Proof. For \(C_T=2I-b_T>0\), the nonnegative symmetric matrix \(b_T\) has spectral radius less than \(2\). Hence \[C_T^{-1}=\frac12\sum_{n\geq0}(b_T/2)^n, \qquad p_T=C_T^{-1}b_{T,\widehat S\setminus T}p_{\widehat S\setminus T}.\] The first formula gives nonnegativity; a path between two vertices in one connected block gives a strictly positive entry in some power, proving strict positivity there. The second formula gives the assertions about harmonic extension.

For the final assertion, the following identity holds for every real vector \(u\) on \(G\): \[ u^{\mathsf T}C_Gu =\sum_{\substack{i<j\\i,j\in G}}b_{ij}v_iv_j \left(\frac{u_i}{v_i}-\frac{u_j}{v_j}\right)^2 +\sum_{i\in G}\frac{(C_Gv)_i}{v_i}u_i^2. \tag{2}\] If one entry of \(C_Gv\) is positive, vanishing of the first sum makes all ratios \(u_i/v_i\) equal, and the corresponding term of the second sum makes that common ratio zero. The form is then positive definite. Otherwise the assumed nonnegative vector \(C_Gv\) is zero. ◻

Proof. Units and tokens pair perfectly. For arrows, pivotal adjunction identifies a map \(H\to D^1_{ji}\otimes L\) with a map \(D^1_{ij}\otimes H\to L\). Pairing it against \(L\to D^1_{ij}\otimes H\) is a semisimple trace pairing, nondegenerate because the trace on each simple matrix block is multiplied by a nonzero dimension. The two orders of a backtrack have the same trace by the pivotal partial-trace identity. Hence \(\tau(ab)=\tau(ba)\), with no sign from internal degree.

The positive-length ideal is nilpotent and its quotient is the semisimple diagonal, proving the radical assertion. A token is annihilated by the radical. A nonzero arrow pairs with an opposite arrow, and a nonzero degree-zero element pairs with a token; neither can lie in the projective socle. This proves the socle assertion.

The map \(f\mapsto\tau\circ f\) gives the first identification in (6). A map of degree \(k\) gives a functional of degree \(k-2\), which accounts for \(\langle2\rangle\). With the bimodule convention \((a f b)(n)=f(na)b\), symmetry gives \[\tau(f(na)b)=\tau(bf(na))=\tau(f(bna)),\] exactly the vector-dual bimodule action. The trace identifies \(D_\mathbb C(P_\alpha)=Q_\alpha\langle-2\rangle\) and \(D_\mathbb C(Q_\beta)=P_\beta\langle-2\rangle\). Reversing the tensor factors and then applying the shift \(2\) proves the second formula. Shifted vector duality squares to the identity. Any contravariant equivalence preserves the additive radical, characterized by invertibility of \(1-gf\) for all opposite maps \(g\). ◻

Proof. The strictly positive vector \(v=\phi(C_T^{-1}\mathbf1)\) satisfies \(\widetilde C_{\widetilde T}v=\phi\mathbf1>0\). Applying (2) to each connected unfolded block proves positive definiteness on the whole unfolded space, including directions outside \(\phi(\mathbb R^T)\). An infinite edge cannot occur inside \(T\), since its two-by-two Cartan matrix is singular. Equation (3) therefore makes every off-diagonal entry of \(\widetilde C_{\widetilde T}\) either \(0\) or \(-1\). The finite Coxeter criterion applies to this simply laced matrix.

There are no arrows between distinct vertices of one color. The corresponding reflections commute, and their product acts on the folded space as \(s_i\), by (8). Choose \(p\) with \(C_Tp>0\). Then \(\phi p\) is strictly inside a full unfolded fundamental chamber. For an arbitrary word representing \(w\), let \(g\) be its lifted product; then \(g\phi p=\phi(wp)\). A root has coefficients either all nonnegative or all nonpositive in the simple-root basis, so pairing it with a chamber-interior point detects its sign. The row at \((i,L)\) of \(\widetilde C\phi(wp)\) is \(w_L(Cwp)_i\). These row signs are the signs of the pairings with \(g^{-1}\epsilon_{(i,L)}\) and \(w^{-1}\epsilon_i\), respectively. Consequently the unfolded left descents are exactly all vertices over the original left descents. Applying the same reasoning to inverse words proves the right-descent assertion.

A word representing the identity fixes \(\phi p\). The stabilizer of a chamber-interior point in a finite Coxeter group is trivial, so its lift is the identity. Thus the lifted product depends only on \(w\). If an original reduced expression is extended by an ascent, every vertex of the added color is an ascent of the lifted prefix. Acting at one of those vertices does not change the signs at the others because they are mutually orthogonal. The whole block is therefore a sequence of ascents. Induction proves reducedness of the lift. The longest original element has every simple descent; its lift has every unfolded simple descent, so is the longest unfolded element. ◻

Proof. We first verify the inverse explicitly. The product \(B_\alpha D_\alpha\) has \(M_\alpha\) in degree \(-1\), \[Z\oplus P_\alpha\otimes_\mathbb C(e_\alpha Ze_\alpha) \otimes_\mathbb CQ_\alpha\langle-2\rangle\] in degree zero, and \(M_\alpha\langle-2\rangle\) in degree \(1\). Write \(\theta_\alpha=t_\alpha/w_L\) when \(\alpha=(i,L)\), so that \(\tau(\theta_\alpha)=1\). Split the middle factor into its unit and \(\theta_\alpha\) lines. The incoming differential onto the token summand is an isomorphism, up to the tensor sign: its coefficient is the dual-basis identity \[\sum_a\tau(qp_a)q_a=q.\] The token has degree \(2\), so this summand is unshifted \(M_\alpha\). Cancel that isomorphism. The remaining unit summand maps identically onto \(M_\alpha\langle-2\rangle\) by multiplication with \(e_\alpha\); cancel it as well. Only the regular bimodule \(Z\) remains. In the opposite product \(D_\alpha B_\alpha\), the dual-basis identity gives the same incoming cancellation, followed by the unit cancellation. These cancellations take place in bimodule complexes, not just after evaluating on projectives.

We next prove the elementary braid relation using the twist-conjugation argument of (Seidel and Thomas 2001, Lemma 2.11 and Propositions 2.12–2.13). For a bounded perfect left complex \(P\), let its evaluation twist be the bimodule cone \[T_P=\mathop{\mathrm{Cone}}\bigl(P\otimes_\mathbb C\mathop{\mathrm{Hom}}_Z(P,Z)\longrightarrow Z\bigr).\] It is unchanged by an internal or cochain shift of \(P\). A homotopy equivalence of \(P\) identifies its evaluation map, and hence its cone, up to homotopy. If \(B\) is a product of the elementary invertible complexes, each of its terms is projective on both sides. Tensor–Hom adjunction identifies its tensor adjoint with \(B^\vee=\mathop{\mathrm{Hom}}_Z(B,Z)\); the evaluation and coevaluation identities hold at chain level. Since \(B\) is invertible, this adjoint is its inverse in the bimodule homotopy category. Lemma 7 keeps its terms in the same class. Tensor–Hom and evaluation now give \[ B T_P B^{-1}\simeq T_{BP}. \tag{11}\]

If distinct vertices \(\alpha,\beta\) have no arrows between them, \(M_\alpha M_\beta=M_\beta M_\alpha=0\), so both \(B_\alpha B_\beta\) and \(B_\beta B_\alpha\) are the simultaneous evaluation cone with projective term \(M_\alpha\oplus M_\beta\). If there is exactly one arrow in each direction, the complexes \[B_\alpha P_\beta =[P_\alpha\langle1\rangle\longrightarrow P_\beta], \qquad D_\beta P_\alpha =[P_\alpha\longrightarrow P_\beta\langle-1\rangle]\] have degrees \((-1,0)\) and \((0,1)\). Both maps are nonzero by the perfect backtrack pairing, and their common arrow space is one-dimensional. Rescaling one term gives \[B_\alpha P_\beta\simeq (D_\beta P_\alpha)\langle1\rangle[1].\] Using \(T_{P_\gamma}=B_\gamma\) in (11) therefore gives \(B_\alpha B_\beta D_\alpha\simeq D_\beta B_\alpha B_\beta\). Multiplication on the left by \(B_\beta\) and on the right by \(B_\alpha\) proves \[B_\alpha B_\beta B_\alpha \simeq B_\beta B_\alpha B_\beta.\]

Vertices within one color are orthogonal. Their twists commute, their inverse products are the \(D_i\), and all mixed terms in the expansion vanish. This proves the simultaneous cone (10). For \(m_{ij}<\infty\), the two alternating color words of length \(m_{ij}\) are reduced expressions for the longest element of the spherical rank-two subsystem. Lemma 8 lifts both to reduced expressions for the same longest unfolded element. Matsumoto’s Theorem relates them by commutations and three-term braid moves, which we have just proved as bimodule homotopies. For \(m_{ij}=\infty\) there is no relation to impose. This proves the action for arbitrary labels.

It remains to verify the natural commutation with labels. On (5), \(T_H\) tensors the label map on the right by \(\mathrm{id}_H\). Evaluation of an edge and its dual always occurs to the left of the label, so composition and path length are preserved. On \(P_{j,D}\), the evaluation term for color \(i\) has the canonical semisimple description \[\bigoplus_{a,L}P_{i,L}\langle a\rangle\otimes_\mathbb C \mathop{\mathrm{Hom}}_{\mathscr C}(L,D^a_{ij}\otimes D) \cong\bigoplus_a P_{i,D^a_{ij}\otimes D}\langle a\rangle.\] Its evaluation is the tautological label map. Tensoring by \(H\) replaces \(D\) by \(D\otimes H\) in this description, naturally in \(D\) and in every projective morphism. Thus \(T_H\) commutes with the positive evaluation cone. Composing this natural identification with the inverse equivalence on both sides proves the inverse commutation without requiring \(T_H\) to be invertible.

Invertible functors preserve all morphism spaces, including both shifts, and hence preserve bigraded Hom cohomology between projective complexes. Finally, tensoring an explicit chain homotopy with any module complex preserves its defining equations with the usual cochain signs. No flatness assertion about the additional module complex is needed. ◻

Proof. Proposition 9 gives \(E_L(x)\simeq T_L(E_{\mathbf1}(x))\). Tensoring labels preserves the positive path length of a minimal differential, and therefore preserves minimality. If a multiplicity object \(D\) occurs at a fixed color and pair of placements in the unit-label complex, its contribution in the \(L\)-member is \(D\otimes L\). For each target simple \(H\), Equation (4) says that the weighted multiplicity is \(w_H\dim_{\mathscr C}D\). Summing over placements of one level proves (13).

Lemma 17 proves that label tensoring commutes with layers, truncations, and the top and socle tests. Its proof uses the exactness of tensoring in the semisimple category, together with (5) and (4). It gives the stated proportionality for functorially constructed subobjects and for tops and socles. Positivity of all label weights then proves the assertion about occurrence at every vertex of a color.

There are finitely many labels and finitely many terms in each complex. To see local finiteness, fix one target label \(H\). For a bound \(R\), Equation (13) expresses \(w_Hm_i^d\leq w_HR\) as a nonnegative integer combination of the finitely many fixed positive numbers \(w_L\). Each coefficient is bounded separately. There are thus only finitely many possible values below \(R\).

For an unfolded vertex \(\beta\) outside color \(i\), \(U_\beta e_\alpha=0\) for every \(\alpha\) in that color. The explicit complexes (9) consequently give \(U_\beta B_i=U_\beta=U_\beta D_i\). Tensoring a minimal left complex with \(U_\beta\) kills its radical differential, leaving exactly its generator spaces at \(\beta\) in their two placements. This proves exterior invariance, and in particular the statement about the \(f\)-generator and its folded counts.

The initial projective \(P_{f,L}\) is indecomposable, and an equivalence preserves indecomposability, proving the first final assertion. A disconnected color support would split its minimal complex into the summands on different components: a differential uses only arrows, which join adjacent colors, and tokens, which stay at one color. Every nonzero minimal summand is noncontractible, since reducing a putative contracting homotopy equation modulo the radical would give an identity equal to zero. Such a splitting contradicts indecomposability. Exterior invariance ensures that \(f\) belongs to the support. ◻

Proof. Differentials do not raise level, so the indicated low terms form a subcomplex. A degree-zero map from a generator of level \(d\) to one of level \(e\) requires a path of length \(d-e\). There are no such maps from levels at most \(0\) to levels at least \(1\). This proves the required orthogonality. The cochain shift \([1]\) lowers levels by one, giving the shift-closure axioms, and the short exact sequence of low subcomplex, full complex and high quotient gives the truncation triangle. Finitely many occupied levels give boundedness.

A degree-zero map between pure complexes at the same level has only length-zero entries. It is precisely a map of graded coefficient representations. A homotopy between two such maps would require path length \(-1\), so there are no homotopies to quotient out. Conversely a finite coefficient representation gives a pure complex, since its backtrack relations say exactly that the differential squares to zero. Kernels and cokernels, formed in each coefficient space, retain the relations. This identifies the heart, its subobjects and its cohomology layers. ◻

Proof. In the positive case apply \(F\) to the triangle \[\tau_{\le d-1}E\longrightarrow\tau_{\le d}E \longrightarrow Y^d(E).\] The first term after applying \(F\) has levels at most \(d\); the last has levels \(d,d+1\). Hence the flanking groups \(H^{d-1}_{\mathrm{lay}}F(Y^d(E))\) and \(H^{d+1}_{\mathrm{lay}}F(\tau_{\le d-1}E)\) vanish, giving a short exact sequence at \(d\). Input levels at most \(d-2\) contribute nothing there, identifying the left term in (15); input levels at least \(d+1\) contribute nothing there, identifying its middle term. For the negative case apply the same argument to \(\tau_{\le d}E\to\tau_{\le d+1}E\to Y^{d+1}(E)\). After \(F\), its first term has levels at most \(d\) and its last has levels \(d,d+1\). Discarding input levels below \(d\) and above \(d+1\) gives (16). ◻

Proof. After the first operation the quotient by the level-\(d\) truncation has levels at least \(d+1\). All subsequent operations preserve that bound. Its long exact layer sequence shows that discarding this quotient does not change the final level \(d\). Repeat after each operation. ◻

Proof. The test of a pure layer has path lengths \(0,1,2\), in total degrees \(d,d+1,d+2\). Its length-zero cycles are the common kernels of the outgoing coefficient maps. At the token end, the images of the incoming differential are exactly the incoming coefficient images, by the perfect opposite-arrow pairings. This proves (17).

Every term of \(e_\alpha E\) comes from a generator at \(\alpha\) or a neighbor, and its total degree is the generator level plus the path length. Under the upper bound, only level-\(d\) tokens can occur at \(d+2\). A token differential into them from length zero would require a generator at level \(d+1\), which is absent. The remaining boundaries are the same incoming linear images. The lower statement follows from the length-zero end in the same way. Finally, coefficient arrows strictly raise cochain placement in a finite array. A nonzero array therefore cannot have zero socle everywhere, or be generated by its incoming images everywhere. Reversing module sides proves the right statements. ◻

Proof. For pure input at \(d\), the evaluation cone inserts generators in the acted color at levels \(d+a-1\), where \(a=0,1,2\) is the path length. The length-zero insertions at \(d-1\) map by identities, modulo the radical, to the old generators in that color. Cancelling those pairs removes every generator below \(d\). No generator was inserted above \(d+1\), and only the acted color can occur above \(d\).

For one vertex \(\alpha\) of color \(i\), put \(A=e_\alpha E\). Distinct vertices of that color are orthogonal, so the acted test is \[ \mathop{\mathrm{Cone}}\bigl(A\oplus A\langle2\rangle \xrightarrow{(\mathrm{id},\theta_\alpha)} A\bigr). \tag{20}\] Projection to the token source \(A\langle2\rangle[1]\) is a chain map with contractible kernel \(\mathop{\mathrm{Cone}}(\mathrm{id}_A)\), proving (19). This is a homotopy calculation for any \(A\), so it neither assumes projectivity nor merely asserts a quasi-isomorphism. The right calculation uses the same cone.

For pure input, the degree-\((d+3)\) output test equals the old degree-\((d+2)\) test. Lemma 14 identifies the latter with the old top. Since the upper layer is supported on a color with no internal arrows, the former counts its generators. This proves the upper-layer assertion.

For the multiplicity formula, count generators with sign \((-1)^d\). The signed contributions of a cancelled scalar pair sum to zero, because its levels are consecutive. In the evaluation cone the inserted signed contribution from a generator at \(\beta\) to an acted vertex \(\alpha\) is minus the alternating path count \[2\delta_{\alpha\beta}-(2I-\widetilde C)_{\alpha\beta}.\] Consequently the total signed vector transforms by \(r_i\). On pure input the bottom vector minus the upper vector is therefore \(r_i\dim M\), proving (18). The same calculation also shows, for general inputs, that \[ \sum_d(-1)^d\dim Y^d(F_iE) =r_i\sum_d(-1)^d\dim Y^d(E). \tag{21}\]

By (19), the degree-\(d\) test at an acted vertex is zero. Lemma 14 identifies it with the socle of the bottom layer. At another vertex \(\beta\), the first term in the tested evaluation triangle is a sum of \((e_\alpha M)\langle1\rangle\), over adjacent acted vertices. It has no degree-\(d\) cohomology, and its degree-\((d+1)\) cohomology is the sum of the old socles at those vertices. If these socles and the old socle at \(\beta\) vanish, the long exact sequence gives zero degree-\(d\) cohomology after the move. This proves preservation of socle vanishing on \(J\). ◻

Proof. Let \(F\) be the equivalence and let \(N\) lie in the heart. For any \(A\) in levels at most \(-2\), its image \(FA\) has levels at most \(-1\), so \[\mathop{\mathrm{Hom}}(A,F^{-1}N)=\mathop{\mathrm{Hom}}(FA,N)=0.\] The orthogonal characterization of the \(t\)-structure implies that \(F^{-1}N\) has levels at least \(-1\). If \(B\) has levels at least \(1\), then so does \(FB\), and \[\mathop{\mathrm{Hom}}(F^{-1}N,B)=\mathop{\mathrm{Hom}}(N,FB)=0.\] Hence \(F^{-1}N\) has levels at most \(0\). Shifting proves the amplitude bound on every pure input. Applying (19) to \(F_i^{-1}E\) and using the bimodule homotopy inverse proves (22), including for nonprojective inputs and for right complexes. Applying (21) and \(r_i^2=1\) proves the inverse reflection formula.

For a vertex \(\beta\) outside color \(i\), the right simple satisfies \[U_\beta B_{\sigma_i}^{\pm1}\simeq U_\beta.\] Indeed the inserted projective bimodule terms have left vertex in color \(i\) and are annihilated by \(U_\beta\). Tensoring a minimal left projective complex with \(U_\beta\) kills its radical differential and records precisely the generators at \(\beta\) with both placements. The displayed equivalence proves the exterior assertion. ◻

Proof. Tensoring a projective label by \(H\) preserves its cochain and internal placements. It preserves radical path lengths, hence commutes with the low-level subcomplex and the high-level quotient. On coefficient representations it is exact, since the label category is semisimple. At a fixed color and placements, the coefficient spaces, tests and their cohomology are obtained by applying \(\mathop{\mathrm{Hom}}_{\mathscr C}(K,-)\) to the corresponding label objects. Exact label tensoring also tensors these objects and their cohomology. The weighted multiplicity identity \[\sum_L w_L[D\otimes L:K]=w_K\dim_{\mathscr C}D\] therefore gives proportionality for generators and for the extremal tests, hence for tops and socles by Lemma 14. All weights are positive, proving the occurrence assertion. Twists and truncations commute with label tensoring, so the argument applies to the constructed families as well. ◻

Proof. The acted test shifts up by one in total degree, and nonneighboring tests are unchanged. At a neighboring vertex \(\beta\) the evaluation triangle is \[ (e_\alpha E)\langle1\rangle\longrightarrow e_\beta E \longrightarrow e_\beta B_\alpha E. \tag{24}\] If the old maximum is \(t\), the last term has no test cohomology above \(t\). Thus precisely the asserted condition produces height \(t+1\), and only at \(\alpha\). If the old test at \(\alpha\) has maximum \(t-1\), its shifted test reaches \(t\). If it has maximum below \(t-1\), an old degree-\(t\) test elsewhere survives: in (24) the map into that degree has zero source. This proves monotonicity. The same triangles preserve total vanishing. Reversing sides proves the right case; a color twist is a product of the commuting unfolded twists in its fiber. ◻

Proof. We prove the left statement by induction on word length. Write \(b=\alpha b'\) and \(N=B_{b'}E\). If the final action raises height, its unique new highest vertex is \(\alpha\) by Lemma 18, and there is nothing to prove. Otherwise \[h_J(N)=h_J(B_\alpha N)=t>h_J(E).\] The vertex \(\alpha\), when it is a highest vertex, already divides \(b\). If a highest vertex \(\beta\ne\alpha\) is not adjacent to \(\alpha\), it was a highest vertex of \(N\). By induction it divides \(b'\), and it commutes past \(\alpha\).

Suppose instead that \(\beta\) is adjacent to \(\alpha\). Since height did not increase, \(H^t(e_\alpha N)=0\). The long exact sequence of (24) gives \[ H^t(e_\beta B_\alpha N) \cong\operatorname{coker}\bigl( H^{t-1}(e_\alpha N)\longrightarrow H^t(e_\beta N)\bigr). \tag{25}\] This cokernel is nonzero, so \(\beta\) is a highest vertex of \(N\). Induction gives a positive factorization \(b'=\beta v\). Put \(N_0=B_vE\), so \(N\simeq B_\beta N_0\).

We now identify the arrow in (25); this is the step that permits nonprojective inputs. Set \(A_0=e_\beta N_0\) and \(C_0=e_\alpha N_0\), and choose opposite arrows \[r\in e_\alpha Ze_\beta,\qquad s\in e_\beta Ze_\alpha,\qquad sr=\kappa\theta_\beta,\quad\kappa\ne0.\] The evaluated cones are \[\begin{split} e_\beta N&=\mathop{\mathrm{Cone}}\bigl(A_0\oplus A_0\langle2\rangle \xrightarrow{(\mathrm{id},\theta_\beta)}A_0\bigr),\\ e_\alpha N&=\mathop{\mathrm{Cone}}\bigl(A_0\langle1\rangle \xrightarrow{r}C_0\bigr). \end{split}\] Let \(\pi:e_\beta N\to A_0\langle2\rangle[1]\) be the token projection in (20). Multiplication by \(s\) gives \((e_\alpha N)\langle1\rangle\to e_\beta N\). On its cone source, the product \(sr\) is \(\kappa\) times the token, whereas its cone target lands in the target \(A_0\), killed by \(\pi\). Therefore \(\pi s\) is, up to the cochain sign and the nonzero scalar \(\kappa\), exactly the connecting projection of the second cone.

Since \(\pi\) is a homotopy equivalence, the image of the map in (25), identified in \(H^{t-1}(e_\beta N_0)\), is \[\ker\bigl(H^{t-1}(e_\beta N_0) \longrightarrow H^t(e_\alpha N_0)\bigr).\] Its cokernel is nonzero, so this last arrow has nonzero image. In particular \(H^t(e_\alpha N_0)\ne0\). By height monotonicity, \(h_J(N_0)\le h_J(N)=t\), hence \(h_J(N_0)=t>h_J(E)\). Applying induction to \(v\) gives \(v=\alpha v'\) in the positive monoid. The simply-laced braid relation now extracts the required vertex: \[b=\alpha\beta\alpha v'=\beta\alpha\beta v'.\] This completes the induction. On the right, use the right evaluation cones and opposite arrows, with the final action on the right. The same proof gives right divisibility. ◻

Proof. First consider a reduced unfolded word. Its test height can increase at most once. Otherwise at a second increase the new letter was a highest-test vertex of the preceding result, whose height already exceeded the starting value. By Lemma 19 it can be extracted on the last-action side of that preceding positive word. Adding it there would give two equal consecutive Coxeter generators, contradicting reducedness.

A pure input at \(d\) has test height at most \(d+2\). Positive twists never produce levels below \(d\). If the output had highest layer \(a>d+1\), exterior invariance would force that layer to be supported on \(\widetilde J\). It has a nonzero top there, so Lemma 14 gives a nonzero test at \(a+2>d+3\). This contradicts the single-increase bound. If the initial tests vanish, they stay zero and give the same contradiction. This proves amplitude \([0,1]\) for reduced unfolded words.

By Lemma 8, reduced color words unfold to reduced words, proving the positive assertion. The inverse of a negative reduced lift is the positive lift of the reversed reduced expression. Lemma 16 therefore proves the negative assertion. The right argument has the same degree bounds. ◻

Proof. The high quotient \(R=\tau_{\ge q+1}E\) is supported outside \(J\cup\partial J\). Its tests at \(J\) vanish termwise, so every evaluation or coevaluation term in \(J\) vanishes and \(GR\simeq R\). Apply \(G\) to the truncation triangle \(\tau_{\le q}E\to E\to R\). Negative twists do not increase levels. The low term therefore remains at levels at most \(q\), while the unchanged quotient is at levels at least \(q+1\). The long exact layer sequence identifies all higher layers with those of \(R\) and identifies level \(q\) with that of \(G\tau_{\le q}E\). Apply \(G\) next to \(\tau_{\le q-1}E\to\tau_{\le q}E\to Y^q(E)\). Its first term stays below \(q\), proving (26). ◻

Proof. Negative preservation is Lemma 23. For the positive assertion use the same high quotient \(R\). It is unchanged. On the low truncation, the only possible level-\(q+1\) output is the upper part of the old level-\(q\) layer. By Lemma 15 this is precisely its top in the operated color; it vanishes by hypothesis.

For accessibility, let \(d>q\) be the greatest occupied level on \(J\). The joint cap at \(d\) holds. At that level the portion on \(J\) is a nonzero direct summand of a pure layer, since the boundary is absent. It has a nonzero top somewhere on \(J\). Apply a negative letter at such a color and continue while level \(d\) remains.

The selected colors form a reduced Coxeter expression. Indeed, suppose the letters already selected form a reduced word \(w\), with the last operation on the left. If the next color \(j\) were a left descent, Matsumoto moves could put \(j\) at the last-action end of a reduced expression for \(w\). The corresponding negative words give the same action. Every intermediate negative operation preserves the joint \(d\)-cap. Immediately before the last inverse twist, the tests in color \(j\) vanish above \(d+2\); after it they vanish at \(d+2\) as well, by the negative test shift. Lemma 14 would then make the current level-\(d\) top at \(j\) zero, contrary to its selection. Thus each choice is a Coxeter ascent. The longest length in \(W_J\) bounds the number of choices before level \(d\) disappears. No larger level is recreated. Repeating at successive excess levels reaches \(q\) in finitely many stages. The empty set \(J\) requires no operations. ◻

Proof. For each \(w\in W_J\) choose a negative reduced lift \(G_w\). Put \(M_L=Y^q(E_L(x))\). The reduced-word amplitude in Corollary 21 places \(G_wM_L\) at levels \(q-1,q\). Apply \(G_w^{-1}\) to its low/high truncation triangle, obtaining \[A_L\longrightarrow M_L\longrightarrow C_L.\] The endpoint bounds are \([q-1,q]\) and \([q,q+1]\). The long exact layer sequence forces the possible level \(q-1\) of \(A_L\) and level \(q+1\) of \(C_L\) to vanish. Thus both are pure at \(q\) and \[ 0\longrightarrow A_L\longrightarrow M_L\longrightarrow C_L \longrightarrow0 \tag{29}\] is exact in the shifted heart. Its subobject is an actual graded subrepresentation by Lemma 11. The low part of \(G_wM_L\) is supported on \(\widetilde J\), since outside-color counts are unchanged, and applying \(G_w^{-1}\) keeps that support.

The inclusion in (29) is obtained by twisting, canonical truncation, and inverse twisting. These operations commute with label tensoring, including their morphisms into \(M_L\). The weighted fusion identity therefore gives (28) for a vector \(s_w\). This does not require compatible choices of Gaussian eliminations among the family members.

The quotient has folded dimension \(m^q(x)-s_w\). Applying \(G_w\) recovers the high part at level \(q\), whose signed generator count is therefore \(w(m^q(x)-s_w)\). Lemma 23 identifies it with the actual level-\(q\) layer at the operated representative. That representative is jointly capped, so (27) and \(wp=p\) give \[ \bigl(w(e-s_w)\bigr)_J\ge0. \tag{30}\] In the strict case all these coordinates are strictly positive.

On the root-coordinate space supported on \(J\), finite Coxeter chambers cover the dual space. Thus every real functional has the form \(\lambda(v)=\sum_{j\in J}c_j(wv)_j\) for some \(w\) and \(c_j\ge0\). Equation (30) gives \[\lambda(e)\ge\lambda(s_w) \ge\min_{s\in\mathcal V}\lambda(s), \qquad \mathcal V=\{s_w:w\in W_J\}\cup\{0\}.\] Separation proves membership in this finite convex hull. If the inequalities are strict and \(\lambda\ne0\), some \(c_j>0\), so the first inequality is strict. A boundary point, including a point of a hull in a proper affine subspace, would admit a nonzero supporting functional with equality. This proves ordinary interior membership. For \(J=\varnothing\) all the vectors are zero. ◻

Proof. First minimize the bimodule complex \(B_h\). Before minimization it has one regular bimodule \(Z\) at \((0,0)\), and all other terms are shifts of \(P_\alpha\otimes_{\mathbb C}Q_\beta\), with indices in \(\widetilde T\); tensoring two such terms has the same form because their middle tensor factor is a graded path space.

No indecomposable regular summand is isomorphic to a projective bimodule of this form. A connected-component regular bimodule is indecomposable: a central idempotent would separate its vertices, contrary to the nonzero arrows connecting them. Modulo the left and right radicals, its top contains the diagonal simple at every vertex of that component. The top of \(P_\alpha\otimes Q_\beta\) contains only the pair \((\alpha,\beta)\). The only remaining possibility is a singleton component with \(\alpha=\beta\); there the regular bimodule has dimension two and the projective bimodule has dimension four. Shifts do not change this obstruction. Thus all regular summands survive, only at \((0,0)\).

A radical differential between projective bimodules is a sum of path tensors with positive length on at least one side: a nonzero scalar component with both lengths zero would be an invertible entry and would have been canceled. Tensoring by \(U_\ell\) kills positive left lengths; every surviving projective differential has positive right length. Maps involving the regular term specialize to radical maps between the simple and a projective. There is no differential between residual simples, since they occur only in cochain degree zero. The specialized right complex is therefore already minimal, and its counts are the bimodule counts with first index \(\ell\).

The Frobenius trace gives adjoint duality \[\mathop{\mathrm{Hom}}_{Z\text{-left}}(M,Z)\cong\mathop{\mathrm{Hom}}_{\mathbb C}(M,\mathbb C)\langle2\rangle.\] It preserves additive radicals and sends \(P_\ell\otimes Q_\beta\) to \((P_\beta\otimes Q_\ell)\langle-2\rangle\). On complexes it reverses cochain placements as well. Thus \((c,u)\) becomes \((-c,-u-2)\). The adjoint of the invertible complex \(B_h\) is its inverse \(B_{h^{-1}}\), by Proposition 9. Minimality and the preceding specialization give (31).

If \(\ell\notin\widetilde J'\), every letter of \(A_{J'}\) fixes \(U_\ell\): its inserted bimodule terms tensor to zero. This proves row invariance. Apply (31) to \((ha)^{-1}=a^{-1}h^{-1}\) to obtain column invariance for \(a\in A_J\). ◻

Proof. An atom \(s\) right-divides a simple \(d\) exactly when \(s\in D_R(d)\), and \(sd\) is simple exactly when \(s\notin D_L(d)\). Failure of (32) therefore transfers a final atom of \(d_i\) to the beginning of \(d_{i+1}\) and enlarges the simple suffix at that extraction stage. This proves necessity.

For sufficiency, induct on prefixes. Suppose \(P=d_1\cdots d_{j-1}\) has greatest simple suffix \(d_{j-1}\) and write \(t(Pd_j)=cd_j\), since \(d_j\) is itself a simple suffix. If \(c\ne1\), choose a final atom \(s\) of \(c\). Then \(sd_j\) is a factor of a simple element, so it is simple. Cancellation in \(Pd_j=bcd_j\) gives \(P=bc\), whence \(s\) right-divides \(P\) and thus \(t(P)=d_{j-1}\). This contradicts \(D_R(d_{j-1})\subseteq D_L(d_j)\). Hence \(c=1\), completing the induction and the successive extraction proof.

Unfold a greedy factorization. By Lemma 8, its factors remain simple and their descent sets become full color preimages. Condition (32) still holds, so the unfolded factorization is greedy. Any final unfolded atom divides its last simple factor. Descent reflection makes its color a final divisor of the last original factor, hence of the original element. Word reversal preserves all positive braid relations and interchanges left and right divisors, proving the left version. ◻

Proof. Fix \(\ell\in\widetilde V\). A right projective has a nonzero map to or from \(U_\ell\) only when it is \(Q_\ell\), via its top quotient or token socle. Equation (33) excludes that projective from \(R_\ell(h)\), so its residual simple splits off as a complex. Tensoring with \(B_h\) preserves indecomposability. Therefore the other summand is zero in the homotopy category, and \(U_\ell B_h\simeq U_\ell\).

Write \(h=h_1h_2^{-1}\) with \(h_1,h_2\) positive, and set \(U_V=\bigoplus_{\ell\in\widetilde V}U_\ell\). Then \[ U_VB_{h_1}\simeq U_VB_{h_2}. \tag{34}\] Use the maximum total test height on \(\widetilde T\), which is zero on \(U_V\). A positive word avoiding \(V\) fixes \(U_V\). If it contains a color of \(V\), all operations before the first such color fix \(U_V\), and that first operation raises height by acting on a highest test. Positive monotonicity then keeps the height positive. Thus final height zero is equivalent to avoiding \(V\).

If the common height in (34) is positive, choose a highest-test vertex. The right version of Lemma 19, valid for bounded tests including right simples, extracts that vertex as a final atom of both unfolded words. Lemma 29 extracts a common final original color \(s\). Write \(h_i=g_i\sigma_s\) and tensor (34) on the right by \(B_{\sigma_s}^{-1}\). The equality persists for \(g_1,g_2\), while \(h_1h_2^{-1}=g_1g_2^{-1}\). Their total positive length decreases by two. Iteration terminates at height zero, when both words avoid \(V\) and their fraction lies in \(A_{T\setminus V}\). ◻

Proof. For an individual unfolded vertex the inverse bimodule is \[D_\alpha=[Z\longrightarrow (P_\alpha\otimes Q_\alpha)\langle-2\rangle],\] with its second term in cochain degree one. A projective at \((c,u)\) and a middle path of length \(a\) insert \(Q_\alpha\) at \((c+1,u+a-2)\), of level \(c+u+a-1\). Only the insertions with \(a=2\) can exceed the old level, and they come from old \(Q_\alpha\) terms.

These offending terms can be canceled simultaneously, even in the presence of the old differential. In \(\mathcal E=\operatorname{Tot} (R D_\alpha)\) let \(A\) be the sum of old \(Q_\alpha\) terms in the \(Z\) column and \(B\) their inserted length-two copies. The latter are at \((c+1,u)\), and the token–unit part of coevaluation gives an invertible block \(\Phi=\pi_Bd|_A:A\to B\), consisting of signed scalar identities. The old differential stays within a tensor column and does not enter this block. The subcomplex \[\mathcal K=A\oplus d(A)\] is contractible: projection onto \(B\) identifies \(d(A)\) with \(B\), and its differential maps \(A\) isomorphically to \(d(A)\) and kills the latter. It is a direct summand as a bigraded module. Writing \(\mathcal E=A\oplus B\oplus W\), the quotient is homotopy equivalent to \(\mathcal E\) and has underlying terms \(W\) with \[ d'_W=d_{WW}-d_{WA}\Phi^{-1}d_{BW}. \tag{37}\] Thus scalar entries induced among the inserted terms cause no problem; their absence was not assumed.

The remaining projectives are the old terms of other vertices and the length-zero and length-one insertions. Their levels are, respectively, the old level, the old level minus one, and the old level. The simple remains at normalized level \(-1\); if its vertex is \(\alpha\), it also inserts \(Q_\alpha\) at \((1,-2)\), again level \(-1\). All remaining terms have level at most \(r\). Subsequent minimization only removes summands and so preserves the bound.

The right-handed positive test identity follows from the same token-cone cancellation as its left-handed version in Lemma 15. Inverting it gives (36), of total shift \(-1\). This is a bimodule homotopy before tensoring, so it holds for simple inputs too. Finally the vertices in one color are orthogonal, and their twists commute; successive application proves the color statements. ◻

Proof. Filter the entire tensor by middle path length. Radical projective entries increase length by one or two. A top quotient into the simple takes a length-zero tensor to its assigned length one; a token inclusion out of it takes assigned length one to length two. A left radical map tensored with a right simple vanishes. Every differential therefore increases this index.

For a pair of normalized levels \((u,v)\) and middle length \(j\), actual total degree is \(u+v+j\). This also holds for simple tensors, since their assigned right level and middle length are \(-1\) and \(1\). The length-one differential preserves the two levels. After its cohomology the only remaining possible differential goes from column zero to column two. In particular every column-one class survives. This assertion can also be seen directly: writing the differential as \(d_1+d_2\), the cycle conditions are \(d_1u_0=0\) and \(d_2u_0+d_1u_1=0\); the only secondary obstruction is the class of \(d_2u_0\) in column two.

Every nonzero tensor uses a left vertex in \(T\cup\partial T\), because right projectives are supported at vertices of \(T\) and only equal-vertex or adjacent paths exist. Hence all relevant left levels are at most \(q\). A secondary differential hitting column two at \((r,q)\) would originate at a level pair of sum \(r+q+1\), since the differential raises total degree by one and length by two. Right levels are at most \(r\) and relevant left levels at most \(q\), so no such source exists. Therefore \(H_2(N,M)\) also survives. It lies in degree \(r+q+2>q\), where (38) is zero.

Project the token tensors at a vertex to the tensor of its two coefficient spaces modulo incoming images. Each length-one boundary has an incoming coefficient on one side, so the projection annihilates boundaries and is surjective onto the tensor of the top quotients. A nonzero right top thus forces every left top at that vertex to vanish. Family proportionality in Proposition 10 makes a left top occurring anywhere in one color occur at every label in some member. Thus the entire left-family top in that color vanishes. ◻

Proof. For a highest right top in \(j\in J\), Lemma 32 licenses the cap-preserving positive move \(z\mapsto\sigma_jz\). It changes \(h\) to \(h\sigma_j^{-1}\) and applies the corresponding inverse twist to the right rows. Lemma 31 shows that \(r\) cannot increase. Candidate data \(p,a\) are unchanged, and the tensor products still compute the same target at \(z'\), so the next move is licensed anew.

At a fixed \(r\), the chosen negative right moves form a reduced word in \(W_J\). Otherwise a right descent could be placed last by braid moves. All intermediate negative moves in that expression preserve the upper bound \(r\). Its final acted-color test shifts down by one, so its degree-\(r+2\) test vanishes, contradicting the currently chosen right top by (35). Sphericity bounds the length of such a sequence. Thus after finitely many moves either the relevant tops disappear or \(r\) decreases. The residual simple bounds \(r\) below by \(-1\), so the procedure terminates. If \(J\) is empty no move is required. ◻

Proof. Column-zero cohomology is a kernel with no incoming boundaries, so inclusion of its vector spaces gives the first assertion.

For the second work row by row and member by member, retaining cochain placements. At a projective vertex of \(\widetilde J\), each right coefficient is a sum of incoming images because its top vanishes. We use this fact inside the token cokernel. Suppose an image at \(\beta\) comes from a coefficient at \(\gamma\) through a right arrow entry. Choose a middle arrow dual to that entry under the backtrack pairing and tensor it with the predecessor coefficient and a left coefficient in \(\mathcal S_\beta\). Its differential has two possible token contributions: the chosen right image paired with the unchanged left coefficient at \(\beta\), and the predecessor paired with the corresponding left-arrow image at \(\gamma\), with its cochain sign. Non-backtrack products vanish; dual arrow bases isolate the selected coefficient even for arrow spaces of dimension greater than one. Modulo boundaries, therefore, the former token tensor equals the latter, up to a nonzero scalar and sign.

The new left coefficient remains in \(\mathcal S\) by the subrepresentation property. Each substitution decreases the right cochain placement by one, so boundedness forces termination. At a projective vertex outside \(\widetilde J\) the left coefficient is zero. An incoming image from the simple is a boundary directly: the assigned length-one simple tensor maps to that token inclusion, and its left differential is zero. Thus every token tensor dies in the cokernel, proving \(H_2(N,\mathcal S)=0\). ◻

Proof. For each hull vector \(s\), Lemma 34 gives \[\chi(n,s)=\frac{\dim_{\mathrm{wt}}H_0(N,\mathcal S) -\dim_{\mathrm{wt}}H_1(N,\mathcal S)}{D_0} \le\frac{\dim_{\mathrm{wt}}H_0(N,M)}{D_0}.\] Linearity gives the same upper bound for \(\chi(n,e)\), since \(e\) is a convex combination of these vectors. Lemma 32 gives \(\chi(n,m)=(\dim_{\mathrm{wt}}H_0(N,M)-\dim_{\mathrm{wt}}H_1(N,M))/D_0\). Subtracting yields \[\frac{\dim_{\mathrm{wt}}H_1(N,M)}{D_0} \ge\chi(n,e)-\chi(n,m)=-\chi(n,p).\] Now \(n\) is supported on \(T\) and \((Cp)_T=-a\), so (40) is exactly (41). ◻

Proof of Proposition 26. Normalize the first representative as above. If \(r>-1\), its highest right layer has nonzero projective coefficients and no residual simple. Some coefficient lies in \(\widetilde V\): otherwise all would lie in \(\widetilde J\), where their tops vanish, and iterating incoming images toward smaller cochain placements would force the entire layer to vanish. All row weights are positive, so \(n_T^{\mathsf T}a>0\). Equation (41) gives nonzero middle cohomology surviving in degree \(r+q+1>q\), contradicting (38). Therefore \(r=-1\).

The surviving middle part now lies in degree \(q\) and has dimension at most the actual degree-\(q\) cohomology. Since \(a'\) is supported on \(V'\), where \(p'_i=m_i^q(z')\), we obtain \[ (a')^{\mathsf T}(p'_T-p_T)\ge n_T^{\mathsf T}a\ge0. \tag{42}\] Interchanging the candidates gives \[ a^{\mathsf T}(p_T-p'_T)\ge0. \tag{43}\] The two normalizations may use different representatives, but their harmonic vectors and open-color vectors belong to the candidate cosets and are unchanged. The exterior values of \(p,p'\) outside \(T\) agree. With \(\delta=p'_T-p_T\) this gives \(a'-a=-C_T\delta\). Adding (42) and (43) yields \(-\delta^{\mathsf T}C_T\delta\ge0\). Positive definiteness forces \(p=p'\), hence \(a=a'\), \(V=V'\) and \(J=J'\), with equality throughout and \(n_T^{\mathsf T}a=0\).

It remains to identify the cosets. Fix any original comparison \(h_0=z'z^{-1}\). Replacing the two representatives by \(bz\) and \(cz'\), with \(b\in A_J\), \(c\in A_{J'}\), replaces \(h_0\) by \(ch_0b^{-1}\). Lemma 28 preserves its block with rows in \(\widetilde V'\) and columns in \(\widetilde V\). These are now the same set \(\widetilde V\).

The forward comparison has no right generators above level \(-1\). At level \(-1\), equality \(n_T^{\mathsf T}a=0\) and (39) imply \(K^{-1}_{\ell\beta}=0\) for \(\ell,\beta\in\widetilde V\), since every coefficient weighting such a count is strictly positive. Block invariance therefore gives the same zero block for \(h_0\) in all degrees \(r\ge-1\). The reverse comparison gives its zero block for \(h_0^{-1}\) in all degrees \(r\ge-1\), regardless of its chosen representatives. Duality (31) transposes this into vanishing for \(h_0\) in all degrees \(r\le-1\). Together the two ranges cover every integer. By Lemma 30, \(h_0\in A_{T\setminus V}=A_J\). Thus \(A_Jz'=A_Jz\), proving uniqueness of the typed candidate. ◻

Proof. The symmetric Frobenius trace pairs opposite paths of complementary lengths, with total internal degree two. On a complex use the cochain supertrace \[\operatorname{Str}_W(a) =\sum_c(-1)^c\tau\bigl(\operatorname{tr}_{\mathrm{mat}}(a|_{W^c})\bigr),\] where the matrix trace sums diagonal path entries and \(\tau\) is the Frobenius trace, not the ordinary trace of the underlying linear map. Composition and this trace give perfect pairings between the corresponding length spaces in Equation (47). Cyclicity gives \[\langle D_1g,h\rangle+(-1)^n\langle g,D_1h\rangle=0\] in complementary degrees. Finite-dimensional duality yields the pairings on cohomology. Units and tokens each contribute the equal-vertex pairing to the length Euler characteristic; arrows contribute minus the adjacency multiplicities. This proves Equation (48). The argument only uses the linear differentials and therefore also applies to subrepresentations.

For Equation (49), take minimal models of \(E_L(z)\) and \(E_H(z)\) and filter their full bigraded Hom complex by paths of length at least \(j\). Its differential is \(D_1+D_2\), with \(D_a\) raising path length by \(a\). The first nonzero differential is \(D_1\), whose cohomology splits by the ordered source and target layer levels. There are only three columns. After this linear page, the only possible differential goes from length zero to length two; every middle-column class is therefore permanent.

The action in Proposition 9 preserves actual bigraded Hom cohomology. Between the initial projectives \(P_{f,L},P_{f,H}\) it is zero for \(L\ne H\); for \(L=H\) it is one-dimensional in each of bidegrees \((0,0)\) and \((0,2)\) and zero elsewhere. Both diagonal dimensions already have permanent representatives. The identity of the full complex is a nonzero length-zero class: a boundary has no length-zero component. For the other class, put a normalized token on one diagonal projective summand of a nonzero minimal model and put zero on the other entries. This is a cycle of bidegree \((0,2)\), because a token multiplied by a radical entry is zero. Its supertrace is \((-1)^c\ne0\) if that summand lies in cochain placement \(c\), whereas every Hom boundary has zero supertrace. It is thus a nonzero actual class in filtration two; filtration three is zero.

The identity and token exhaust the actual diagonal dimensions, and the off-diagonal cohomology vanishes. Hence every permanent middle class is zero, proving Equation (49). A direct summand of a pure layer gives a direct summand of its linear Hom complex, so the same vanishing holds for it. This last assertion does not require the summand to split off the full transported complex. ◻

Proof. Extend a length-zero linear cocycle \(g:M_L\to Y^d(E_H(z))\) by zero on the other source summands. Because \(M_L\) is a direct summand for the linear differential, this embeds its class into the linear page of \(\mathop{\mathrm{Hom}}(E_L(z),E_H(z))\). No length-zero boundary is present. The secondary differential is represented by \[d_{E_H}^{(2)}g-(-1)^n g d_{E_L}^{(2)},\] where the superscript denotes the length-two part of the differential. The two possible ordered layer pairs of its image are \[(q,d-1)\quad\text{and}\quad(q+1,d),\] respectively. The latter term would require a level-\((q+1)\) source at a vertex of \(\widetilde I\), since a length-two entry is a token at a single vertex. The cap excludes it. Thus the secondary image lies in the first block and gives Equation (52).

If this image were zero, the length-zero class would be permanent: there are no incoming differentials and no later possible outgoing differential. Its total degree is \(n+v=d-q<0\), but actual Hom cohomology between the transported family members vanishes in every negative total degree by the initial projective calculation in Lemma 37. The class is therefore zero. Finally, sum the resulting dimension inequalities with weights \(w_Lw_H\) and use Equation (47), interchanging the two label indices. This proves Equation (53). ◻

Proof. Write \(\mathcal Y=\mathcal Y^d\), and fix one of the subfamilies \(\mathcal S\). A nonzero homogeneous length-zero cocycle from a member of \(\mathcal S\) to one of \(\mathcal Y\) has a nonzero graded image supported on \(\widetilde J\). That image has a socle: in its highest occupied cochain placement every outgoing coefficient arrow is zero, since arrows raise that placement. This gives a socle of the target on \(J\), contrary to the hypothesis. Hence \[h_0^{\mathrm{wt}}(\mathcal S,\mathcal Y)=0.\] Inclusion of \(\mathcal S\) in \(\mathcal M\) also gives \[h_0^{\mathrm{wt}}(\mathcal Y,\mathcal S) \le h_0^{\mathrm{wt}}(\mathcal Y,\mathcal M),\] because length-zero cocycles have no boundaries to quotient out. Apply Equation (51) to these two families. Only nonnegativity of \(h_1^{\mathrm{wt}}(\mathcal S,\mathcal Y)\) is needed; for \(\mathcal M\), middle vanishing does hold by Lemma 37. Subtraction gives \[ D_0(m-s)^{\mathsf T}Cy \ge h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y)\ge0. \tag{55}\]

This inequality is affine-linear in \(s\), so it holds on the entire hull. At its interior point \(e=m-c\), its left side is \(D_0c^{\mathsf T}Cy\). Symmetry of \(C\) and the null equation give \[ c^{\mathsf T}Cy =-\sum_{a\notin I}\sum_{i\in I}b_{ai}c_i y_a\le0. \tag{56}\] It is strictly negative if \(y\) meets \(\partial I\). Comparing with Equation (55) proves boundary vanishing, \(c^{\mathsf T}Cy=0\), and \(h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y)=0\).

For each \(j\in J\), ordinary interior membership supplies the two nearby hull points \(e\pm\varepsilon\mathbf e_j\). Their inequalities, whose value at \(e\) is zero, force \((Cy)_j=0\). Therefore \[m^{\mathsf T}Cy =c^{\mathsf T}Cy+(m-c)_J^{\mathsf T}(Cy)_J=0.\] The Euler identity for \((\mathcal M,\mathcal Y)\) now gives the remaining Hom vanishing. When \(J\) is empty, \(m=c\) and no coordinate variation is required. ◻

Proof of Proposition 36. Assume Equation (45) holds at every representative capped on \(I\). Since \(\partial J\subseteq(I\setminus J)\cup\partial I\), cap accessibility in Lemma 24 supplies a representative capped on \(J\) and hence on all of \(I\). We will use only cap-preserving moves in \(J\), so the exterior data in Equation (44) remain fixed throughout.

The convex data at each capped representative. Let \(M_L\) be the part of \(Y^q(E_L(z))\) supported on \(\widetilde I\). It is a direct summand of the pure layer, because the level-\(q\) coefficients on \(\partial I\) vanish. This does not assert a splitting of \(E_L(z)\). Its weighted generator array is \(\phi m\), where \(m\) agrees with \(m^q(z)\) on \(I\) and is zero elsewhere. In particular, \(m-c\) is supported on \(J\).

The harmonic prescription on \(J\) is \(c_J\): its boundary data are those of \(c\), and \(C_Ic_I=0\). Remote coordinates outside \(I\cup\partial I\) do not enter those equations. The strict part of Proposition 27 and the universal hypothesis Equation (45) therefore give \[ (m-c)_J\in\operatorname{int}_{\mathbb R^J} \operatorname{conv}\{s_J\}, \tag{57}\] with the subrepresentation families required in Lemma 39. They lie in \(\mathcal M\) because their support is in \(\widetilde J\). The assertion is renewed after every capped move: both \(\mathcal M\) and its hull may change, but the strict hypothesis is quantified over every capped representative.

An occupied boundary below \(q\). The set \(I\setminus J\) is nonempty, since \(C_J\) is positive definite whereas \(C_I\) has a positive null vector. Choose a color there. Its prescribed positive level-\(q\) multiplicity occurs in some member \(E_L(z)\). By the family properties in Proposition 10, the minimal color support of that member is connected and contains \(f\), which lies outside \(I\). A support path from the chosen color to \(f\) meets \(\partial I\). There is consequently an occupied boundary level. Let \(d_b\) be the lowest such level over the family. The boundary cap and zero boundary data at level \(q\) give \[ d_b<q. \tag{58}\] All boundary colors lie outside \(J\), so \(d_b\) is fixed in the coset. No adjacency of \(f\) to \(I\), and no sign restriction on \(q\), was used.

The upward induction. Choose \(d_{\min}\) below or equal to every initially occupied layer. Positive amplitude preserves this lower bound. At the beginning of a stage \(d\in\{d_{\min},\ldots,d_b\}\), maintain, for every \(k<d\), \[ \begin{gathered} \mathop{\mathrm{soc}}_{\widetilde J}\mathcal Y^k=0,\qquad (Cm^k)_J=0,\\ h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y^k) =h_0^{\mathrm{wt}}(\mathcal Y^k,\mathcal M)=0. \end{gathered} \tag{59}\] Here \(\mathcal M\) always refers to the current level-\(q\) summand. Initially these conditions are vacuous below the lower bound. Since \(d\le d_b<q\), Lemma 38 and the condition at \(d-1\) imply \[ h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y^d)=0. \tag{60}\]

Suppose \(\mathcal Y^d\) has a socle in color \(j\in J\). Then the top of \(\mathcal M\) at that color vanishes. Otherwise the proportionality of tops across the full label family supplies a simple coefficient-array quotient at the same unfolded vertex as the chosen socle, in some member of \(\mathcal M\). Composing that quotient with the socle inclusion gives a nonzero homogeneous length-zero cocycle. Explicitly, a functional killing the incoming coefficient images followed by a vector killed by the outgoing coefficients has both differential compositions zero. This contradicts Equation (60). The top of \(\mathcal M\) there is the top of the full level-\(q\) layer. Thus \(F_j\) preserves the cap by Lemma 24.

Preservation of earlier stages. For this single move, compare the old and new data by ascending through \(k<d\). Suppose the old top at \(j\) in level \(k-1\) has already been shown to vanish; this is vacuous below the lower bound. The adjacent layer sequence in Lemma 12 identifies \[ \mathcal Y^k_{\mathrm{new}} \cong Y^k\bigl(F_j\mathcal Y^k_{\mathrm{old}}\bigr). \tag{61}\] There is no contribution from the old level \(k-1\), precisely because its top at \(j\) vanishes. The zero socle on \(J\) is preserved by this unshifted operation, by Lemma 15.

The new representative is capped and satisfies Equation (57). Lemma 39 therefore gives all the new conditions in Equation (59) at level \(k\). In particular the old and new Cartan rows at \(j\) both vanish. Multiplicities outside \(j\) are unchanged, and \(C_{jj}=2\), so \(m^k_{\mathrm{new},j}=m^k_{\mathrm{old},j}\). If \(t_j^k\ge0\) is the folded weighted old top at \(j\) in level \(k\), the pure positive layer formula gives \[m^k_{\mathrm{new},j} =m^k_{\mathrm{old},j}-(Cm^k_{\mathrm{old}})_j+t_j^k.\] Hence \(t_j^k=0\). Positivity of all label weights makes every old top in that color zero. This supplies the premise for the next level; in particular, it proves Equation (61) also at \(k=d\). It also restores the preceding-stage conditions needed to apply Lemma 38 after this move.

This order avoids circularity: top vanishing at \(k-1\) first identifies the operation at \(k\); separation and the row comparison then establish top vanishing at \(k\).

Termination and contradiction. Continue firing a color of \(J\) whenever a socle remains at level \(d\). At that level the operations are the unshifted pure-layer operations. By Lemma 13, they satisfy the positive braid relations, and by Lemma 15 a last acted color has zero socle after its operation. Write the most recent letter on the left. If the next chosen color were a left descent of the reduced Coxeter word already fired, Matsumoto’s Theorem would move that letter to the left end, presenting the current layer as the result of its operation acting last. Its socle would then be zero, a contradiction. Every choice is therefore a Coxeter ascent. Sphericity of \(J\) bounds the number of choices by the length of its longest element.

When the firing stops, the socle on \(J\) at level \(d\) vanishes. Lemma 39 supplies Equation (59) for this level and, if \(d<d_b\), permits the next stage. There are only finitely many stages. At \(d=d_b\) the same lemma gives \(m^{d_b}|_{\partial I}=0\), contradicting its fixed occupied boundary level. If \(J=\varnothing\), there are no firing steps: the hull has its zero-dimensional meaning and the separation lemma already gives the same contradiction. This proves the proposition. ◻

Proof. Insert the vertices in the auxiliary order. A vertex with no earlier neighbor starts a new component. Otherwise all its earlier neighbors belong to one existing component, so insertion enlarges that component and cannot merge two components. At a limit stage, any finite path has already appeared at an earlier stage: its finitely many vertices have indices bounded below the limit. Thus components cannot merge at a limit either. Every component has a least vertex, which has no earlier neighbor, proving the assertion. ◻

Proof. For \(T=\varnothing\), there is no proper type. The closed exterior data would prescribe \(m^q=c\) and a cap on \(I\cup\partial I\), contrary to the empty-set case of Proposition 36.

In the nonempty closed case, Lemma 24 supplies a representative capped on \(T\), hence on \(I\). If the closed sublevel were empty, every representative capped on \(I\) would satisfy \(m_j^q>c_j\) for every \(j\in T\): a coordinate with \(m_j^q\le c_j\) could be opened alone, giving type \(T\setminus\{j\}\). This contradicts Proposition 36. Thus this sublevel is nonempty.

We prove both connectivity assertions by induction on \(|T|\). For a vertex \(v=(J,A_Jz)\) in either sublevel, let \(p_v\) be the harmonic interpolation of \(m^q(z)\) on \(J\), and set \[u_v=(p_v)_T-b_T^*.\] The open inequalities and nonnegative interpolation give \[ u_v\le0,\qquad (u_v)_J=-C_J^{-1}C_{J,T\setminus J}(u_v)_{T\setminus J}, \qquad (Cp_v)_T=C_Tu_v. \tag{63}\] Order vertices first by increasing \(\sum_{i\in T}(u_v)_i\). At an equal negative sum put larger types first; at sum zero put smaller types first. Refine the remaining ties by any well-order. Only finitely many sums occur: open counts lie in bounded intervals with finitely many possible values, the exterior is fixed, and there are finitely many types. The resulting order is well founded. Remaining ties are between incomparable vertices, since distinct comparable cosets of the same type cannot occur.

Consider first \(u=u_v\ne0\), and write \(V=T\setminus J\). Put \[N=\{i\in V:(C_Tu)_i\ge0\}.\] An upper extension has the form \((K,A_Kz)\), with \(J\subsetneq K\subsetneq T\), and is still allowed. The harmonic change is zero outside \(K\) and on \(K\) equals \[ (p_{(K,A_Kz)}-p_v)_K=-C_K^{-1}(C_Tu)_K. \tag{64}\] The old rows on \(J\) vanish. If \(K\setminus J\) avoids \(N\), the nonnegative inverse with positive diagonal makes the sum strictly increase. If \(\varnothing\ne K\setminus J\subseteq N\), the change is nonpositive; a strict decrease makes the extension earlier, and equality favors its larger type at the negative sum. Moreover \(N\ne V\), because otherwise \(C_Tu\ge0\) would imply \(u=C_T^{-1}C_Tu\ge0\).

Every earlier upper extension therefore contains a new color in \(N\). Removing its other new colors gives a comparable earlier extension whose type is contained in \(J\cup N\). These extensions all connect through the maximum \((J\cup N,A_{J\cup N}z)\). Thus the earlier upper part is empty or connected.

An earlier lower neighbor of \(v\) must strictly decrease the sum. Write it as \((R,A_Rw)\), where \(R\subsetneq J\), \(w\in A_Jz\). The difference of its harmonic vector and \(p_v\) is zero outside \(J\), has prescribed values \[m_i^q(w)-(p_v)_i\qquad(i\in J\setminus R),\] and is harmonically interpolated on \(R\). If all these values were nonnegative, the sum could not decrease. Hence the set of newly opened colors satisfying \(m_i^q(w)<(p_v)_i\) is nonempty. Close the other newly opened colors. This produces a comparable earlier lower vertex in the strict sublevel \(\mathcal R_{<}(J,A_Jz;q)\), with harmonic bound \(p_v\).

The boundary of \(J\) is capped: its colors lie in \(V\) or \(\partial T\). Every vertex of this inner strict sublevel is conversely an allowed earlier lower vertex in the original sublevel, because \(p_v\le b^*\) and its harmonic difference from \(p_v\) has strictly negative sum. Induction makes the inner sublevel empty or connected. Since each earlier lower vertex is comparable with a vertex in it, the whole earlier lower part is empty or connected as well.

If \(u_v=0\), we are in the special closed case. An upper extension keeps the same harmonic vector and has larger type, so is later. Every allowed lower neighbor is earlier, and these neighbors form \(\mathcal R_{\le}(J,A_Jz;q,I,c)\). Its exterior data are supplied by Equation (62) together with the equalities and caps on \(V\). The empty \(J\) case is impossible by Proposition 36; otherwise induction makes this lower part nonempty and connected.

If both earlier parts are nonempty, every earlier lower vertex is below every earlier upper vertex, so their union is connected. Lemma 40 now applies. A vertex with no earlier neighbor must have \(u_v\ne0\), \(N=\varnothing\), and empty inner strict sublevel. Therefore \[-(Cp_v)_i>0\quad(i\in V),\qquad (Cp_v)_J=0.\] At every representative \(w\in A_Jz\) capped on \(T\) one also has \(m_J^q(w)\ge(p_v)_J\); otherwise opening a failing coordinate alone would give a vertex in the inner strict sublevel. Together with the caps on \(V\), these are exactly the candidate conditions of Proposition 26. There is at most one such vertex in the fixed spherical residue. Every component has one by Lemma 40, so a nonempty sublevel is connected. ◻

Proof. All entries of \(p^k\) are nonnegative by Lemma 6. Bounding a block’s first coordinate bounds every exterior count for its type. Those counts have finitely many possible values in a bounded interval by Proposition 10. There are finitely many types, their harmonic interpolations are fixed linear maps, and there are finitely many sign masks. Thus the block alphabet has finite initial segments. Its least element is zero.

An infinite strictly descending sequence of heights would have no nonzero block above the largest occupied level of its first term. It would therefore involve only finitely many levels \(k\ge d_*\). At the highest level the block can decrease only finitely often, so eventually stays fixed. The same argument then applies to the next level, and successively to all remaining levels, contradicting strict descent. Equivalently, if \(r(B)\) is the finite rank of a block in its alphabet, the heights are ordered by the finite Cantor sums \(\sum_{k\ge d_*}\omega^{k-d_*}r(B_k)\), with decreasing exponents. ◻

Proof. An upper neighbor has the form \(v'=(U,A_Uz)\), where \(T\subsetneq U\) is spherical. Compare at the highest incidence level of its new colors. Above that level, the new colors and their neighbors have zero entries, so interpolation changes neither vectors nor signs. At the comparison level \(k\), the change is zero outside \(U\) and is \[ (p^k(v')-p^k(v))_U=-C_U^{-1}(Cp^k(v))_U. \tag{68}\] Rows on \(T\) vanish. If no new color is active, all the new rows at this level are nonpositive. A negative row strictly increases the sum. If all vanish, the vector and its intrinsic signs stay fixed, and the incident positive new coordinates have sign \(+1\). Closing them increases the sign mask. If the new colors have no recorded incidence at all, the heights tie. Consequently every descending upper neighbor adds an active color.

Conversely, suppose only active colors are added. At their highest incidence level all new rows are nonnegative. A positive row strictly decreases the sum; if all vanish, the vector is fixed and the positive new coordinates have sign \(-1\), so the sign mask decreases. Thus such an extension is descending whenever its type is spherical.

Let \(G\) be a connected component of \(T\) together with all active colors, and suppose \(G\) contains an active color. Choose the highest recorded level \(k\) incident to \(G\). All full rows on \(G\) are nonnegative: closed rows vanish, while an active color either has its highest incidence at \(k\), or has zero data on itself and its neighbors there. Hence \[ C_Gp_G^k=(Cp^k)_G+b_{G,\widehat S\setminus G}p_{\widehat S\setminus G}^k \ge0. \tag{69}\] The vector \(p_G^k\) is nonzero, since otherwise incidence would give a negative full row. It is strictly positive by connectedness: a zero coordinate adjacent to a positive one would again have a negative row.

If \(G\) were nonspherical, Lemma 6 would give \(C_Gp_G^k=0\). Both nonnegative terms on the right of Equation (69) would vanish. Thus \(G\) is the entire positive-support component, and higher levels vanish on \(G\cup\partial G\) by the choice of \(k\). Equation (65) assigns sign \(+1\) to \(G\). Every active color in it has highest incidence \(k\) and zero full row, contradicting Equation (67). Therefore \(G\) is spherical. Components with no active color lie in the old spherical type \(T\). The whole union is consequently spherical.

Every descending upper neighbor can now be joined to the one obtained by removing all its nonactive new colors. The latter is descending and still contains a new color. All these reduced extensions connect through the extension by every active color, which is spherical by the preceding argument. If there are no active colors, there is no descending upper neighbor. ◻

Proof. Write a lower neighbor as \((R,A_Rw)\), with \(R\subsetneq T\) and \(w\in A_Tz\). Its vector difference from \(p^k(v)\) is zero outside \(T\), has actual-minus-prescribed values on the newly opened colors \(T\setminus R\), and is harmonically interpolated on \(R\). Consider the highest recorded level at which an opened coordinate has either prescribed or actual positive data. Above it both vectors and sign masks agree.

Suppose no opened color is eligible. At the comparison level the actual-minus-prescribed values are all nonnegative. Indeed any positive prescribed coordinate is at its component’s highest visible level, and its actual cap holds, because there is no higher incidence among the opened coordinates. Failure of eligibility then gives the nonnegative difference. A strict difference increases the sum after interpolation. If all differences vanish, the full vectors and intrinsic signs agree, and every positive opened coordinate has sign \(-1\). Its mask changes from \(-1\) to zero, increasing the block. Without a comparison level there is a tie. This proves that a descending lower neighbor must open an eligible color.

If all opened colors are eligible, the reverse comparison applies. At the highest visible level among them their differences are nonpositive, and the other opened coordinates have zero difference. A strict difference decreases the sum. With equality, some positive opened coordinate has sign \(+1\), and every nonzero mask change is from \(+1\) to zero. Thus the neighbor is descending.

From any descending lower neighbor, close its ineligible opened colors. This gives a comparable descending lower neighbor opening only eligible colors. It remains to connect all such choices.

First take a visible component with sign \(-1\) at its highest level \(q\). The proper eligible choices in that component are exactly its strict spherical sublevel, with the fixed exterior cap at \(q\). This factor is empty or connected by Proposition 41.

For sign \(+1\), components must be grouped more carefully. Let \(I\) be their whole positive-support component at level \(q\); group together all the diagram components of \(T\) contained in \(I\). Their union is precisely \(T\cap I\). No other closed color is in \(\partial I\): a closed zero harmonic entry cannot neighbor positive support. The sign rule says that \(I\subseteq S\), that its positive vector \(c_I=p_I^q(v)\) satisfies \(C_Ic_I=0\), and that all higher incident data vanish. Since \((I\setminus T)\cup\partial I\) is outside \(T\), its harmonic data are the fixed actual data. They therefore satisfy Equation (62) for the spherical block \(T\cap I\). Its proper eligible choices form the special closed sublevel for this entire block, which is nonempty and connected by Proposition 41. No separate closed-sublevel assertion for its possibly disconnected diagram components is being used.

All these blocks are unions of components of \(T\); nonvisible components stay closed. The Artin presentation and standard parabolic embedding give the direct product of the block groups. Thus a subcoset in \(A_Tz\) is specified by its block subcosets acting from the common representative \(z\). Operations in another block change no counts in a given block. Consequently the all-eligible lower choices are exactly the product of the proper-choice factors just described, each with its top (the choice to open nothing) adjoined, with the all-top tuple deleted.

This product is empty or connected. Every tuple can be joined upward to one with a single non-top coordinate. Such tuples connect within a fixed coordinate because its proper-choice factor is connected. Tuples supported in two different coordinates connect through the tuple carrying both choices. Empty factors merely force their coordinates to stay top. Hence all-eligible neighbors are empty or connected; adjoining the other descending neighbors through their comparable all-eligible choices proves the result. ◻

Proof. Lemmas 43 and 44 handle the two parts. If both are nonempty, each lower neighbor is below each upper neighbor, so they are joined by edges. ◻

Proof. Write \(v=(T,A_Tz)\) and \(v'=(U,A_Uz)\), with \(T\subsetneq U\). All recorded vectors agree. Equality of each sign-mask coordinate forces every new closed color in \(U\setminus T\) to have zero entry at every recorded level: at a positive entry its mask would change from zero to \(+1\) or \(-1\). Harmonicity on \(U\) then forces all its neighboring entries to vanish too. Hence \(U\setminus T\subseteq K(v)\), the invisible sets agree, and \[T\setminus K(v)=U\setminus K(v).\] A visible closed color is positive at some recorded level; it therefore cannot neighbor an invisible color. This proves the first assertions.

Active colors are common, because they depend only on the recorded vectors and their signs; no new closed color is active. If there is an active color, its upper extension from the larger coset is descending and comparable with both vertices. Corollary 45 then places both entire descending neighborhoods in the same strict-sublevel component.

Suppose instead that there are no active colors. Both upper descending parts are empty, so only lower neighbors matter. Their visible components, highest levels, signs, and grouped eligibility blocks agree. If the larger coset has a descending lower neighbor, it has an eligible opening after a move \(g\in A_U\). The nonadjacency already proved gives \[A_U=A_{T_+}\times A_{U\cap K}.\] Write \(g=g_+g_0\) in these factors. Omitting \(g_0\) cannot change any actual count at a visible color: it commutes with \(g_+\) and operates only in invisible colors. Since \(g_+\in A_T\), the same eligible color can be opened in the smaller coset, using the representative \(g_+z\). Opening that single color gives a descending lower neighbor of the smaller vertex by Lemma 44. Conversely every descending lower neighbor of the smaller vertex is also one of the larger, by transitivity of coset inclusion and equality of their heights. Thus nonemptiness agrees and, when nonempty, the two neighborhoods share a lower vertex. Corollary 45 finishes the proof. ◻

Proof. A minimum exists by Lemma 42. Take a finite path between two minimum-height vertices in the connected full graph. If its greatest height is above the minimum, consider a maximal contiguous segment at that height. Consecutive vertices of the segment are comparable and tied. The segment has descending attachments at its two ends. Lemma 46 and Corollary 45 put these attachments in one connected component of the strict sublevel. Replace the segment by a finite path in that component.

Doing this for all segments at the current greatest height removes that height entirely and introduces no vertex at or above it. If the resulting path is still not a minimum path, repeat. An infinite repetition would give a strictly descending sequence of greatest heights, contrary to Lemma 42. The process therefore ends with a path entirely among minima. The constancy of \(K,T_+\) along its edges is Lemma 46. ◻

Proof. The set \(T_+\) is contained in the spherical type \(T\). Every visible closed color is positive at some recorded level, so Equation (71) excludes adjacency to \(K\).

Along any minimum path starting at \(v\), the sets \(T_+,K,D\) are constant by Proposition 47, and every type is contained in \(D\). All cosets on that path lie in \(A_Dz\). Indeed this holds initially because \(T\subseteq D\). A downward inclusion preserves it; for an upward inclusion, the larger coset has type contained in \(D\) and contains a point of \(A_Dz\), so the whole larger coset also lies in \(A_Dz\).

Let \(a\in H(Y,x)\) and set \(r=x^{-1}ax\). Right multiplication by \(r\) is an automorphism of the coset poset, preserving types. At a representative \(cx\), where \(c\in A_Y\), \[E_L(cxr)=E_L(cax)\simeq B_cE_L(ax) \simeq B_cE_L(x)=E_L(cx).\] It therefore preserves all layer counts and heights. The translated vertex \((T,A_Tzr)\) is a minimum, so Proposition 47 joins it to \(v\) by a minimum path. The coset containment along that path gives \(zr\in A_Dz\), hence \(r\in z^{-1}A_Dz\), proving Equation (73). The equivalent formulation follows from \(z=bx\). ◻

Proof. By Lemma 8, the graph on \(\widetilde Y\) is a disjoint union of finite simply laced graphs whose Cartan matrices are positive definite. We first establish a path identity on one connected component with an edge. Write \(\mathsf B\) for its adjacency matrix.

There is a strictly positive vector \(\mu\) with \[\mathsf B\mu=\delta\mu,\qquad 0<\delta<2.\] For completeness, maximize the Rayleigh quotient of \(\mathsf B\) on the unit sphere. Replacing a maximizing vector by its coordinatewise absolute value does not decrease the quotient. The resulting nonnegative eigenvector is strictly positive: the eigenvalue equation at a zero coordinate would force its neighbors to be zero, and connectedness would then force the entire vector to vanish. An edge makes the maximum eigenvalue positive, and positivity of \(2I-\mathsf B\) makes it less than \(2\).

Choose opposite arrow bases on each edge so that the trace of their product, in either order, is \(\sqrt{\mu_i\mu_j}\). The nondegenerate opposite-arrow pairing and ordinary symmetry of the Frobenius trace allow this choice. For coefficient maps of a pure layer, the length-two component of the square of the differential is therefore the relation \[ \sum_{j\sim i}\sqrt{\mu_j/\mu_i}\,(i\,j\,i)=0. \tag{74}\] Here a path denotes the corresponding composition of coefficient maps, at successive cochain placements. Indeed, before division by \(\mu_i\), its coefficient in the token trace is \(\sqrt{\mu_i\mu_j}\). Minimality excludes length-zero differential entries, so only two length-one entries contribute to this part of \(d^2\). The relation is valid on the prescribed interval because every intervening generator there is in \(\widetilde Y\).

We prove that all sufficiently long abstract paths lie in the span of insertions of (74). Let \(\mathcal V_n\) be the finite-dimensional Hilbert space with orthonormal basis the paths \((a_0,\ldots,a_n)\) of length \(n\). For \(n\geq2\) and \(1\leq i\leq n-1\), define \[c_i^{(n)}:\mathcal V_n\longrightarrow\mathcal V_{n-2}\] by zero unless \(a_{i-1}=a_{i+1}\). On a backtrack \((a,b,a)\) at these positions, delete its two edges and multiply the shortened path by \(\sqrt{\mu_b/\mu_a}\). The adjoint inserts exactly the weighted sum in (74), with the surrounding path fixed. All these maps preserve the two endpoints.

The contraction and insertion operators satisfy \[\begin{align*} c_i^{(n)}(c_i^{(n)})^* &=\delta\,\mathrm{id}_{\mathcal V_{n-2}}, \tag{75}\\ c_i^{(n)}(c_{i-1}^{(n)})^* &=c_{i-1}^{(n)}(c_i^{(n)})^* =\mathrm{id}_{\mathcal V_{n-2}} \quad(2\leq i\leq n-1). \tag{76}\end{align*}\] In (75), the scalar at the insertion vertex \(a\) is \(\sum_{b\sim a}\mu_b/\mu_a=\delta\). For (76), the subsequent contraction forces the inserted detour to use the adjacent edge already present in the path. Its two factors are reciprocal square roots, so their product is \(1\). In particular, both identities in (76) use the same ambient length \(n\).

Let \[\mathcal E_n=\bigcap_{i=1}^{n-1}\ker c_i^{(n)}, \qquad \mathcal E_0=\mathcal V_0,\quad \mathcal E_1=\mathcal V_1,\] and let \(\Pi_n\) be the orthogonal projection onto \(\mathcal E_n\). Since these spaces decompose by endpoints, \(\Pi_n\) preserves endpoints. We may therefore extend \(\Pi_k\) to longer path spaces by applying it to the first \(k\) edges while keeping the tail fixed.

Write \[\delta=2\cos\theta,\qquad 0<\theta<\pi/2,\qquad [u]=\frac{\sin(u\theta)}{\sin\theta}, \qquad h=\min\{j\in\mathbb Z_{\geq1}:j\theta\geq\pi\}.\] Then \([n]>0\) for \(1\leq n<h\), and \([n]=\delta[n-1]-[n-2]\). For \(2\leq n<h\), denote by \(\mathsf Q_n\) the extension of \(\Pi_{n-1}\) to the first \(n-1\) edges of a length-\(n\) path. We claim \[ c_{n-1}^{(n)}\mathsf Q_n(c_{n-1}^{(n)})^* =\frac{[n]}{[n-1]}\Pi_{n-2} \quad\hbox{on }\mathcal V_{n-2}, \tag{77}\] and \[ \Pi_n=\mathsf Q_n- \frac{[n-1]}{[n]}\, \mathsf Q_n(c_{n-1}^{(n)})^* c_{n-1}^{(n)}\mathsf Q_n. \tag{78}\]

At \(n=2\), \(\mathsf Q_2\) is the identity, and (77) is (75), since \([2]/[1]=\delta\). To pass from \(n-1\) to \(n\), let \(\mathsf R_n\) be the extension of \(\Pi_{n-2}\) to the first \(n-2\) edges. Extending (78) for \(n-1\) by one edge gives \[\mathsf Q_n=\mathsf R_n- \frac{[n-2]}{[n-1]}\, \mathsf R_n(c_{n-2}^{(n)})^* c_{n-2}^{(n)}\mathsf R_n.\] Endpoint preservation gives the identities \[c_{n-1}^{(n)}\mathsf R_n =\Pi_{n-2}c_{n-1}^{(n)},\qquad \mathsf R_n(c_{n-1}^{(n)})^* =(c_{n-1}^{(n)})^*\Pi_{n-2}.\] They concern \(\mathsf R_n\), whose prefix stops before the last pair, not \(\mathsf Q_n\). Substitution, followed by (75) and the two adjacent identities, yields \[\begin{align*} c_{n-1}^{(n)}\mathsf Q_n(c_{n-1}^{(n)})^* &=\delta\Pi_{n-2} -\frac{[n-2]}{[n-1]}\, \Pi_{n-2}c_{n-1}^{(n)}(c_{n-2}^{(n)})^* c_{n-2}^{(n)}(c_{n-1}^{(n)})^*\Pi_{n-2}\\ &=\left(\delta-\frac{[n-2]}{[n-1]}\right)\Pi_{n-2}\\ &=\frac{[n]}{[n-1]}\Pi_{n-2}. \end{align*}\] This proves (77).

To justify the projection formula, set \(T_n=c_{n-1}^{(n)}\mathsf Q_n\). The identity just proved reads \[T_nT_n^*=\frac{[n]}{[n-1]}\Pi_{n-2}.\] In finite-dimensional Hilbert spaces, an operator and its product with its adjoint have the same image: their orthogonal complements are both the kernel of the adjoint. Thus \(T_n\) maps onto \(\mathcal E_{n-2}\). Restricted to \(\operatorname{im}\mathsf Q_n\), its kernel is \(\mathcal E_n\), because the prefix projection has already imposed all contractions except the last one. To verify the projection explicitly, put \(\lambda=[n]/[n-1]>0\). The established range identity gives \[(T_n^*T_n)^2=\lambda T_n^*T_n,\qquad \mathsf Q_nT_n^*T_n=T_n^*T_n =T_n^*T_n\mathsf Q_n.\] Consequently \(\mathsf Q_n-\lambda^{-1}T_n^*T_n\) is self-adjoint and idempotent. It has image in \(\operatorname{im}\mathsf Q_n\cap\ker T_n\) and is the identity on that subspace. This is the projection (78), completing the induction.

It remains to prove that \(\mathcal E_n\) vanishes for some \(n\). The endpoint-dimension recurrence below will give that finite path bound. Let \(\mathsf d_n(i,j)\) be the dimension of \(\mathcal E_n\) with endpoints \(i,j\). Extending a path of length \(n-1\) by one edge makes the endpoint-dimension matrix of \(\operatorname{im}\mathsf Q_n\) equal to \(\mathsf d_{n-1}\mathsf B\). The restricted surjection \(T_n:\operatorname{im}\mathsf Q_n\to\mathcal E_{n-2}\) preserves endpoints and has kernel \(\mathcal E_n\). Hence, for \(2\leq n<h\), \[\mathsf d_n=\mathsf d_{n-1}\mathsf B-\mathsf d_{n-2}, \qquad \mathsf d_0=I,\quad\mathsf d_1=\mathsf B.\] Multiplication by \(\mu\) and induction give \(\mathsf d_n\mu=[n+1]\mu\). In particular, the recursion is valid at \(n=h-1\). No denominator \([h]\) has been used. Since \(\pi\leq h\theta<\pi+\theta<3\pi/2\), \[0\leq\mathsf d_{h-1}\mu=[h]\mu\leq0.\] The entries of \(\mathsf d_{h-1}\) are nonnegative and \(\mu\) is strictly positive. It follows that \(\mathsf d_{h-1}=0\), and therefore \(\mathcal E_{h-1}=0\). The same equality also forces \([h]=0\); no classification of the graph was needed.

In finite-dimensional inner product spaces, \[\mathcal E_n^\perp =\sum_{i=1}^{n-1}\operatorname{im}(c_i^{(n)})^*.\] Thus every individual path of length \(h-1\) is a linear combination of insertions of (74). Evaluate such an identity on the coefficient maps, starting at any fixed cochain placement in the prescribed interval. Each inserted relation is then a valid length-two part of \(d^2=0\) at the corresponding placement. The coefficient maps may depend on the placement; no translation invariance of them is required. The entire composition vanishes.

For a connected component without an edge, use the bound \(N=1\). There are finitely many components; take the maximum of their bounds. A longer path factors through an initial segment of the relevant vanishing length, and no path crosses components. The empty set \(Y\) is vacuous and may also be assigned \(N=1\). ◻

Proof. For an acted vertex, Lemma 15 gives the test shift \(\langle2\rangle[1]\), which lowers the ordinary cochain bound by one. At an unacted vertex \(\beta\), applying the vertex test to the evaluation cone gives a triangle \[\bigoplus_{\alpha\in\widetilde{\{i\}}} e_\beta P_\alpha\otimes_{\mathbb C}e_\alpha E \longrightarrow e_\beta E \longrightarrow e_\beta F_iE .\] When \(\beta\) is adjacent to color \(i\), the first term is a sum of acted tests shifted only in internal degree, because each \(e_\beta P_\alpha\) is an arrow space. It consequently has ordinary cochain cohomology at most \(m\). The long exact sequence gives zero above \(m\) for the new test and a surjection \[H^m_{\mathrm{coch}}(e_\beta E) \longrightarrow H^m_{\mathrm{coch}}(e_\beta F_iE),\] where each group retains its internal grading. For a nonneighbor the first term vanishes and the test is unchanged.

Thus once cohomology at degree \(m\) has vanished at a vertex, later operations at other colors cannot restore it: the later groups in that degree are quotients of zero. Another operation at its own color still gives zero in degree \(m\), using the bound above \(m\). After every color has been operated, degree \(m\) has vanished everywhere. The vertices of a single color are pairwise nonadjacent, so the simultaneous color operation satisfies the same conclusions. ◻

Proof. Fix a vertex and a generator coefficient in layer \(d_2\) at placement \(p\), with \(l<p\leq r\). Multiplying its projective generator by its token gives a cycle in the corresponding vertex test. Every differential entry is of positive path length and every product of a token with such an entry is zero. The tested cohomology vanishes at \(p\), so this token coefficient is a boundary from placement \(p-1\).

Project such a boundary onto its token coefficient space. A length-one differential entry can contribute only through an opposite arrow in the test, giving an incoming arrow-coefficient image from the same layer. A length-two entry can contribute only through a unit path, and its source generator would lie in layer \(d_2+1\). This latter layer is absent. These exhaust the possibilities, since minimality excludes length-zero entries. The perfect opposite-arrow pairing therefore shows that the full coefficient space at placement \(p\) is spanned by incoming arrow-coefficient images at \(p-1\).

For \(p\geq l+N\), iterate this spanning assertion \(N\) times. Every path involved has its placements in \([l,r]\), so Lemma 49 makes its coefficient composition zero. Layer \(d_2\) is therefore empty for \(l+N\leq p\leq r\).

Proceed downward through the layers. Suppose that the higher layers have already been removed on \([l+jN,r]\). For layer \(d_2-j\) and \(p>l+jN\), a possible token source from the next higher layer at \(p-1\) is absent on that interval. The same argument expresses the coefficient space by incoming arrows, and then by \(N\)-step paths. It removes this layer on \([l+(j+1)N,r]\). The successive conclusions are \[\begin{array}{c|c} \text{layer}&\text{interval proved empty}\\ \hline d_2 &[l+N,r]\\ d_2-1 &[l+2N,r]\\ \vdots&\vdots\\ d_1 &[l+N(d_2-d_1+1),r]. \end{array}\] On the last interval every possible generator layer has been removed, proving the assertion. ◻

Proof. Suppose that \(h(a_k)\leq h_*\) for all \(k\). Choose a common lower bound \(d_1\) on the generator levels of the initial family \(E_L(x)\). Positive amplitude, Lemma 15 and the bounded-layer formalism of Lemma 11, preserves this bound at every stage.

All generator placements outside \(Y\) are fixed by the family properties in Proposition 10. Let \(d_{\mathrm{ext}}\) be their largest occupied level. The exterior includes the unchanged f generator at placement \((0,0)\), so this is defined. If a current global highest layer has degree \(d>d_{\mathrm{ext}}\), it is supported entirely in \(\widetilde Y\). A nonzero such layer has a nonzero top: incoming coefficient maps increase cochain placement in a finite array and cannot generate a nonzero array indefinitely. Lemma 14 then gives a nonzero test in total degree \(d+2\). It follows that every current generator has level in the fixed interval \[[d_1,d_2],\qquad d_2=\max(d_{\mathrm{ext}},h_*-2).\]

By Lemma 50, the common upper bound on ordinary cochain test cohomology drops by at least one at every step \(a_k\to a_{k+1}\). It therefore tends to \(-\infty\). Let \(p_{\mathrm{ext}}\) be the least exterior generator cochain placement, over the initial finite family. It is fixed, and \(p_{\mathrm{ext}}\leq0\) because of the f generator. Set \[N=N(Y),\quad K_0=N(d_2-d_1+1),\quad r=p_{\mathrm{ext}}-1,\quad l=r-K_0.\] At a sufficiently late stage all tested cohomology on \(\widetilde Y\) lies in cochain degrees strictly below \(l\). Every generator in \([l,r]\) lies in \(\widetilde Y\), since \(r<p_{\mathrm{ext}}\). Lemma 51 consequently shows that every member of the family has no term at cochain placement \(r=l+K_0\).

Some member nevertheless has nonzero test cohomology, by positive height monotonicity from \(h(1)>-\infty\). For that member there is a term in a cochain degree below \(l\), while its unchanged f generator is in degree \(0>r\). The absent term in degree \(r\) splits its minimal complex into two nonempty bounded minimal complexes, one on either side of that degree. Neither summand is contractible. Indeed a contracting homotopy would express its identity as \(dh+hd\); every component of the latter lies in the additive radical because all entries of \(d\) do. Reduction modulo the radical would give a nonzero identity equal to zero. This contradicts the indecomposability of \(E_L(a_kx)\) from Proposition 10. ◻

Proof. The finite-type fraction property in Theorem 5 first gives \(u=pa=qb\) with \(p,q\) positive. Whenever \(p\) and \(q\) have a common left atom, write \(p=\sigma_i p'\), \(q=\sigma_i q'\) and cancel it. This replaces the common extension by \(p'a=q'b\). The sum of positive lengths decreases by two, so the process terminates with no common left atom.

Suppose the resulting extension satisfies \(h(u)>\max(h(a),h(b))\). Choose one label \(L\) and one unfolded vertex \(\alpha\in\widetilde Y\) attaining \(h(u)\) for \(E_L(ux)\). For this same family member, its starting test height on each side is smaller than \(h(u)\). If one starting set of tests vanished, persistence of vanishing under positive words would contradict the final nonzero test. Otherwise Lemma 19 applies to each side. It makes \(\alpha\) a left divisor of both unfolded positive words.

Lemma 29 reflects this divisibility to a common original color. In its right-divisor form this lemma is transferred to left divisors by word reversal: every Artin braid relation is preserved by reversal, and a color unfolds to a product of commuting letters whose reversal represents the same element. Thus the original positive elements \(p,q\) have a common left atom, a contradiction. This proves the stated inequality. Repeated pairwise application constructs a common extension of any finite family, with height no larger than the maximum of its starting heights. ◻

Proof. If no such \(p\) existed, then for every positive \(p\) with \(h(p)\leq h_0\) the union of supports of all its permitted continuations would equal \(Y\). For each color \(i\in Y\), choose a continuation \(b_i\) whose support contains \(i\) and for which \(h(b_i p)\leq h_0\). Lemma 53 gives a common positive left extension of these endpoints: \[u=c_i b_i p\quad(i\in Y),\qquad c_i\in A_Y^+,\qquad h(u)\leq h_0.\] Cancelling \(p\) shows that \(w=up^{-1}=c_i b_i\) is a positive element independent of \(i\). Every \(b_i\) is a right divisor of \(w\), so \(\operatorname{supp}(w)=Y\). This constructs a new positive endpoint \(wp\) with height at most \(h_0\). At every intermediate letter of this operation the height also stays at most \(h_0\), by positive monotonicity.

Starting from \(p=1\) and repeating gives an infinite sequence forbidden by Proposition 52. Therefore some \(p\) has a proper union of the supports of all its permitted continuations. Take \(J\) to be precisely that union. This establishes one common proper subset, not merely a separately missing color for each continuation. ◻

Proof. Suppose first that some test is nonzero. Use \(p,J\) from Lemma 54. For \(s\in H(Y,x)\), the equivalences of family members are preserved by every prefix of the action. In particular \[h(ps)=h(p),\qquad h(bps)=h(bp) \quad(b\in A_Y^+).\] Apply Lemma 53 to \(p,ps\), obtaining \[ap=bps,\qquad a,b\in A_Y^+,\qquad h(ap)=h(bps)\leq h(p)\leq h_0.\] Thus \(a\) is a permitted continuation of \(p\). The equality \(h(bp)=h(bps)\) shows that \(b\) is also permitted. Both have support in \(J\), so \[s=p^{-1}b^{-1}ap\in p^{-1}A_Jp.\] This proves (80).

If all the tests are acyclic, the evaluation term for each positive color twist is a bounded acyclic complex of graded vector spaces tensored with projectives. It is contractible, since a bounded acyclic vector-space complex splits degreewise, in each internal degree. The evaluation cone is consequently equivalent to the original family member. Every positive generator fixes the family, and its inverse fixes it as well by applying the inverse equivalence. The generators therefore give \(H(Y,x)=A_Y\). This argument includes \(Y=\varnothing\), when there are no tests and the group is trivial. ◻

Proof of Corollary 2. If \(Q=gA_Yg^{-1}\) and \(P=hA_Xh^{-1}\), put \(x=g^{-1}h\). The proof of Theorem 1 and Theorem 57 give \[Q\cap P=g c^{-1}A_Rc g^{-1} \quad\text{for some }R\subseteq Y\text{ and }c\in A_Y.\] If this intersection is strictly smaller than \(Q\), then \(R\ne Y\), since conjugation by \(c\in A_Y\) preserves \(A_Y\). Thus a strict intersection step decreases the size of the displayed defining set.

Start with \(Q=A=A_S\) and no selected members of \(\mathcal F\). Whenever \(Q\) is not contained in some \(P\in\mathcal F\), replace \(Q\) by \(Q\cap P\) and select that member. The preceding strict decrease permits at most \(|S|\) steps. The resulting \(Q\) is contained in every member of \(\mathcal F\) and is the intersection of the selected subfamily, proving the displayed equality and parabolicity. Finally, the family of parabolics containing \(E\) is nonempty because it contains \(A\); its intersection is the unique smallest such subgroup. ◻

Proof of Corollary 4. In the presentation graph in which every finite label is an edge, the infinite-label pair gives the visual splitting over \(A_{S\setminus\{s,t\}}\) of (Charney et al. 2026, Definition 2.4). Theorem 1 makes intersections of two conjugates of this edge group parabolic in \(A\). Hence (Charney et al. 2026, Theorem 3.2) gives acylindrical hyperbolicity and weak malnormality of the edge group, and (Charney et al. 2026, Proposition 4.2) gives weak malnormality of every proper parabolic.

The finite intersection supplied by weak malnormality is parabolic by Theorem 1. A nontrivial parabolic contains a conjugate of some \(\sigma_s\), which has infinite order: the exponent-sum homomorphism \(A\to\mathbb Z\) sends every standard generator to \(1\). Thus a finite parabolic is trivial, giving the stated witness. ◻

Proof. Theorem 1 places \(A\) in the class of groups whose parabolics are closed under pairwise intersections. Apply (Möller et al. 2024, Proposition 1.3 and Theorem 1.7). ◻

Proof of Corollary 3. Fix the supplied \(M\) and \(X\), and use their relative construction. Let \(N\) be the least common multiple of the integers \(2m\) for finite edge labels \(m\ge4\), and take \(K=\mathbb Q(\exp(2\pi i/N))\); take \(K=\mathbb Q\) if there is no such label. Then \(K\) contains every parameter \(q_m=\exp(\pi i/m)\). This field and its exact arithmetic are computable from the finite matrix \(M\).

We realize each Temperley–Lieb–Jones factor in its planar model. Loop-free pairing diagrams give finite bases for each required underlying planar morphism space; stacking diagrams evaluates every closed loop as \(q_m+q_m^{-1}\). The Jones–Wenzl recursion computes the simple idempotents \(p_0,\ldots,p_{m-2}\) over \(K\), with nonzero denominators in this range (Chen 2014, Definition 2.1.1, Remark 2.1.3, and Theorem 2.3.2). Negligible maps are kernels of finite trace-pairing matrices, so their quotient spaces and basis reductions are obtained by exact linear algebra. For idempotents \(p,q\), their corner morphism space is the image of \(f\mapsto qfp\) in that quotient (Chen 2014, Definition 2.4.1, Lemma 2.4.3, and Definition 5.2.1). Composition, tensoring and pivotal evaluation are consequently computable in these finite diagram spaces. These finite kernels, quotients and corner images commute with scalar extension to \(\mathbb C\), recovering the model used in Section 2.

The known simple list also makes decomposition effective. For any object \(U\) needed in a product, compute bases of \(\mathop{\mathrm{Hom}}(p_a,U)\) and \(\mathop{\mathrm{Hom}}(U,p_a)\). Their composition pairing takes values in \(\mathop{\mathrm{End}}(p_a)=K\) and is nondegenerate; invert its matrix to obtain dual inclusions and projections. The sum of the resulting simple-block projections is \(\mathrm{id}_U\), by the complete semisimple decomposition of (Chen 2014, Theorems 5.4.4–5.4.5). These identities hold over \(K\): they hold after extension to \(\mathbb C\), and scalar extension is faithful. Thus no search for an unspecified categorical decomposition is required. Finite products of the factors and direct sums are handled by tensor products and matrices of these maps.

It follows that every space \(\mathop{\mathrm{Hom}}(L,D^a_{ij}\otimes H)\) and every multiplication map defining \(Z\) has computable structure constants over \(K\). Let \(Z_K\) denote this finite algebra. Its trace weights belong to \(K\), and matrix inversion computes trace-dual bases, hence the coevaluation maps as well as the evaluation maps. All terms and differentials of \(B_i,D_i\) are now explicit over \(K\). Unnormalized planar bases suffice; the square-root normalization used only in the proof of Lemma 49 is not needed for this construction.

Given the input words \(u,v\), form the word \(w=u^{-1}vu\) and compute the bounded projective complex \(B_wP_{f,\mathbf1}\) by finite tensor products. Cancel invertible scalar differential entries between equal shifted projectives until none remain, as in Section 3. Each cancellation uses exact arithmetic in \(K\) and removes two summands, so the process terminates.

The positive-path-length ideal of \(Z_K\) is nilpotent and its quotient is a product of copies of \(K\). It is therefore the radical, and it remains the radical after extension to \(\mathbb C\). The computed minimal complex stays minimal under that extension. By uniqueness of minimal models, it is equivalent over \(\mathbb C\) to \(P_{f,\mathbf1}\) precisely when its only generator is that projective in placements \((0,0)\). This is a finite check of its summands; when it succeeds there is no differential left. Equation (81) identifies success with \(w\in A_X\), equivalently \(v\in uA_Xu^{-1}\). Taking \(X=\varnothing\) decides whether a word is the identity, and applying this test to the quotient of two words decides their equality. ◻