Reviewed remote mathematical baseline: 18d324703969b5bcc9aa94043cfd90a6e8211f34 (PR76), using the coordinator’s refreshed isolated worktree. Read repository instructions, current checkpoints and ZERO-GEOMETRY scope/freeze, NS100/101, PR76’s derivation and PR74’s nonlinear certificate review. This is a paper applicability assessment, not a historical proof or certificate replay. Both original-evidence recovery groups remain OPEN.
Wall check: Same open gap. Closest: NS100/101, PR74/76. No new arithmetic estimate is supplied. The concrete question is whether complete-kernel convolution, half-line convexity, or positive Gaussian mixtures yield an independently estimable sufficient condition. The two specified positive-mixture constructions fail necessary hypotheses; general positive factorization remains open.
Keep the original complete square-lattice coefficients:
phi(u)=sum_(n>=1) (2*pi^2*n^4*exp(9u/2)-3*pi*n^2*exp(5u/2))
*exp(-pi*n^2*exp(2u)), u>=0,
phi(-u)=phi(u),
X(r)=integral_R phi(u)*exp(i*r*u)du,
C(t)=integral_R s^2*phi(s+t)*phi(s-t)ds.
The extension is the complete smooth Jacobi kernel, never a reflected finite prefix. All polynomial moments and derivatives needed below are integrable. The requirement is positive definiteness of C on the whole real line: every finite matrix C(x_i-x_j), arbitrary real nodes and arbitrary complex coefficients, must be positive semidefinite. There are no parity/source restrictions. Its exact consequence is
L1[X](r)=X'(r)^2-X(r)*X''(r)=4*integral_R C(t)*cos(2*r*t)dt>=0
for EVERY real r,
J(r)=16*L1[X](r)>=0.
The identity includes zeros and r=0; no division by a transform occurs. First-Laguerre positivity alone is not RH. PR76’s polynomial correction is already included by working directly with C.
An independently derived equality C(t)=sum_j (g_j*tilde(g_j))(2t), with g_j in L2(R), tilde(g)(u)=conj(g(-u)), and sum_j ||g_j||_2^2<infinity, would suffice: arbitrary finite quadratic forms become sums of squared L2 norms. Fourier positivity follows almost everywhere, then everywhere from the continuous complete transform. But constructing g_j from the square root of the unknown Fourier transform assumes the target. No such original-coefficient factor is obtained here.
Put f_j(u)=u^j*phi(u) for j=0,1,2, and h_+=f_2+f_0, h_-=f_2-f_0. Writing y=s+t and expanding (y-t)^2 gives
C(t) = (1/2)*(f_2*f_0-f_1*f_1)(2t)
= (1/8)*(h_+*tilde(h_+)-h_-*tilde(h_-)
+4*f_1*tilde(f_1))(2t). (1)
The oddness of f_1 is essential: tilde(f_1)=-f_1. Each autocorrelation in (1) is positive definite, but its negative coefficient cannot be discarded. Its Fourier domination is exactly
|X+X''|^2 <= |X-X''|^2+4*|X'|^2, every real r,
which expands to L1[X]>=0. Thus (1) gives no independently estimating lemma. It uses complete integrable functions, so no forbidden full-line termwise theta unfolding occurs. NS101 already exposes the same signed-comparison issue for the tilted autocorrelation derivative; the polynomial feature decomposition does not escape it.
The precise Pólya criterion is continuity, evenness, normalization at zero, convexity on t>0, and decay to zero; it implies positive definiteness. The cited source itself warns that its smooth admissible kernels fail the convexity hypothesis. This is not an all-purpose curvature theorem. Csordas, arXiv:1309.0055v2, Theorem 3.4 and following paragraph.
For our actual C, smooth evenness gives C'(0)=0, positivity gives C(0)>0, and the complete tails give C(t)->0. If C were convex on the positive half-line, its derivative would be nondecreasing from zero; hence C(t)>=C(0) there, a contradiction. This also excludes a nonzero positive triangular-mixture representation
C(t)=integral_(0,infinity) (1-|t|/a)_+ dnu(a), nu>=0,
whose members are convex on the positive half-line. The usual Fejér/Pólya cosine-positivity criterion based on decreasing convex input therefore cannot apply directly. Eventual convexity alone does not supply its missing near-origin hypothesis. A different Fejér-type theorem would need separately stated hypotheses and transfer; none is asserted here.
The complete original theta tail yields constants K,c>0 with
phi(u)<=K*exp(-c*exp(2*|u|)) on all of R. Since
exp(2*|s+t|)+exp(2*|s-t|)
>= (exp(2*|s|)+exp(2*|t|))/2,
integration gives 0<C(t)<=B*exp(-(c/2)*exp(2*|t|)), with finite positive
B=K^2*integral_R s^2*exp(-(c/2)*exp(2*|s|))ds.
Suppose instead C(t)=integral_[0,infinity) exp(-a*t^2)dmu(a), mu>=0. Its finite nonzero mass equals C(0). Some finite A has m=mu([0,A])>0, implying C(t)>=m*exp(-A*t^2), contradicting the bound above. This excludes exactly positive Gaussian scale mixtures, not general autocorrelations or positive-definite kernels with faster-than-Gaussian decay.
NS100 excludes per-slice positivity; none is assumed. NS101’s reciprocal control shares smoothness and double-exponential decay, so the two rejections above also apply to it: they do not distinguish original arithmetic. PR74 additionally supplies an analytic first-Laguerre failure range for that family, with its original coefficients/Jacobi initial data changed. Those changes leave an original-coefficient factorization unexcluded. PR74’s failed local drift certificate is preserved; no modified drift, divided-difference, or transport ansatz is proposed here.
Decision: admit no new estimating lemma. Stop the two stated mixture constructions. The remaining missing input is an explicit factor/comparison derived from the original square-lattice coefficients, with complete tails and all coefficient combinations retained. Merely demanding such a factor is the existing gap.
Budget: at most 1–4 hours for paper admission of one externally supplied explicit factor formula; zero scan/certification budget without that formula. Success would establish the complete first-Laguerre target, with no further transfer estimate, but not RH/G2. Failure would reject only that formula. No automatic adjacent variation, new row, thaw, manuscript version, computation, commit or outreach.
Final wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR74/76. Scoped known walls exclude direct convex-half-line and positive-Gaussian-mixture certificates. The original complete sign and cofinal signed-arithmetic lower bound remain open.