Independent mathematical proof review — 2026-10-17
Reviewer: separate AI subagent; independent derivation and source inspection, not external human peer review.
Scope: original main.tex; verify_transition.py; verify_rational_replay.py; ordered-pair center verifier through moment assembly. No original files modified. Full certificate reruns are assigned separately; no numerical rerun claimed here.

Verdict
No fatal mathematical defect found in the principal interval-transition or minimum-degree argument. The main theorem follows correctly from the uniform certified inequalities if the accompanying verifiers establish them. One repairable gap affects the stated unbounded adaptive-search implementation/proof connection, not the fixed 1000-cell theorem. Two minor wording improvements are advisable.

Independent derivations
1. Normalization: inserting pi*exp(4s)=x into the summand gives pi*exp(5s-x)*n^2*(2n^2*x-3)*exp(-(n^2-1)*x). At s+t/2 the additional exp(5t/2) cancels in the product. Thus the printed A_x and p_x=A_x(t)A_x(-t)/A_x(0)^2 are correct, including the factor 1/2 in t.
2. Derivatives: d/dt of R(z)exp(-z), z=n^2*x*exp(2t), is 2z(R'-R)exp(-z). This independently recovers R recurrence. B_k is A_x^(k)(0); differentiating its center dependence gives exactly C_k. Leibniz rule recovers the binomial signed-moment formula and quotient differentiation recovers dm.
3. Complete tails: R_k degree k+1, k<=10. B monomials have powers n^(2j+2)<=24. C monomials have powers at most 26. For n>=8 the ratio (1+1/n)^26 exp(-(2n+1)x) is bounded by (9/8)^26 exp(-17x). The T and V majorants correctly dominate every omitted summand. All product interactions are retained through interval multiplication. No omitted tail-tail term found.
4. Schur calculation: v^T G^-1 v=(w^3-2vwy+uy^2)/(uw-v^2). Independently differentiating gives the printed F', Delta', S'. Expanding det H gives uwz-uy^2-v^2z+2vwy-w^3 and its gradient exactly agrees with the replay D'.
5. Inertia/minimum degree: parity splits M5 into E and H. Congruence H~diag(G,S) gives the stated inertias. All degree<=4 polynomials use E and G, so strict positivity of both really excludes even quartics as well as odd cubics. The null direction is correctly unique and monic at S=0.
6. Translation-Gram consequence: the difference transform tends to A_jk=(-1)^k p^(j+k)(0). With U=diag(i^j), U A U*=M5, since odd derivatives vanish and i^(j+k)(-1)^((j+k)/2)=1 for even j+k. A negative eigenvalue persists under the finite-dimensional limit. Thus sufficiently small six-point translation Grams fail positivity. Specifying U A U* would remove ambiguity in 'conjugation'.
7. Arithmetic inspection: interval add, multiply, reciprocal, and negation in the inspected verifiers are enclosing operations. The positive Taylor remainder uses the correct first omitted term and ratio u/(L+2). Nine squarings correctly map exp(-t/512) to exp(-t). Separate rational interval replay shares analytic assembly, which it discloses; it is not an independently developed proof.

Findings
F1 — moderate, repair before release: the termination corollary permits unbounded theta truncation, but the only described exponential evaluator requires t<512 and nine squarings. Since t=(n^2-1)x, larger N eventually violates that range. State adaptive range reduction: for any t>=0 choose integer r with t/2^r<1, use the same Taylor enclosure there, and square r times. Allow the arithmetic grid/precision and Taylor order to increase so total enclosure widths tend to zero. The reported N=7 results remain within the existing range and are unaffected.
F2 — low: abstract says 'Below it' without immediately limiting centers to I. Change to 'Within this interval, below it'. The theorem itself is scoped correctly.
F3 — low: the strict-negative ellipse is empty at S=0. 'At x_* it contracts to the unique null coefficient pair' should say its closure shrinks to that pair as x approaches x_* from below; at x_* the nonpositive set is the singleton. Current subsequent empty-above statement is correct.

Scope and limits
The main theorem is local to [3.64,3.66]; no global transition uniqueness or higher-degree positivity follows. Quintic minimum degree holds below x_* inside that interval. No result about RH or center-averaged kernels follows. Originality/priority requires literature review and cannot be established from algebra inspection. Mathematical value arises from the theta-specific certified classification and lower-degree exclusion, not novelty of Schur algebra. This review is not a formal proof-assistant verification and does not eliminate common mathematical assumptions shared among AI reviewers.

Separate expansion: stability theorem
Let S be C1 on I, S'>=kappa>0, and S(x_*)=0. If S_tilde is continuous, sup_I |S_tilde-S|<=epsilon, and y in I is any root of S_tilde, then |y-x_*|<=epsilon/kappa. Proof: |S(y)|<=epsilon and integration of S'>=kappa bounds |S(y)-S(x_*)| below by kappa|y-x_*|. If S_tilde is C1 with S_tilde'>0 and retains opposite endpoint signs, it has exactly one root. Endpoint signs are guaranteed by epsilon<min(-S(left),S(right)). Perturbations of the moment data must separately preserve G>0 and E>0 to transfer the degree/inertia classification. This is a standard stability consequence; it is a useful expansion route, not a new breakthrough or a new numerical certification.

Cleanup
No scratch files or processes were created. Only these requested permanent review reports were written. Earlier blocked scratch was not retried or altered.
