The Jordan obstruction for the Jones–Temperley–Lieb endpoint-sector seeds

H. Negami, Quantum gates from the middle convolution of twisted Burau representations, arXiv:2610.00293v1, builds braid representations for quantum gates by the Katz–Long–Moody construction: a seed representation of $F_n\rtimes B_n\cong B_{1,n}$ on $V$ gives operators $S_i$ on $V^{\oplus n}$, $S_i=s_i^{\oplus n},\mathrm{diag}(I,R_i,I)$ with $R_i=\begin{pmatrix}0&g_i\I&I-g_{i+1}\end{pmatrix}$, and a quotient by $K+L_\lambda$, $K=\bigoplus_j\mathrm{Ker}(g_j-I)$. Remark 6.7(b) tests seeds from the endpoint sectors of the Jones–Temperley–Lieb path representations ($\rho(\sigma_i)=\alpha_\ell I+\alpha_\ell^{-1}E_i$, $\alpha_\ell=ie^{-\pi i/(2\ell)}$) for $n=3$, $\ell\in{4,5,6,7,8,10,12,16}$ and $n=4$, $\ell\in{5,7}$: in sectors with $K=0$ finite-precision rank tests suggest non-trivial Jordan blocks of the ambient $S_i$, which "by themselves … establish neither non-semisimplicity nor unboundedness". Problem 6.8 ends: "Also prove the Jordan obstruction for the Jones--Temperley--Lieb endpoint-sector seeds". The verbatim statements are in the literature note.

Issue #13581 fixes the reading: the obstruction is a Jordan chain of length two of $S_1$ (all $S_i$ are conjugate to it in the braid group) in a sector with $K=0$, with the standard path-model operators $E_i$ (which the source names without writing their entries), and it is proved for every $n\ge3$, $\ell\ge4$ and every endpoint sector of dimension at least two.

A non-semisimple braid generator rules out any invariant positive-definite Hermitian form, so these seeds cannot produce unitarizable KLM gate representations; the paper's large sampled word norms are the numerical shadow of this. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.result proves the obstruction exactly and uniformly.

Issue #13581 records the literature check before any Lean: the paper has a single version (2026-09-25); its summary table says "An exact obstruction remains open"; the related KLM papers 2609.11793 and 2608.01727 treat the Hecke scalar family and general triangularization, not path-model endpoint sectors; the repository had only the Temperley–Lieb projection laws of the path model (WeightedLegalPathTemperleyLieb). Zenodo title searches ("Temperley-Lieb" "Long-Moody", "Long-Moody" Jordan, Negami) and a web search for the obstruction (2026-10-06) returned no settlement. not-found-in-searched-scope.

Let $a=\alpha_\ell$, $\delta=2\cos(\pi/\ell)=-(a^2+a^{-2})$, $b=-a^{-3}$.

1. Temperley–Lieb relation. With $\mu_c=\sin(c\pi/\ell)$, $\sum_{d\sim c}\mu_d=\delta\mu_c$ (the missing boundary neighbours would contribute $\sin0=\sin\pi=0$), so the frozen theorem gives $E_j^2=\delta E_j$, hence $(\rho_j-a)(\rho_j+a^{-3})=0$.
2. $K=0$. Every $g_j$ is conjugate to $\rho_0^2$, so $(g_j-a^2)(g_j-a^{-6})=0$; for $\ell\ge4$ neither $a^2$ nor $a^{-6}$ equals 1, so $g_j$ has no fixed vector.
3. A local block. A sector with two paths contains a path beginning $1,2,1$ (loop_path) and its partner with vertex 2 replaced by 3; on their span, in a basis $f_0,f_1$, $\rho_0=\begin{pmatrix}-a^{-3}&a\0&a\end{pmatrix}$ and $\rho_1=\begin{pmatrix}a&0\a^{-3}&-a^{-3}\end{pmatrix}$ (local_vectors).
4. Jordan chain. On slots 1 and 2 of that span, $S_1=\begin{pmatrix}0&UG\U&U(I-H)\end{pmatrix}$ with $U=\rho_1$, $G=\rho_0^2$, $H=UGU^{-1}$; an explicit Laurent-polynomial vector $w$ gives $(S_1-b)^2w=0$ and $(S_1-b)w\ne0$ because $a^6\ne1$ (sector_jordan).

The proof would fail if $a^2$ or $a^{-6}$ were 1 (it is at $\ell=3$, where $3\pi/\ell=\pi$), or if a sector of dimension at least two contained no path beginning $1,2,1$.

The canonical source is D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.lean. Its public declarations are adj, mu, delta, v1, Sector, E, alpha, rho, s, seed, g, S1, b, claim and result, with instances for DecidableRel (adj ℓ), Fintype and DecidableEq on the sectors. It imports the frozen D5/S3/Quantum/Algebra/WeightedLegalPathTemperleyLieb (LegalPath, pathProjection, weighted_legal_path_temperley_lieb) and the frozen D5/S3/QuantumBounds/ReferenceFrameTax (sine_reference_eigenvector, the zero-boundary sine eigenvector relation behind the Perron–Frobenius weights); the adjacency is Mathlib's SimpleGraph.pathGraph, and the proof applies Mathlib's sine and cosine positivity, Real.sin_two_mul, Complex.exp_nat_mul and the nonsingular-inverse lemmas. It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom. The module statement is sha256:7c54306aac355610e4a98f524609bd8b78367e684156ec02ef42326826b20363, the result statement sha256:81d8490d2baa9fc1f1cb395e70d32a3e56207ba543060f724de8738bd0a22563 and the claim statement sha256:092bc5b9520961e1ac46c1856a375b050c33a631c6613c16fbf68525883bc8d4. The Freeze event is sha256:940f0d828cb2930549906e443194a8633509c459d457791124cfece5e883f85b; its project-level prerequisites are sha256:4596e4b3f62f1bf4c28dd053935ac6ed2f262f7926b61592d319479015e27798 and sha256:57be3e0134327e85e5920fac333ec9d9338b42abf2c25b0982e639a15929c2b8, the frozen WeightedLegalPathTemperleyLieb and ReferenceFrameTax. The event also records v1.congr_simp and lmBlock.congr_simp, congruence lemmas that Lean realizes while simp elaborates, and seed.eq_def, the equation lemma Lean generates for the recursive definition seed; none is an authored declaration.

Numerical check (60-digit arithmetic for the remark's seeds, double precision for the wide scan): all 20 endpoint sectors of dimension at least two for the remark's ten $(n,\ell)$ have $K=0$ and $\dim\ker(S_1-b)^2>\dim\ker(S_1-b)$; a wide scan $n=3,\dots,7$, $\ell=4,\dots,14$ (141 sectors) has no exception; the braid relations and $E_i^2=\delta E_i$ hold to $10^{-60}$, and every other eigenvalue of $S_1$ in the sampled sectors is semisimple.

Tier 1 explicit problem of a 2026 paper, preregistered in issue #13581 before any Lean. theorem; resolution proved.

| declaration | proof_shape | escape_witness | admission_basis | | --- | --- | --- | --- | | result | content | loop_path | open-problem-resolution |

The private theorems loop_path, local_vectors, sector_jordan and no_fixed_seed are content; every other private theorem (μ_pos, perron_frobenius, δ_pos, E_square, α_data, E_triple, ρ_inverse, seed_zero_polynomial, first_projection, block_action) is bind-only and is used on the proof path of result (CLAUDE.md §3.2 「有消费的辅助声明」). The direct frozen dependencies are WeightedLegalPathTemperleyLieb and ReferenceFrameTax. Utility is none: the module proves a universal statement and contains no finite enumeration, checker, numeric reduction or certified instance. There is no digestion atom.

What the settlement shows

Proved by result: for every $n\ge3$, $\ell\ge4$ and every endpoint sector of dimension at least two, $K=0$ and $S_1$ has a Jordan chain of length two at $-\alpha_\ell^{-3}$; in particular for all twenty sectors of Remark 6.7(b).

Established inside the proof. The whole obstruction lives on a four-dimensional subspace spanned by two paths in slots 1 and 2, uniformly in $n$, $\ell$ and the sector; its coupling is the factor $a^6-1$, so the chain is a genuine non-diagonal block, not a coincidence of eigenvalues (local_vectors, sector_jordan).

Argued, not formalized.

• Non-unitarizability and unboundedness: $S_1^kw=b^kw+kb^{k-1}v$ with $|b|=1$ and $v=(S_1-b)w\ne0$, so $|S_1^kw|\to\infty$ linearly and no positive-definite invariant Hermitian form exists; every $S_i$ is conjugate to $S_1$, so each braid generator is non-semisimple.
• The chain survives every KLM quotient with $\lambda\notin{0,1}$: the four-dimensional subspace has zero components in slots $3,\dots,n$, and the recurrence $z_j=g_{j+1}z_{j+1}$ describing $L_\lambda$ then forces slots 2 and 1 to vanish, so it meets $K+L_\lambda=L_\lambda$ only in 0.

Open. The irreducibility locus and image-closure questions of Problem 6.8 are not addressed.

Effect on the paper. The table entry "An exact obstruction remains open" is settled: the Jones–Temperley–Lieb endpoint-sector seeds give non-semisimple, hence non-unitarizable, ambient Long–Moody representations in every endpoint sector of dimension at least two, for every $n\ge3$, $\ell\ge4$; the paper's other results are unaffected.

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.