoint fresh interiors and share only original endpoints. Lemma 67 therefore protects their covers. The same argument protects the exact \(M_{16}\) intervals: a shape insertion could enter one only if its two endpoints lay inside it, whereas its fresh bottom and atoms are never shape endpoints. Fences persist by Lemma 68.

We next check the final batch. A new vertex of \([0,t]\) belongs to no comparison interval \([a,b]\) with \(a>0\), by (48). A new vertex of \([t,1]\) can belong to such a comparison only if \(b=1\) and \(a\leq t\). The lower endpoints of the latter insertions are in \(\mathcal R_C\), and consequently lie below proper original upper bounds.

In particular, a reverse fence cannot subdivide a fresh forward coatom edge \(c\prec1\): that would require its lower endpoint to be exactly \(c\), which lies below no proper original vertex. A forward cover with a proper upper endpoint cannot be subdivided by an insertion ending at \(1\). Thus all forward shortcut edges and all reserved coatom witnesses survive. A forward \(M_{16}\) interval with proper upper endpoint cannot be entered by a reverse fence. If its upper endpoint is \(1\), its fresh bottom lies below no proper original vertex; hence no point of \(\mathcal R_C\) lies above that bottom. So this \(M_{16}\) interval also remains exact. Forward base insertions cannot enter any of these positive-bottom intervals.

All reverse comparison intervals have two proper original bounds in the forward order. Neither kind of final insertion can add a vertex to such an interval: an insertion beginning at \(0\) cannot do so, and an insertion ending at \(1\) cannot do so. Their covers, witnesses and exact \(M_{16}\) intervals are therefore unchanged. The final reverse fences label every vertex in each reverse comparison interval by the definition of \(\mathcal R_C\). All required forward principal vertices belong to \(\mathcal F_C\); the sole omitted proper target is the detector \(X\), whose lower interval was fenced at the outset. This proves (i) and (ii).

For (iii), an old vertex in a forward comparison belongs to \(\mathcal F_C\), so its lower interval is fenced, with the stated exception for the global top. The only possible new vertex is an interior vertex \(w\) of a final reverse fencing insertion into \([s,1]\). If \(x_i\leq w\), then \(x_i\leq s\) by (48). Every vertex below \(w\) either lies below \(s\) or lies on the same private chain above \(s\). This assertion also includes vertices of other insertions in the final batch: a point from another branch can be below \(w\) only by passing through the old endpoint \(s\). Thus \(s\) is comparable to every vertex of \([0,w]\). Moreover \(s\in\mathcal F_C\), since \(x_i\leq s<1\), and its lower interval is fenced. This is exactly alternative (b), including the requirement that the bottleneck lie above the lower chosen chain vertex. It proves closure of the normal-group tests without any new round of insertions.

For (iv), no shape insertion introduces a point strictly below \(X\). Forward fresh points lie below no proper original vertex, and reverse shape insertions in the detector are contained in \([X,z]\). A final insertion ending strictly below \(X\) introduces only points below that endpoint, so it cannot create a new coatom of \([0,X]\). A final insertion ending above \(X\) introduces no new point below \(X\). The only remaining possibility would be an insertion into \([0,X]\), which was omitted. Hence the original covers \(q_i\prec X\) are precisely the coatoms of the lower half. Its two separate original branches, and those of the upper half, still give fences by Lemma 68.

Finally, Lemma 69 bounds the size before the final batch by \(q=q_0+27hN^2\). Each of \(\mathcal F_C\) and \(\mathcal R_C\) has at most \(q\) members, and each fencing insertion adds four vertices. The final size is at most \(q+4q+4q=9q\). This also proves finiteness directly from the displayed batches. ◻

Proof of Proposition 66. Let \(B\) be the absolute chain threshold of Theorem 14, and choose the single integer \(N=\max\{2,\lceil B\rceil\}\). Construct \(L=L_N\) by the finite operations above. This choice is independent of every possible representing group. Finiteness follows from Lemma 70 and (53); no limiting lattice is used.

All original lattice operations survive by Lemma 67. Every proper original Boolean vertex is an endpoint of a forward requested pair and of a reverse requested pair, so its two principal intervals receive the required fences. The detector statements follow from Lemma 70. This gives (i) and (ii).

For (iii), take a finite representation \([D,G]\). By Lemma 5, the global fence and the required principal fences provide all forward labels. In every forward requested test, the sequence \(x_0<\cdots<x_N\) and its private shortcuts have the shape required by Theorem 14, including its nonconsecutive shortcuts. The interval \([x_i,x_j]\) is fenced; every target in (49) has its bottom saturation, fence, and exact positive-bottom \(M_{16}\) interval; and all the specified principal vertices are meets of coatoms below the largest endpoint, with \(v_{iN}\) themselves coatoms. Finally, Lemma 70(iii), together with the global fence, verifies condition [cls:test-normal] at every vertex of every comparison: either its lower interval is fenced or it has the specified labelled bottleneck above the lower chain vertex. In the latter case, Corollary 6 supplies the required soluble-normal and commuting-normal consequences. All hypotheses of the chain Theorem hold, and \(N\geq B\) forces equality of its endpoint types. The forward requested pairs are precisely (43) and (47), giving the asserted equalities.

For (iv), fix one of the indicated actual upper-filter representations. Its least lattice vertex is the former \(1\), and its group bottom is \(A\). The entire reverse test for \((u,v)\) lies inside this upper filter. Its principal labels come from intervals \([w,1]^{\mathrm{op}}\), which are fenced by the final reverse batch. All vertices of its comparison intervals have those fences, by Lemma 70(ii). The reverse shape construction supplies all the other hypotheses of the same chain Theorem. Consequently the two endpoint labels, now taken relative to \(A\), are equal. The proof for \((z,X)\) is identical. If the actual intervals are compatible, their common subgroups and their labels are the same on overlaps; the argument makes no additional choice of a representation and thus applies simultaneously.

Finally, the order reversal exchanging the paired components in (53) supplies the same Boolean and detector data in the other orientation. This proves the last assertion. ◻

Proof. Put \(E=D\cap S\). For every \(Y\in[D,G]\), the identity \(G=DS\) gives \[Y=D(Y\cap S).\] Thus \(Y\mapsto Y\cap S\) is an isomorphism from \([D,G]\) to the interval of \(D\)-invariant subgroups between \(E\) and \(S\); its inverse is \(K\mapsto DK\).

First suppose that \(E\) is subdirect in \(S\). By the subdirect-power description in Lemma 7, its simple factors are full diagonal strips on a partition \(\Pi\) of \(I\). Every subgroup between \(E\) and \(S\) is subdirect. Its strips split the blocks of \(\Pi\), and their diagonal identifications are forced by their containment of \(E\). Therefore the preceding interval is anti-isomorphic to the lattice of \(D\)-invariant equivalence relations refining \(\Pi\).

Fix \(i_0\in I\), let \(A=D_{i_0}\), and let \(B\) be the stabilizer in \(D\) of the block of \(\Pi\) containing \(i_0\). The block correspondence for a transitive action identifies these equivalence relations with \([A,B]\): the block containing \(i_0\) associated with \(U\in[A,B]\) is \(U i_0\), and all its other blocks are its \(D\)-translates. Conversely, the setwise stabilizer of the block containing \(i_0\) recovers \(U\). It follows that \(\mathcal M^{\mathrm{op}}\cong[A,B]\). But \(|B|\le |D|<|G|\), contrary to the choice of \(G\) over both orientations.

The coordinate projections of \(E\) have equal order, since \(D\) is transitive on \(I\). If they are proper and nontrivial, Proposition 9, applied with the possibly unlabelled lower endpoint \(D\), and the global fence imply that \(E\) is the product of those projections. Choose a proper original Boolean vertex \(Y\) of the decorated lattice. Its forward label equals the top label, so \(Y\cap S\) is subdirect by Proposition 8. It is proper, because \(Y<G\).

A product of nontrivial coordinate subgroups cannot lie in a proper subdirect subgroup: the latter has a diagonal link between two coordinates, whereas the former contains an element that is nontrivial in one of those coordinates and trivial in the other. This contradicts \(E\le Y\cap S\). The only remaining possibility is that all coordinate projections of \(E\) are trivial, and then \(E=1\). ◻

Proof. Use left conjugation for the given action and let \(\pi_0:S\to T\) be the chosen coordinate projection. The map \[s\longmapsto f_s,\qquad f_s(d)=\pi_0(d\cdot s),\] identifies \(S\) with \[\mathcal F_\alpha =\{f:D\to T:f(ad)=\alpha(a)(f(d)) \text{ for }a\in A,d\in D\}.\] Indeed, values on representatives of \(A\backslash D\) are independent coordinates of \(S\). Under this identification the action is \((g\cdot f)(d)=f(dg)\).

An extension \(\beta\) defines \[\mathcal F_\beta =\{f:D\to T:f(ud)=\beta(u)(f(d)) \text{ for }u\in U,d\in D\}.\] This is a \(D\)-invariant subgroup of \(\mathcal F_\alpha\), with independent values on the cosets in \(U\backslash D\). Consequently \(\mathcal F_\beta\cong T^{[D:U]}\), and it projects onto every coordinate of \(\mathcal F_\alpha\). Set \(S(U,\beta)=\mathcal F_\beta\) in these coordinates.

Conversely, let \(K\le\mathcal F_\alpha\) be a \(D\)-invariant subdirect subgroup. Its diagonal-strip partition is invariant under the transitive action on \(A\backslash D\). The block containing \(A\) has stabilizer \(U\) for some \(A\le U\le D\). Projection at \(1\) identifies the diagonal factor on that block with \(T\). The induced action of \(U\) on this factor gives a homomorphism \(\beta:U\to\operatorname{Aut}(T)\) extending \(\alpha\). For \(f\in K\) and \(u\in U\) one has \(f(u)=\beta(u)(f(1))\). Applying this identity to \(d\cdot f\) gives \[f(ud)=\beta(u)(f(d)).\] Thus \(K\le\mathcal F_\beta\). Both groups have one freely varying diagonal factor for each coset of \(U\), so they are equal.

Containment of two subdirect groups reverses refinement of their strip partitions. Hence \(S(U,\beta)\le S(V,\gamma)\) forces \(V\le U\). For \(v\in V\), varying \(f(1)\) over \(T\) in the two defining equations then gives \(\beta(v)=\gamma(v)\). The converse follows immediately from those equations. Finally, every subgroup containing a subdirect subgroup is subdirect, so all members of the stated interval have already been included. ◻

Proof. Let \(X\) be the detector vertex of Proposition 66. Its forward type equals \(T\), so \(E_X=X\cap S\) is subdirect in \(S\). The action of \(D\) on the simple factors of \(E_X\) is transitive. Its factor-stabilizer action is \(\beta_X:U_X\to\operatorname{Aut}(T)\) by the construction in Proposition 74.

Suppose that this action preserved a subgroup \(1<P<T\). Transporting \(P\) among the factors of \(E_X\) would give a \(D\)-invariant product \(Q\) of proper nontrivial coordinate subgroups of \(E_X\). Thus \(DQ\) is strictly between \(D\) and \(X\). Choose a coatom \(Y\) of \([D,X]\) containing \(DQ\). The detector’s coatom property gives a designated vertex \(d\le Y\) whose intersection \(d\cap S\) is subdirect in the global coordinates. Hence \(Y\cap S\) projects onto every simple factor of \(E_X\): projection onto such a factor can be identified with projection onto any one global coordinate in its diagonal support. Thus \(Y\cap S\) is a proper subdirect subgroup of \(E_X\) containing the product \(Q\), which is impossible by the diagonal-link argument in Proposition 73.

The automorphism lemma recalled by Pálfy (Pálfy 2019, Lemma 2.11) states that a subgroup of \(\operatorname{Aut}(T)\) not containing \(\operatorname{Inn}(T)\) preserves some proper nontrivial subgroup of \(T\). It follows that \[\beta_X(U_X)\ge\operatorname{Inn}(T).\]

The reverse detector interval is the actual interval \([A,U_X]\) in (56), and it is fenced. Set \(K=\ker\beta_X\). Lemma 4 gives either \(AK=U_X\) or \(K\le A\). The first possibility gives \(\beta_X(U_X)=\beta_X(A)=\alpha(A)\), as required.

In the second possibility \(K\le C_X=\operatorname{core}_{U_X}(A)\), and \([A,U_X]\) is represented as \[[\alpha(A),\beta_X(U_X)].\] Write \(H=\beta_X(U_X)\) and \(B=\alpha(A)\). If \(\operatorname{core}_H(B)\) is nontrivial, it contains \(\operatorname{Inn}(T)\) by Lemma 71. Quotienting by that core leaves a soluble top, contradicting Lemma 72. Otherwise \(C_X=K\), and \(U_X/C_X\cong H\) has socle \(T\) with multiplicity one. The reverse detector supplies a strictly smaller labelled domain in \([A,U_X]\) whose simple type is equal to that of \(U_X/C_X\). The equality is supplied by the reverse tests in Proposition 66, applied through (56). Proposition 8 would make its positive socle multiplicity strictly smaller than one. This final contradiction excludes \(K\le A\). ◻

Proof. The normal-subgroup test gives the asserted supplementation unless \(\ker\beta\le A\). In the latter case \([A,U]\) is isomorphic to \([\alpha(A),\beta(U)]\). Proposition 75 puts the inner group in its bottom, contrary to Lemma 72. Supplementation gives the equality of images. ◻

Proof. Use the minimal representation already chosen. Index the original Boolean component in its reverse order by subsets \(J\) of \(\{1,\ldots,4\}\). Let \(Y_J\) be the corresponding forward group, so \(Y_\varnothing=G\) and \(Y_{\{1,\ldots,4\}}=D\). Every proper nontrivial \(Y_J\) has the top forward type and hence a subdirect intersection with \(S\). Proposition 74 provides \((U_J,\beta_J)\).

The reverse labels and tests in Proposition 66 apply in the actual filters (56). They give the fences and equal simple types on comparable proper Boolean nodes. All the nonempty proper nodes therefore have one common simple type \(R\): connect two atoms through their proper two-atom join, and connect any other node to an atom below it. Lemma 5 and Proposition 8 prove [ext:e1] and the transport assertions in [ext:e2]. The order relation and restrictions follow from (55).

For the assertion about a proper join, let \(K\) be the union of a nonempty family of index sets, and put \(V=\langle U_J:J\text{ in the family}\rangle\). Then \(A\le V\le U_K\), and \(\beta_K|_V\) is an extension of all the \(\beta_J\). Its subdirect group is contained in every \(Y_J\cap S\) and contains \(Y_K\cap S\). Original Boolean meets are preserved in the decorated lattice, so the intersection of those \(Y_J\) is exactly \(Y_K\). The subdirect groups are equal; the dictionary gives \(V=U_K\).

Proposition 75 and Lemma 76 give [ext:e3]. Finally, a common extension for a family with full union would give a nontrivial subdirect subgroup of \(S\) contained in all the \(Y_J\cap S\). Their intersection is \(D\cap S=1\), because their reverse Boolean join is the top. This proves [ext:e4]. ◻

Proof. For two normal subgroups, \[f(B_1\cap B_2)\ \supseteq\ f([B_1,B_2]) =[f(B_1),f(B_2)]\ \supseteq\ [P,P]=P.\] Induction proves the assertion for the finite family. ◻

Proof. Write \(J=\operatorname{Inn}(T)\) and let \(\iota:T\to J\) be the isomorphism \(t\mapsto c_t\). Consider \[Q=\bigcap\{B\trianglelefteq A:\alpha(B)\ge J\}.\] The family is nonempty. Since \(J\) is perfect, Lemma 78 shows that its finite intersection still maps over \(J\). The subgroup \(\alpha^{-1}(J)\) belongs to the family, so in fact \[ \alpha(Q)=J. \tag{57}\] Thus \(Q\) is the least normal subgroup of \(A\) with image containing \(J\).

Fix \(i\) and put \(C_i=\operatorname{core}_{U_i}(A)\). The hypothesis gives \(Q\le C_i\). Make the analogous construction in \(C_i\): \[Q_i=\bigcap\{B\trianglelefteq C_i:\alpha(B)\ge J\}.\] The same intersection argument, using \(C_i\cap\alpha^{-1}(J)\), gives \(\alpha(Q_i)=J\). For \(u\in U_i\) and \(c\in C_i\), \[\alpha(ucu^{-1}) =\beta_i(u)\alpha(c)\beta_i(u)^{-1}.\] Since \(C_i\trianglelefteq U_i\) and \(J\trianglelefteq\operatorname{Aut}(T)\), conjugation by \(U_i\) permutes the defining family for \(Q_i\). Thus \(Q_i\trianglelefteq U_i\), and in particular \(Q_i\trianglelefteq A\). Minimality of \(Q\) gives \(Q\le Q_i\). Conversely, \(Q\le C_i\), \(Q\trianglelefteq C_i\), and (57) give \(Q_i\le Q\). Consequently \(Q_i=Q\) for every \(i\).

It follows that \(Q\trianglelefteq K\) and \(Q\le\operatorname{core}_K(A)\), proving the common-core assertion. Set \(L=Q\cap\ker\alpha\). Since \(L=Q\cap\ker\beta_i\) for every \(i\), the subgroup \(L\) is normalized by each \(U_i\) and therefore by \(K\). The map \[\vartheta:Q/L\longrightarrow T, \qquad \vartheta(qL)=\iota^{-1}(\alpha(q))\] is an isomorphism. Transporting the conjugation action of \(K\) on \(Q/L\) through \(\vartheta\) defines a homomorphism \(\beta:K\to\operatorname{Aut}(T)\). For \(u\in U_i\) and \(q\in Q\), \[\alpha(uqu^{-1}) =\beta_i(u)\alpha(q)\beta_i(u)^{-1} =\iota\bigl(\beta_i(u)(\vartheta(qL))\bigr).\] As \(\vartheta\) is onto, this proves \(\beta|_{U_i}=\beta_i\). These restrictions determine \(\beta\) uniquely because the \(U_i\) generate \(K\). ◻

Proof. By [ext:e3], \(U_x=A\ker\beta_x\). Since \(A<U_x\) and \(C_x\le A\), the kernel is not contained in \(C_x\). The assumption makes \(\beta_x\) a homomorphism on \(U_x/C_x\), so its nontrivial kernel there contains the unique minimal normal subgroup \(N_x/C_x\). Consequently \(\beta_x(N_x)=1\).

Let \(P\) be the kernel of the action of \(U_x\) on \(I_x\). Modulo \(C_x\), the faithful conjugation action on \(S_x\) embeds \(P/N_x\) into \(\operatorname{Out}(R)^{I_x}\). To justify faithfulness, the centralizer of \(S_x\) in \(U_x/C_x\) is normal and disjoint from \(S_x\); uniqueness of the nonabelian minimal normal subgroup makes it trivial. Schreier’s Theorem therefore makes \(P/N_x\) soluble. Its image under \(\beta_x\) is a soluble normal subgroup of \(H=\alpha(A)\). Lemma 71 makes that image trivial, so \(P\le\ker\beta_x\), as claimed. ◻

Proof. Fix a proper pair join \(w=u\cup v\). Equal-type transport gives \[ N_u=(U_u\cap N_w)C_u. \tag{60}\] The first factor on the right acts by inner automorphisms on \(N_w/C_w\), hence preserves every coordinate subproduct there. The second factor satisfies \(C_u\le C_x\le U_x\). Since \(C_x\) acts trivially on \(I_x\), equivariance of \(p_{wx}\) says that \(C_u\) fixes the set \(F_{w,j}\). It too normalizes \(N_w(F_{w,j})\). Thus \(U_u\cap N_w(F_{w,j})\) is normal in \(N_u\) and lies in \(N_u\).

In \(N_w/C_w\), the subdirect group \((U_u\cap N_w)/C_w\) has its diagonal strips indexed by \(I_u\). Their supports are the fibres of \(p_{wu}\). Because \(p_{wx}=p_{ux}p_{wu}\), intersecting with the \(F_{w,j}\)-support product selects exactly the strips indexed by \(F_{u,j}\). The isomorphism from this subdirect group to \(N_u/C_u\) therefore shows that \(U_u\cap N_w(F_{w,j})\) maps onto \(\prod_{i\in F_{u,j}}S_{u,i}\) under \(q_u\). The first group in (59) is normal in \(N_u\) and has the same image. This image is perfect. Lemma 78 proves (i).

Every domain occurring in (59) contains \(U_x\), and all the factor maps are \(U_x\)-equivariant. Its conjugation therefore permutes the whole definition with \(j\), proving (ii). The images in (i) generate \(N_u/C_u\), and \(C_u\le A\); since \(U_u=AN_u\), this proves (iii).

For \(u\ne v\), the two lifts in (iv) lie in disjoint coordinate support products of \(N_{u\cup v}/C_{u\cup v}\). They commute in that quotient. For \(u=v\) the identical argument takes place in \(N_u/C_u\). Core inclusion along \(U_x\le U_u\le U_{u\cup v}\) gives the asserted containment in \(C_x\).

Finally suppose that \(\alpha(C_x)=1\). Then \(\beta_u(C_u)=1\). The kernel of \(\beta_u\) supplements \(A\) and is not contained in \(C_u\), so it contains \(N_u\) by the unique-minimal-normal-subgroup argument in Lemma 80. This proves (v). ◻

Proof. Let \(\pi:R_0\to R_0/C\) be the quotient map. Start with \(B=R_0\). At every stage \(B\) will be normal in \(R_0\) and satisfy \(\pi(B)=R_0/C\). For a fixed \(t\), replace \(B\) by \[ F_t(B)=\bigcap_{\ell\in L_t}(B\cap D_t)^\ell, \qquad K^\ell=\ell^{-1}K\ell. \tag{61}\] The subgroup \(B\cap D_t\) still maps onto \(R_0/C\) by Lemma 78. Put \(K=B\cap D_t\). The subgroup \(K\) is normal in \(D_t\), because \(B\trianglelefteq R_0\) and \(D_t\leq R_0\). Since \(L_t\) normalizes \(D_t\), every conjugate \(K^\ell\) lies in \(D_t\) and is normal in \(D_t\). For \(d\in D_t\) and \(\ell\in L_t\), the assumption \([D_t,L_t]\leq D_t\cap C\) gives \(\pi(d^\ell)=\pi(d)\). Consequently \(\pi(K^\ell)=\pi(K)=R_0/C\) for every \(\ell\). This applies the quotient map only to elements of \(D_t\leq R_0\); it does not require \(C\) to be normalized by \(L_t\). Applying Lemma 78 inside \(D_t\) shows that \(F_t(B)\) has full image.

The new subgroup is normal in \(R_0\). Indeed, \(R_0\) normalizes \(B\), \(D_t\), and \(L_t\). For \(r\in R_0\) one therefore has \((K^\ell)^r=(K^r)^{\ell^r}=K^{\ell^r}\), and \(\ell^r\in L_t\). Thus \(R_0\) permutes precisely the conjugates intersected in (61). The operation is descending, since the identity conjugate occurs. Apply \(F_1,\ldots,F_s\) in repeated rounds. Finiteness forces stabilization, and at a stable round every intermediate inclusion is equality. The resulting subgroup \(B_*\) is therefore normalized by each \(L_t\) as well as by \(R_0\). Thus \(B_*\trianglelefteq W\) and \(B_*\le R_0\), while \(\pi(B_*)=R_0/C\). Since \(B_*\le\operatorname{core}_W(R_0)\), the asserted core also has full image. ◻

Proof. For \(u\in\mathcal U_x\) set \[D_{u,k}=U_x\cap N_u(F_{u,k}).\] The transported subdirect group \((U_x\cap N_u)/C_u\) maps isomorphically onto \(N_x/C_x\). Selecting its strips over \(k\) shows that \[ D_{u,k}\le R_k,\quad D_{u,k}C_x=R_k, \quad D_{u,k}\cap C_x=C_u. \tag{62}\] It is normal in \(R_k\): all elements of \(R_k\le N_x\) fix the factor labels \(I_x\), and the support-preimage construction is equivariant.

For \(j\ne k\), both \(D_{u,k}\) and \(L_{u,j}\) lie in \(N_u\) and have disjoint supports modulo \(C_u\). Thus \[[D_{u,k},L_{u,j}]\le C_u.\] Since \(C_u\le D_{u,k}\), this also proves that \(L_{u,j}\) normalizes \(D_{u,k}\). Moreover \(R_k\) normalizes \(L_{u,j}\) by Lemma 81(ii), since \(R_k\) fixes every label of \(I_x\). We can therefore apply Lemma 82 to \(R_k\), its normal subgroup \(C_x\), and all the lifts \(L_{u,j}\) with \(j\ne k\), using the subgroup \(D_{u,k}\) for the lift labelled by \((u,j)\).

Define canonically \[W_k=\langle R_k,L_j:j\ne k\rangle, \qquad B_k=\operatorname{core}_{W_k}(R_k).\] That lemma proves (i); the definition proves (ii). Since \(C_x\le R_k\le W_k\), it normalizes \(B_k\). Conjugation by \(A\) permutes the pairs \((W_k,R_k)\) and hence their cores, proving (iii). Their images in \(N_x/C_x\) are the distinct nontrivial coordinate factors, so the \(B_k\) are distinct. ◻

Proof. Write \(g\) as a word whose letters belong to the individual \(P_\lambda\); inverses are allowed because these are subgroups. Interchange adjacent letters with different labels to move the \(k\)-labelled letters to the end. Each interchange introduces a commutator in \(C\). Move that error to the right through the remaining word, using \[c\ell=(c\ell c^{-1})c.\] The conjugated letter remains in its original \(P_\lambda\), so its label does not change. Each interchange reduces the number of pairs consisting of a \(k\)-letter preceding a non-\(k\)-letter. The process terminates with the asserted expression. This is a word calculation; it does not form a quotient by \(C\) or require \(C\trianglelefteq L\). ◻

Proof. The subgroup \(\alpha(C_x)\) is normal in \(\beta_x(U_x)=H\). Lemma 71 reduces the assertion to excluding \(\alpha(C_x)=1\). Assume that equality and use the lifts and detecting subgroups just constructed. Put \[L=\langle L_j:j\in I_x\rangle.\] The group \(A\) normalizes \(L\).

Let \(a\in A\cap L\). We show that \(a\) fixes every member of \(I_x\). For subgroup conjugation use the right-action notation \(K^g=g^{-1}Kg\). If \(a\) sent \(k\) to \(h\ne k\), equivariance would give \(B_k^a=B_h\). Lemmas 81 and 84, with \(C=C_x\) and the individual \(L_{u,j}\) as labelled subgroups, give \[a=pqc,\qquad p\in\langle L_j:j\ne k\rangle,\quad q\in L_k,\quad c\in C_x.\] The element \(p\) normalizes \(B_k\), \(q\) normalizes \(B_h\), and \(c\) normalizes both. It follows successively that \[B_k^{qc}=B_h, \qquad B_k^q=B_h, \qquad B_k=B_h,\] contradicting Lemma 83. Thus \(a\) fixes the factor set. Lemma 80 now gives \(\alpha(a)=1\).

Consequently \(A\cap L\le\ker\alpha\). Because \(A\) normalizes \(L\), the formula \[\theta:AL\longrightarrow\operatorname{Aut}(T), \qquad \theta(a\ell)=\alpha(a)\] defines a homomorphism. It is well defined by the intersection inclusion; multiplication is respected since \(L\trianglelefteq AL\). For each \(u\in\mathcal U_x\), Lemma 81(iii),(v) says that \(U_u\) is generated by \(A\) and lifts killed by both \(\theta\) and \(\beta_u\). Hence \(\theta|_{U_u}=\beta_u\).

The three sets \(u\in\mathcal U_x\) have full union \(\{1,\ldots,4\}\). Their generated domain is \(AL\), so this common extension contradicts [ext:e4]. The assumption \(\alpha(C_x)=1\) is impossible. ◻

Proof. The assertion is immediate for an atom, so suppose that \(|z|\ge2\). Every \(C_x\) in the intersection contains \(C_z\). Pass to \(U_z/C_z\) and use bars for the images of \(A,U_x,C_x\). Put \[N=N_z/C_z,\qquad E_x=(U_x\cap N_z)/C_z.\] The transport isomorphism in [ext:e2] gives \(E_x\cap\overline C_x=1\). Both are normal in \(\overline U_x\), so \([E_x,\overline C_x]=1\).

Since \(U_z=AN_z\) and \(A\le U_x\), one has \(\overline U_x=\overline A E_x\). Let \(M=\langle E_x:x\subseteq z,\ |x|=1\rangle\). Each \(E_x\) is normalized by \(\overline A\). The compatible-join generation in [ext:e2] therefore gives \[\overline U_z=\overline A M,\] so \(M\trianglelefteq\overline U_z\). It is a nontrivial subgroup of the minimal normal subgroup \(N\), and hence \(M=N\). Thus \(\bigcap_x\overline C_x\) centralizes \(N\). The centralizer is normal in \(\overline U_z\) and intersects \(N\) in \(Z(N)=1\). If it were nontrivial it would contain the unique minimal normal subgroup \(N\), a contradiction. The intersection is therefore trivial, which proves the equality of cores.

Each \(C_x\) is normal in \(A\) and maps over \(\operatorname{Inn}(T)\) by Proposition 85. Apply Lemma 78 with \(P=\operatorname{Inn}(T)\) to obtain the image assertion. ◻

Proof. Take the two proper Boolean nodes \[z=\{1,2,3\},\qquad w=\{2,3,4\}.\] Lemma 86 gives \(\alpha(C_z),\alpha(C_w)\ge\operatorname{Inn}(T)\). Lemma 79 therefore extends \(\beta_z\) and \(\beta_w\) to \(\langle U_z,U_w\rangle\). Since \(z\cup w=\{1,\ldots,4\}\), this contradicts [ext:e4]. ◻

Proof of Theorem 2. Choose the finite decorated lattice of Proposition 66. If either it or its dual had a finite-group interval representation, Proposition 77 would produce the configuration excluded by Theorem 87. Hence this lattice has no such representation, proving Theorem 2. ◻