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

Abstract

Negami (arXiv:2610.00293, Problem 6.8) asks for a proof of the Jordan obstruction observed numerically for Katz-Long-Moody seeds built from the endpoint sectors of the Jones-Temperley-Lieb path representations. For every n >= 3, every level ell >= 4 and every endpoint sector of dimension at least two, every seed generator g_j has no fixed vector, so K = 0, and the ambient Long-Moody operator S_1 has a Jordan chain of length two at the eigenvalue -alpha^(-3).

Definition 1.1 (The A graph).

$$\forall \ell :\mathbb{N},\mathrm{adj}\left(\ell \right)=\mathrm{Adj}\left(\mathrm{pathGraph}\left(\ell -1\right)\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.adj (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

Vertices are the elements of Fin(ell - 1); vertex a is the paper’s vertex a + 1 of the A_(ell-1) graph. adj is the adjacency relation of Mathlib’s path graph pathGraph(ell - 1), under which two vertices are adjacent exactly when their labels differ by one (pathGraph_adj).

Definition 1.2 (The Perron-Frobenius weights).

$$\forall \ell :\mathbb{N},\forall a:\mathrm{Fin}\left(\ell -1\right),\mathrm{mu}\left(\ell ,a\right)=\sin \left(\frac{(\mathrm{val}\left(a\right)+1)\cdot \pi }{\ell }\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.mu (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

mu(a) = sin((a + 1) pi/ell), the standard path-model weight of vertex a + 1.

Definition 1.3 (The loop value).

$$\forall \ell :\mathbb{N},\mathrm{delta}\left(\ell \right)=2\cdot \cos \left(\frac{\pi }{\ell }\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.delta (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

delta_ell = 2 cos(pi/ell), the value with E_i^2 = delta_ell E_i.

Definition 1.4 (The left endpoint).

$$\forall \ell :\mathbb{N},\forall h:(2\le \ell ),\mathrm{val}\left(v1\left(\ell ,h\right)\right)=0$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.v1 (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

The first vertex, the paper’s vertex 1, at which every path starts.

Definition 1.5 (The endpoint sectors).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),\mathrm{Sector}\left(\ell ,n,h,t\right)=\mathrm{LegalPath}\left(\mathrm{adj}\left(\ell \right),n+1,v1\left(\ell ,h\right),t\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.Sector (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

Sector(ell, n, t) is the basis of the endpoint sector of the length-(n + 1) paths on the A graph that start at the left endpoint and end at t, the frozen LegalPath type.

Definition 1.6 (The path-model Temperley-Lieb operators).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),\forall j:\mathbb{N},E\left(\ell ,n,h,t,j\right)=\mathrm{ite}\left(j<n,\mathrm{delta}\left(\ell \right)\cdot \mathrm{pathProjection}\left(\mathrm{adj}\left(\ell \right),\mathrm{mu}\left(\ell \right),\mathrm{delta}\left(\ell \right),n+1,v1\left(\ell ,h\right),t,j+1\right),0\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.E (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

E_j, for j < n, is delta times the frozen weighted path projection at interior vertex j + 1: it vanishes unless the vertices j and j + 2 of the path agree at some c, and then replaces vertex j + 1 by each neighbour d of c with coefficient sqrt(mu(h) mu(d))/mu(c). For j >= n it is 0.

Definition 1.7 (The braid phase).

$$\forall \ell :\mathbb{N},\mathrm{alpha}\left(\ell \right)=i\cdot \exp \left(\frac{-\pi i}{2\cdot \ell }\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.alpha (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

alpha_ell = i exp(-pi i/(2 ell)), with beta_ell = alpha_ell^(-1) and no additional scalar twist.

Definition 1.8 (The braid generators).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),\forall j:\mathbb{N},\mathrm{rho}\left(\ell ,n,h,t,j\right)=\mathrm{alpha}\left(\ell \right)\cdot 1+\mathrm{alpha}{\left(\ell \right)}^{-1}\cdot E\left(\ell ,n,h,t,j\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.rho (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

rho(sigma_j) = alpha I + alpha^(-1) E_j on an endpoint sector.

Definition 1.9 (The seed braid generators).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),\forall i:\mathbb{N},s\left(\ell ,n,h,t,i\right)=\mathrm{rho}\left(\ell ,n,h,t,i\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.s (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

s_i = rho(sigma_i), the action of the braid generators of B_n inside B_(n+1).

Definition 1.10 (The seed free generators).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),(\mathrm{seed}\left(\ell ,n,h,t,0\right)=\mathrm{rho}\left(\ell ,n,h,t,0\right)\cdot \mathrm{rho}\left(\ell ,n,h,t,0\right))\wedge (\forall j:\mathbb{N},\mathrm{seed}\left(\ell ,n,h,t,j+1\right)=s\left(\ell ,n,h,t,j+1\right)\cdot \mathrm{seed}\left(\ell ,n,h,t,j\right)\cdot s{\left(\ell ,n,h,t,j+1\right)}^{-1})$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.seed (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

The images of x_1 = sigma_0^2 and x_(j+1) = sigma_j x_j sigma_j^(-1), zero-based: seed(0) = rho(0)^2 and seed(j + 1) = s(j + 1) seed(j) s(j + 1)^(-1).

Definition 1.11 (The free-group images).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),\forall j:\mathrm{Fin}\left(n\right),g\left(\ell ,n,h,t,j\right)=\mathrm{seed}\left(\ell ,n,h,t,\mathrm{val}\left(j\right)\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.g (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

g(j) for j in Fin(n) is seed(j), the paper’s g_(j+1) = rho(x_(j+1)).

Definition 1.12 (The ambient Long-Moody operator).

$$\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall h:(2\le \ell ),\forall t:\mathrm{Fin}\left(\ell -1\right),\forall k:\mathrm{Fin}\left(n\right),\forall l:\mathrm{Fin}\left(n\right),\forall x:\mathrm{Sector}\left(\ell ,n,h,t\right),\forall y:\mathrm{Sector}\left(\ell ,n,h,t\right),S1\left(\ell ,n,h,t\right)\left((k,x),(l,y)\right)=\mathrm{ite}\left((\mathrm{val}\left(k\right)=0)\wedge (\mathrm{val}\left(l\right)=1),(s\left(\ell ,n,h,t,1\right)\cdot \mathrm{seed}\left(\ell ,n,h,t,0\right))\left(x,y\right),\mathrm{ite}\left((\mathrm{val}\left(k\right)=1)\wedge (\mathrm{val}\left(l\right)=0),(s\left(\ell ,n,h,t,1\right))\left(x,y\right),\mathrm{ite}\left((\mathrm{val}\left(k\right)=1)\wedge (\mathrm{val}\left(l\right)=1),(s\left(\ell ,n,h,t,1\right)\cdot (1-\mathrm{seed}\left(\ell ,n,h,t,1\right)))\left(x,y\right),\mathrm{ite}\left((2\le \mathrm{val}\left(k\right))\wedge (k=l),(s\left(\ell ,n,h,t,1\right))\left(x,y\right),0\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.S1 (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

Eq. (braid-matrices) at i = 1 on n slots, zero-based: on slots 0 and 1 the block s_1 (0, g_1; I, I - g_2), and s_1 on every other slot. In the formula S1 is written with entries indexed by (slot, path).

Definition 1.13 (The eigenvalue).

$$\forall \ell :\mathbb{N},b\left(\ell \right)=-(\mathrm{alpha}{\left(\ell \right)}^{-1}{)}^3$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.b (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

b(ell) = -alpha_ell^(-3), the eigenvalue of rho(sigma_j) on the image of E_j.

Definition 1.14 (The Jordan obstruction).

$$claim\iff (\forall \ell :\mathbb{N},\forall n:\mathbb{N},\forall t:\mathrm{Fin}\left(\ell -1\right),\forall hl:(4\le \ell ),(3\le n)\Rightarrow (\mathrm{let}h:(2\le \ell ):=\mathrm{trans}\left(\mathrm{decide},hl\right);((2\le \mathrm{card}\left(\mathrm{Sector}\left(\ell ,n,h,t\right)\right))\Rightarrow ((\forall j:\mathrm{Fin}\left(n\right),\forall v:\mathrm{Sector}\left(\ell ,n,h,t\right)\to \mathbb{C},(\mathrm{mulVec}\left(g\left(\ell ,n,h,t,j\right),v\right)=v)\Rightarrow (v=0))\wedge (\exists w:(\mathrm{Fin}\left(n\right)\times \mathrm{Sector}\left(\ell ,n,h,t\right))\to \mathbb{C},(\mathrm{mulVec}\left(S1\left(\ell ,n,h,t\right)-b\left(\ell \right)\cdot 1,\mathrm{mulVec}\left(S1\left(\ell ,n,h,t\right)-b\left(\ell \right)\cdot 1,w\right)\right)=0)\wedge (\mathrm{mulVec}\left(S1\left(\ell ,n,h,t\right)-b\left(\ell \right)\cdot 1,w\right)\ne 0))))))$$

Formalization. D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.claim (✓ std3).

Citation. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

For every ell >= 4, n >= 3 and end vertex t whose sector has at least two paths (as in the Lean statement, h is let-bound to the proof of 2 <= ell obtained from hl : 4 <= ell, and the sector and the operators take it as an argument): no seed generator g_j has a nonzero fixed vector (so K = 0, the sectors admitted by the remark’s test), and S_1 is not semisimple at b: some w has (S_1 - b)^2 w = 0 and (S_1 - b) w != 0.

Theorem 1.15 (The Jordan obstruction holds for every endpoint sector).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.result (✓ std3). ∎

Resolves. Problems/negami-2026-jones-temperley-lieb-jordan-obstruction (proved) by D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.result.

Source. Repository-derived.

Acknowledgement. H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Commentary.

Put a = alpha_ell, delta = 2 cos(pi/ell) = -(a^2 + a^(-2)). The weights satisfy sum over neighbours of mu = delta mu (the boundary terms sin 0 and sin pi vanish), so the frozen Temperley-Lieb theorem gives E_j^2 = delta E_j and (rho_j - a)(rho_j + a^(-3)) = 0. Hence every g_j satisfies (g - a^2)(g - a^(-6)) = 0, and since ell >= 4 neither a^2 nor a^(-6) is 1, so g_j has no fixed vector. A sector with two paths contains a path beginning 1, 2, 1 and the same path with its vertex 2 replaced by 3 (vertex 3 exists as ell >= 4); on their span rho_0 and rho_1 act, in a suitable basis f_0, f_1, by (-a^(-3), a; 0, a) and (a, 0; a^(-3), -a^(-3)). On the four-dimensional subspace of slots 0 and 1 the operator S_1 is (0, UG; U, U(I - H)) with U = rho_1, G = rho_0^2, H = U G U^(-1), and an explicit Laurent-polynomial vector w gives (S_1 - b)^2 w = 0 and (S_1 - b) w = (a^6 - 1) times a nonzero vector, which is nonzero as a^6 != 1.

References

• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.E
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.S1
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.Sector
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.adj
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.alpha
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.b
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.claim
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.delta
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.g
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.mu
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.result
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.rho
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.s
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.seed
• Truth anchor: D5/S3/Quantum/Algebra/JonesTemperleyLiebLongMoodyJordanObstruction.v1
• Dependency: D5/S3/Quantum/Algebra/WeightedLegalPathTemperleyLieb
• Dependency: D5/S3/QuantumBounds/ReferenceFrameTax