The stabilisation conjecture for the constraint polytopes of m-dependence tail dependence matrices is false for every m ≥ 3

The source's LP characterisation of T_d (Theorem FABC applied to B = T/d, via Theorem BCTD) formalised as IsTDM: a nonnegative weight x on the 2ᵈ subsets of [d] with sum x = 1, sum over S containing i of x_S = 1/d, and sum over S containing i,j of x_S = Tᵢj/d; the d-scaled form IsTDM' in which the total-mass row is slack; and their equivalence isTDMᵢff (for d > 0)

Soundness of the two finite certificate formats for T_d membership: a nonnegative weight list over subsets given as bitmasks (isTDM'ₒfₚackW, with cov1ₚackW / cov2ₚackW reducing the 2ᵈ-fold coverage sums to sums over the list), and a separating functional M -> sumᵢ uᵢ Mᵢi + sum_(i<j) zᵢj Mᵢj that is <= 0 at every rank-one 0/1 matrix 1_S 1_S^T and > 0 at M (notᵢsTDM'ₒfₛep); plus the objective identity fval = sum u + zsum₁ p + zsum₂ q + zsum₃ r for the m-dependence family (fvalₘdep3, fvalₘdep2)

Control against the source's Proposition MA2: the three published vertices (alpha,beta) = (2/3,1/3), (0,1/2), (1/2,0) of the d >= 6 two-dependence region are realised at d = 6 by explicit rational weight vectors

Control against the source's Proposition MA2: (alpha,beta) = (0,3/5), which violates alpha + 4 beta <= 2, and (4/5,0), which violates 2 alpha - beta <= 1, are proved NOT to be 6-dimensional TDMs by explicit separating functionals

Control against the source's Remark ma2-d<6: (alpha,beta) = (1,1) is realised at d = 3 and refuted at d = 4, so the source's three different small-d two-dependence regions genuinely differ

Non-vacuity control: alpha = 0 – the identity matrix – is realised at d = 9, so IsTDM' is not the empty predicate

Compute-first gate: the source's four published two-dependence H-descriptions – d = 3 (alpha >= 0, 0 <= beta <= 1, 2alpha - beta <= 1), d = 4 (alpha,beta >= 0, alpha+beta <= 1, 2alpha-beta <= 1), d = 5 (alpha >= 0, 0 <= beta <= 1/2, alpha+beta <= 1, 2alpha-beta <= 1) and d >= 6 (alpha,beta >= 0, alpha+4beta <= 2, 2alpha-beta <= 1, re-checked at d = 7 and d = 8) – reproduced EXACTLY, both inclusions, from the source's own definitions by exact rational linear programming

For every d with 4 <= d <= 16 there is an explicit rational alpha = (alpha₁,alpha₂,alpha₃) whose 3-dependence Toeplitz matrix is a d-dimensional TDM and is not a (d+1)-dimensional TDM; hence thirteen strict inclusions P_(d+1)³ subsetneq P_d³, each carried by two finite certificates (a nonnegative weight list at d, a separating functional checked over all 2ᵈ⁺¹ subsets at d+1)

No d₀ <= 16 is a stabilisation dimension for the 3-dependence constraint polytopes: for every d₀ <= 16 there are d >= d₀ and a rational (p,q,r) with P d (p,q,r) not equivalent to P d₀ (p,q,r). Equivalently, if the source's conjectured d₀ exists at m = 3 then d₀ >= 17 – against d₀ = 6 at m = 2

The constraint polytope P₈³ of the 3-dependence tail dependence matrices at d = 8: fifteen valid inequalities, each proved for ALL rational (alpha₁,alpha₂,alpha₃) by its own separating functional, together with twelve realisable points. Exact linear programming shows the fifteen are precisely the facets and the twelve precisely the vertices

The exact H- and V-descriptions of P_d³ for d = 4,...,9: 7, 9, 10, 12, 15, 16 facets and 7, 9, 8, 11, 12, 12 vertices, all listed explicitly. The facet count grows throughout, where at m = 2 it settles at 4 by d = 6

For every d >= 7 and every alpha in P_d³: alpha₂ + 2 alpha₃ <= 1 + 1/floor((d-5)/2). Hence the support value h_d = maxalpha₂ + 2 alpha₃ tends to 1, the value it takes on the moving-maximum polytope C³, so the 3-dependence polytopes converge to C³ in that direction

maxalpha₂ + 2 alpha₃: alpha in P_d³ = 1 + 1/ceil(3(d-4)/2) for every d from 8 to 32, and this value is strictly decreasing at every step from d = 6 to d = 32 – so P_(d+1)³ is strictly smaller than P_d³ at every one of those steps

The moving-maximum polytope Cᵐ = conv (n₁(R)/|R|,..., nₘ(R)/|R|): R a nonempty subset of 0,...,m, where nₖ(R) = #r in R: r+k in R, is contained in P_dᵐ for EVERY d, by truncated shifts of a single window pattern. C² is exactly the source's stabilised d >= 6 region alpha,beta >= 0, alpha+4beta <= 2, 2alpha-beta <= 1; C³ has 7 vertices and 9 facets, computed exactly; C⁴ has 13 vertices and 23 facets

Theorem A: for every m = r+1 >= 3 and every d, an explicit rational weight vector – one closed formula, two truncated mirror windows (u+-m,-m+1,0) cap [0,d-1] and (u+-m,-1,0) cap [0,d-1] on a linear ramp of slope 1/N, N = 3d+2m-1 – realises the m-dependence Toeplitz matrix with alpha₁ = d/N, alphaₘ₋₁ = (d+m)/N, alphaₘ = (d+m-1)/N and all other alphaₖ = 0 as a d-dimensional tail dependence matrix; hence alphaₘ₋₁ + 2 alphaₘ = 1 + (m-1)/N > 1 at every dimension

Theorem B: for every m = r+1 >= 3, every d and every K with 2m + K(m-1) < d, an explicit nonnegative combination of 4K+5 constraint rows proves alphaₘ₋₁ + 2 alphaₘ <= 1 + 1/(K+1) on ALL of P_dᵐ. At m = 3 with K = floor((d-7)/2) this is exactly the founding's alpha₂ + 2 alpha₃ <= 1 + 1/floor((d-5)/2) for every d >= 7 (boundₘ3), which was row TU1 and was prose

The Shyamalkumar-Tao stabilisation conjecture is FALSE at every m >= 3: for every m >= 3 and every d₀, the point of TA1 lies in P_(d₀)ᵐ and not in P_(3 d₀ + 4m)ᵐ, so no dimension is a stabilisation dimension (notₛtabilisesM); at m = 3, not Stabilises d₀ for EVERY d₀ (notₛtabilisesₐll), superseding the founding's d₀ >= 17. With Proposition MA2 (m = 2) and the m = 1 computation the conjecture holds exactly for m <= 2

The gap to the limit is Thetaₘ(1/d): (m-1)/(3d+2m-1) <= h_dᵐ - 1 < (m-1)/(d-2m-1) for m >= 3 and d > 2m+1, where h_dᵐ = maxalphaₘ₋₁ + 2 alphaₘ: alpha in P_dᵐ. So the m-dependence constraint polytopes approach their limit at rate exactly 1/d in the direction (0,...,0,1,2) and never reach it

Controls at m <= 2 by exact rational outer approximation: P₂¹ = 0 <= alpha₁ <= 1 and P_d¹ = 0 <= alpha₁ <= 1/2 for every 3 <= d <= 9 (stabilisation at d₀ = 3); and the source's four two-dependence regions at d = 3, 4, 5 and d = 6,7,8,9 – the MA2 region now re-confirmed at d = 9, one dimension past the founding's gate

Negative controls for the ramp/dual pair, all kernel-clean: the m >= 3 hypothesis is load-bearing (at m = 2 the two mirror windows coincide and the ramp's first off-diagonal equation gives 2/3 where 2/7 is claimed, at d = 6); the fit hypothesis 2m + K(m-1) < d of Theorem B is load-bearing (at d = 4 the point (4,7,6)/17 IS a 3-dependence TDM with alpha₂ + 2 alpha₃ = 19/17 > 10/9); the bound has teeth (alpha = (0,1,1) is refuted at d = 9 by the bound alone); and Theorems A and B are consistent at d = 24 (1 + 2/77 <= 1 + 1/9), proved independently of both

Control against a PUBLISHED m >= 3 theorem – the family's first: S. Tao, 'On Complete Positivity of Sparse Symmetric Toeplitz Matrices', J. Comp. Pure Appl. Math. 3(1) (2025) 1-9, DOI 10.33790/cpam1100114, Theorem 3.2 Case 2 says that for the two bands m-1 and m with d >= 2m+1 the matrix is a TDM iff alphaₘ₋₁, alphaₘ >= 0 and 2 alphaₘ₋₁ + 2 alphaₘ <= 1. The family's exact simplex reproduces that region at all 16 tested points (d = 7..10 by four parameter pairs), including the boundary (0,1/4,1/4) and the violating (0,1/4,13/50)

The exact-rational verification behind TA1, TB1 and TR1: every coverage equation of TA1's weight vector for m = 3..8 and d = m+1..30 (zero violations; d-1 violations at m = 2); the dual certificate of TB1 checked over ALL 2ᵈ subsets for m = 3,4,5 and d <= 18 (max_S g(S) = 0, attained only at the empty set, diagonal sums exactly -(K+1), 0,..., 0, K, 2K); the 38-pair end-to-end run of TR1 (m <= 6, d₀ <= 14, d₁ <= 66); the Lean parametrisation Kₗean = Kₛtrat - 1 confirmed for m = 3..6 and d < 40 and the fit plus strict bound for m = 3..9 and d₀ <= 59; and the independent simplex confirmation of the exclusion at (m,d₀) = (3,4), (3,5), (3,6), (3,8)