Positive roots of the flag-manifold Einstein polynomials of Arvanitoyeorgos, Sakane and Statha: an unproved remark for SU(N), the sharp constant for SO(n), and exact root counts

The degree-16 SU(N) polynomial F₃ re-derived by exact resultant elimination, reproducing the paper's four printed coefficients a₀, a₁, a₁₅, a₁₆ exactly

Erratum: the displayed F₃(1) in arXiv:2502.18491v1 carries exponent 3 where it should be 1

The paper's own two signs for F₃: F₃(0) > 0 and F₃(1) < 0 for k₁ > k

F₃(2) > 0 for all real k ≥ 2, p ≥ 3, k₁ ≥ 6kp — step (1) of the unproved Remark 1, proved, with the constant sharpened from 8kp

Two positive roots of F₃, localised in (0,1) and (1,2)

Negative controls for the SU(N) half: the constant cannot go below 5.5583, the (2,3) threshold is exactly 26, and k < k₁ is not removable

The degree-8 SO(n) polynomial H₁ re-derived by exact resultant elimination, reproducing the paper's printed leading coefficient, constant coefficient and H₁(1)

Erratum: the displayed H₁(0) in arXiv:2409.18990v1 is not the value of H₁(0)

Four signs of the SO(n) chain — H₁(0) > 0, H₁(1) < 0, H₁(2) > 0, 0 < γ < 2/3 — for k ≥ 3, p ≥ 3, k₁ ≥ (29/4)kp

H₁(2/3) > 0 for all real k ≥ 3, p ≥ 3, k₁ ≥ (29/4)kp — the binding step of Theorem 6.3, with 10kp sharpened to 7.25kp

Two positive roots of H₁, localised in (2/3, 1) and (1, 2), for all real k ≥ 3, p ≥ 3, k₁ ≥ (29/4)kp

Negative controls for the SO(n) half: 29/4 is within 1.2 % of the best constant available to the paper's own route

Remark 1 of arXiv:2502.18491 proved in full, with 8kp sharpened to (45/8)kp: four positive roots of F₃ for all real k ≥ 2, p ≥ 3

The optimal constants bracketed, and the structural reason the paper's β cannot be simplified

Remark 1 of arXiv:2502.18491 proved at six explicit (k,p), each for every real k₁ above a threshold well below the paper's 8kp

Measured: Lean's binop% elaborator is superlinear in the length of one +-chain, and chunking a large polynomial definition is what makes these files elaborate at all

Theorem 6.3 of arXiv:2409.18990 with k₁ ≥ 10kp sharpened to k₁ ≥ (29/4)kp: four positive roots of H₁, in the paper's own chain, for all real k ≥ 3, p ≥ 3

The four auxiliary signs of the SO(n) chain at the sharpened constant: H₁(0) > 0, H₁(1) < 0, H₁(2) > 0 and 0 < γ < 15/32, for all real k ≥ 3, p ≥ 3, k₁ ≥ (89/16)kp

H₁(15/32) > 0 for all real k ≥ 3, p ≥ 3, k₁ ≥ (89/16)kp, and two positive roots of H₁ in (15/32,1) and (1,2) — Theorem 6.3's binding step with the separating point moved off 2/3 and 10kp sharpened to 5.5625kp

The SO(n) perfect square: the top k₁-degree part of k₁⁸·H₁(u/k₁) is ((kp-2)u - A)², so the paper's γ is a forced vertex and no simpler lower separating point exists

Negative controls for the sharpened SO(n) constant: 89/16 is within 0.03 % of the best constant available to the point 15/32, the hypotheses are non-vacuous, and the lower separating point cannot be simplified

Theorem 6.3 of arXiv:2409.18990 kernel-checked with k₁ ≥ 10kp sharpened to k₁ ≥ (89/16)kp: H₁(γ) < 0, and four positive roots of H₁ for all real k ≥ 3, p ≥ 3

The optimal constant in k₁ ≥ c·kp for the SO(n) four-root theorem bracketed in [5.56004, 5.5625], with the exact asymptotic optimum 5.5602472…

Measured: a 1,077,987-monomial ring identity done as twenty of at most 5,566, by splitting a Handelman certificate by the shifted variable's degree and expanding one variable at a time

The SU(N) and SO(n) flag-manifold constructions have the *same* asymptotic four-root threshold: their k → ∞ limit polynomials differ by x ↦ 1/x and a strictly positive factor

The SU(N) constant 45/8 = 5.625 sharpened to 89/16 = 5.5625 by moving the chain's upper separating point from 2 to 17/8, and the SU(N) optimum bracketed in [5.5600, 5.5625]

F₃(17/8) > 0 for all real k ≥ 2, p ≥ 3, k₁ ≥ (89/16)kp, and two positive roots of F₃ in (0,1) and (1,17/8) — the SU(N) chain's upper separating point moved off the paper's 2, and 8kp sharpened to 5.5625kp

Negative controls for the sharpened SU(N) separating point: the constant is load-bearing, the sign turns exactly at the certificate's own limit, and the gain over the point 2 is not an artefact

Remark 1 of arXiv:2502.18491 proved in full on the slice p = 3: four positive roots of F₃ for every real k ≥ 2 and every real k₁ ≥ (89/16)·3k = 16.6875k, in the paper's own chain with 2 replaced by 17/8

Remark 1 of arXiv:2502.18491 proved in full on the slice k = 2, at k₁ ≥ (137/32)·2p = 8.5625p against the paper's 16p, with the optimal constant for that slice bracketed in [4.16667, 4.28125]

Negative controls for the two slices: both slice constants are within 0.045 % and 2.7 % of the best possible, and both hypotheses are non-vacuous

Measured and parked: the general SU(N) F₃(β) < 0 certificate costs 1,214,664,642 monomial expansions in one ring and 7,847,931 after the best known decomposition — about 37 CPU-hours and a 30 MB source — while fixing either parameter collapses it by a factor of 117

The SU(N) four-root constant depends on the slice, and the pattern confirms the SU/SO limit bridge from the outside

Remark 1 of arXiv:2502.18491 in full for all real k ≥ 2, p ≥ 3 at k₁ ≥ (89/16)kp = 5.5625kp, certified in exact integer arithmetic — the paper's 8kp and row efr-13's (45/8)kp improved by moving the upper separating point, with five of the seven ingredients now kernel-bound

The seventeen Handelman signs of the coefficients of (1+x)⁹·H₁ on k ≥ 3, p ≥ 3, k₁ ≥ 11kp, giving signVariations ≤ 4 with the sign of g₂ left free

At most four positive roots of the SO(n) polynomial H₁, with multiplicity, for all real k ≥ 3, p ≥ 3, k₁ ≥ 11kp — against the Descartes bound of eight

Control: the Descartes bound for H₁ is exactly eight on the same region — its nine coefficients strictly alternate in sign, each by a Handelman certificate

Exactly four positive roots of H₁ at (k,p,k₁) = (3,3,104), entirely inside one module

At most four positive roots of the SU(N) polynomial F₃ for all real k ≥ 2, p ≥ 3, k₁ ≥ 50kp — the same argument as efr-36 against the Descartes bound of sixteen, certified in exact integer arithmetic but with its Lean check stopped on price

Measured: the exact positive-root census of both polynomials — the count is 0, 2 or 4 and never more, and the coefficient signs alternate in every cell

The structural reason for the Descartes deficiency: the Newton polygon of H₁ splits its eight roots 2 + 4 + 2 by size, and the middle four are never positive

Exactly four positive roots of H₁, with multiplicity, for all real k ≥ 3, p ≥ 3, k₁ ≥ 11kp — the first exact count for this construction

Exactly four positive roots of the SU(N) polynomial F₃, with multiplicity, at four explicit parameter points — (k,p,k₁) = (2,3,26), (2,4,33), (3,3,45), (5,5,300) — the first exact root counts anywhere in this construction, at ratios k₁/(kp) as low as 4.125 against the paper's 8

The SU(N) four-root threshold bracketed by two kernel theorems: exactly four positive roots at (1000,3,16681), ratio 5.56033 — *below* the certificate constant 89/16 = 5.5625 — and exactly two at (2,3,24), ratio 4, so no four-root theorem on the slice k = 2 can use a constant c ≤ 4

Measured and corrected: the two uniform SU(N) routes are memory-bound, not CPU-bound — SUSliceP3 costs 32.65 GB and half the at-most-four certificates 27.1 GB, both above the family's 10 GB per-statement cap — and a bigger Poincaré exponent does not lower the at-most constant

Controls for the kernel-bound counts: the two-root point refutes every four-root theorem below ratio 4 on k = 2, the four-root points refute "at most three", and the route's own failure near a double root is measured rather than assumed