Sharp constants for Ptolemaic negative type: the p-anchor ladder, an impossibility window above q=1, and Bernstein cube inequalities on both sides of q=1

the Bernstein cube inequality of arXiv:2608.20606v1 (its Theorem A.3): Hₓ >= 3, H_y >= 3, N >= 0 and Delta >= 0 on [0,1]⁶, with the Bernstein certificate recomputed inside the proof from the paper's own equations (A.1)-(A.8) rather than transcribed from its ancillary file

the correlation bound (47) and its exponent range: -(1+2h)/3 <= -N/sqrt(Hₓ H_y) <= 0 on the cube; h_q = 2^q - 2 lies in [0,1] exactly for 1 <= q <= log₂ 3; and at q = log₂(9/4) the constant is exactly -1/2

a five-variable strengthening the source does not state: Hₓ >= 3, H_y >= 3 and N >= 0 hold already for (x,y,g,k,l) in [0,1]⁵ with g, k, l free of the coupling g = he, k = hf, l = hd – certificates of 213, 213 and 226 terms against 699, 699 and 782 – and Delta does NOT relax: each of the three couplings g <= h, k <= h, l <= h is individually necessary, Delta = -305/4 at (1,1,1/2,1,1/2,1/2), (1,1,1/2,1/2,1,1/2) and (1,1,1/2,1/2,1/2,1)

exact ranges on the cube: sup Hₓ = sup H_y = sup N = 27 and inf Hₓ = inf H_y = 3, inf N = 0, every one attained at an explicit point; and the sup of Delta is sandwiched in [3125/16, 11807/54], attaining 3125/16 at (1,1,1,1/2,1/2,1/2)

Lemma A.4 (metric specialization) and Theorem A.2 (the symmetric projective star formula) of arXiv:2608.20606v1: H₀ = x² Hₓ, H₁ = x⁴ y² H_y, Nₛtar = -x³ y N under the chart substitution, hence the star partial correlation Nₛtar/sqrt(H₀ H₁) obeys the same bound -(1+2h)/3 <=. <= 0

negative controls: the coefficient test is not vacuous (a single negative coefficient is rejected, and the certificates of Hₓ >= 4, N >= 1, 9 Delta >= 1 and of the relaxed Delta all fail); the constants 3, 0, 0 and 27 are all attained so none of the four inequalities can be tightened; and each of the twelve bound hypotheses is load-bearing, with one explicit failing point per variable

the source's own numbers, replicated: all 24305 rational Bernstein coefficients derived independently from the printed equations (A.1)-(A.8) agree term for term with anc/bernstein_certificate/certificate.json, and every entry of its Appendix-A table reproduces – multidegrees (2,2,2,2,2,2)/(2,2,2,2,2,2)/(2,2,3,2,2,2)/(4,4,6,4,4,4), counts 729/729/972/21875, zero counts 0/0/190/415, least positive 3/3/(1/12)/(4/9), largest 27/27/27/(11807/54)

engineering: homogenising to uⱼ = tⱼ, vⱼ = 1 - tⱼ turns 'all Bernstein coefficients are nonnegative' into 'this polynomial in twelve variables has nonnegative coefficients', which removes both the stored certificate and the basis-conversion layer – 1288 lines, 1.8 CPU-minutes, no shared library – against the stored-tensor plus de-Casteljau design of grunbaum-equipart (1872 lines, 1.75 CPU-hours, a dylib, and a transcription gap between the paper's polynomials and the checked data)

inversion covariance at n points: hat G = Lambda G Lambda for the Schoenberg matrix of the inverted metric based at the same point, the matching positive-diagonal congruence of a Schur complement in witness form, and the resulting invariance of the p-anchor partial correlation under inversion at the base, for every anchor count

base change of the Schoenberg matrix, gram P B X Y = gram P A X Y - gram P A B X - gram P A B Y + gram P A B B for every pair including X = A, needing only P B B = 0; and the invariance of a Schur complement onto a fixed pair under every block-lower-unitriangular congruence of the anchor block

the p-anchor ladder: kappaᵖ_q = (p 2^q - (p+1))/(p+1), the complete-split exponent csq p r = r(p+1)/(p(r-1)), the identity kappaᵖ_q = 1/(r-1) iff the CS(p,r) roundness gap vanishes, the table reproducing P(3) = 2, P(4) = log₂ 3, P(5) = log₂(9/4), P(6) = 1 and the two open targets 16/9 and 5/3 with kappa³ = 1/3 and 1/4 exactly, and the uniform computation chi(CS(p,2)) = -kappaᵖ_q for every p and every exponent

the three-anchor partial correlation of CS(2,3) based at an independent vertex is exactly (3-2t)/(2(3-t)) with t = 2^q; it equals -kappa³_q exactly at t = 1 and t = 2; and at t = 9/4 it is -1 against -kappa³ = -11/16 – so no three-anchor analogue of the four-point correlation estimate can hold above q = 1, and the window of the missing lemma is exactly 0 < q <= 1

Schoenberg's identity for a zero-sum vector with a distinguished base, and the p-anchor gap decomposition a^T Dm a - 2 (g a).c + c^T H c = c^T (H - g X) c + (a - X c)^T Dm (a - X c) for any witness X with Dm X = g^T, with the source's one-anchor eq:gap-decomposition as the p = 2 case

negative controls for the ladder: the two targets need different constants and the source's own kappa² = 5/27 misses the 3+4 threshold 1/3; the bound is attained on CS(3,2), so no smaller constant is possible; the q > 1 refutation is absent at t = 3/2 and worse at t = 5/2; and positivity of the congruence diagonal, P B B = 0, symmetry of the anchor block and the zero-sum condition each have an explicit failing instance

the Bernstein-cube relaxation below q = 1 is repaired by a PINCHED rational envelope and by nothing else: a uniform band |e - Phi_fit(sigma)| <= delta fails for every delta > 0 (measured cell -1.00e-06 at delta = 1e-4), while sigma <= Phi_q(sigma) <= m sigma/(1+(m-1)sigma) – valid on a 200001-point grid with m = 2 at q = log₂(5/3) and m = 3 at q = 2 log₂(4/3) – gives nonnegative cells at both k = 3 and k = 4

the source's endpoint principle does not generalise: for the Schur pencil K(t) = K₀ - eᵗ u u^T - e⁻ᵗ v v^T with u the all-ones direction, conditioned onto a fixed pair, interior negative local minima of the partial correlation occur in 11 of 957 random instances at p = 4 and 16 of 799 at p = 5, against 0 of 1226 for the source's proved p = 2 case and 1 of 1113 at p = 3

the Bernstein pipeline of this family lifted into a shared, family-independent checker LeanProblemSpec/Cert/BernsteinCore,Bernstein.lean — arity a parameter, Bernstein coefficients computed rather than stored, binary box subdivision by two integer linear substitutions in the homogenised (u,v) variables, an exact-integer negative-witness mode, and one kernel-clean soundness theorem (checkTreeₛound, checkNegₛound, checkCertₛound, all [propext, Classical.choice, Quot.sound], no native_decide in the proof); all four cells of thm:cube of arXiv:2608.20606v1 re-proved through it with certificate sizes identical to this family's private pipeline (699, 699, 782, 21460 terms at (2,2,2,2,2,2) x3 and (4,4,6,4,4,4))

sup Delta over [0,1]⁶ narrowed from [3125/16, 11807/54] (11.9 % wide, dossier row PN4) to [3125/16, 1761/9] (0.18 % wide), i.e. 1757 < 9 sup Delta <= 1761; the upper end from a 64-leaf subdivision tree spent entirely on the three defect axes e, f, d, the lower end from the checker's exact-integer negative-witness mode at (1,1,1,1/2,1/2,1/2); with the measured thresholds root 1968, (x,y,h) 8-leaf 1968 (no gain), (e,f,d) 8-leaf 1769, all-six 64-leaf worse than (e,f,d) 64-leaf, (e,f,d) 64-leaf 1761

negative controls for the shared checker on this family's cube: the root expansion REJECTS 9 Delta <= 1761 and an 8-leaf tree on (e,f,d) rejects it too while accepting its own threshold 1769; an 8-leaf tree on the wrong axes (x,y,h) still rejects the root's own 1967, i.e. gains nothing; the checker rejects Hₓ >= 4 and 9 Delta >= 1; the negative-witness mode confirms 9 Delta >= 1 is false at (1,1,1,1,1,1) where Delta = 0, does NOT refute the true bound 1761, and rejects a witness with a coordinate 3/2 outside the box

the exact validity window of the cube relaxation in h: Delta(1,1,h,0,0,0) = 64 h (h+1) for every real h, while Hₓ = H_y = 12 and N = 4 at the same point for every h; hence the fourth cell of thm:cube of arXiv:2608.20606v1 fails at that single point for EVERY exponent 0 < q < 1 (where h_q = 2^q - 2 in (-1,0)), holds for h >= 0 with equality at h = 0, and the two floating-point witnesses of the strategy journal are the rationals -128/9 at 2^q = 5/3 and -896/81 at 2^q = 16/9

controls for PN20

the R1.3 price gate of journal/2026-08-29-ptolemaic-ladder.md sec 7, paid: the four-leaf-star Delta under the pinched envelope needs a dense Bernstein array of 6.19e9 coefficients at multidegree (66,64,64,6,4,4,4,4,4) in the better of two charts (1.96e10 in the chain chart of the plan), with its constituent cells H_U, H_V, N already at 3.5e7 each – three orders of magnitude over the 4 CPU-h / 10 GB cap, so K1b fires and the arm is PARKED ON COST; while the three-leaf cell one rung down (Ercan's own cube under the same envelope, the h < 0 extension) is 27012 sparse monomials / 1.90e5 dense at (38,38,4,4,4), only 8.7x the family's own 21875 and comfortably in cap – and, probed further, splits three ways: its first two cells H_U >= 0 and H_V >= 0 have ZERO negative Bernstein coefficients at the root and are landable immediately, while its third cell has 247 (resp. 212) negatives out of 190125, all at the balanced-star corner x1 = x2 = 1 where the inequality is an equality, and subdivision reduces the worst one by a measured factor of exactly 2 per split with exactly one bad box surviving, so NO finite subdivision tree certifies it at the sharp constant [referee 2026-09-03: this is a CHART-DEPENDENT statement – true in the direct chart used here; the correction lives in PN25 and v3 Remark 13.5]

the h < 0 analogue of thm:cube of arXiv:2608.20606v1, certified at the ROOT: at 2^q = 5/3 (h = -1/3, kappa²_q = 1/9, m = 2) and 2^q = 16/9 (h = -2/9, kappa²_q = 5/27, m = 3) – the two exponents q = log₂(5/3) and q = 2log₂(4/3) at which BHKL's conjecture predicts P(8) = q(9) and P(7) = q(8), both BELOW q = 1 – the source's own six-variable chart with the three defect ratios pinched to the envelope e = w(2 + 2(m-1)(1-w)² t)/(1+w²) at the three branch ratios w = x, y, xy of its chain chart satisfies Hₓ >= 3, H_y >= 3 and Delta >= 0 on [0,1]⁵, hence |N|/sqrt(Hₓ H_y) <= kappa²_q; sharp, with equality at the balanced star, which the pinch envE m 1 t = 1 puts at the 3-dimensional face x = y = 1 of the box (referee 2026-09-03; the ledger said "a corner"); Delta certificate 65750 terms at multidegree (22,22,4,4,4), no subdivision, 7 min 48 s / 7.14 GB for the whole module

negative controls for the h < 0 cube: the constant cannot be lowered (polDelta 1 1 h 1 1 1 = 0 for every real h, cells 25/3, 25/3, 25/27 at h = -1/3, so every K < 1/9 fails at the balanced star); Hₓ >= 4 is rejected by the certificate route and is genuinely false (measured inf Hₓ = 3.8667 resp. 3.9199); and the source's third cell N >= 0 is FALSE below q = 1 on this locus – rejected by the checker at both exponents and refuted by exact rational points of the box, (x,y,t) = (1/4, 1, (1,0,1)) with denominator 4 at 2^q = 5/3 and (1, 1/8, (1/4,1,1)) with denominator 8 at 2^q = 16/9, turned into exists a box point with polN < 0

the upper envelope w(2+2(m-1)(1-w)²)/(1+w²) DOMINATES the m sigma/(1+(m-1)sigma) of dossier row PN15 pointwise on w >= 0, m >= 1, by the exact identity w(2+2(m-1)(1-w)²)(1+w²+2(m-1)w) - 2mw(1+w²) = 2(m-1)w(1-w)²(w²+2(m-1)w), so it is valid for free and its denominator is 1+w² instead of (1+w²)(1+w²+2(m-1)w) – which cuts the k = 3 Delta cell from 27012 monomials at (36,36,4,4,4) (dense 171125) to 9366 at (22,22,4,4,4) (dense 66125); and the measured chart dependence that corrects dossier row PN22: the SAME cell under the SAME envelope has 247 resp. 212 negative Bernstein coefficients in the direct chart alpha = (1,x1,x2) and ZERO in the chain chart alpha = (1,x,xy) (the source's own), because the cell's Hessian at the balanced star is a(y1² - y1 y2 + y2²) exactly (measured over Fraction: A = B = C at equal slack, so b = -a/2) and the chain chart is the shear that turns the negative cross term into a nonnegative one