Decision: no candidate admitted; no unresolved MAJOR issue found in the three stop decisions. The nonlocal attempt has a concrete sufficient estimate and a complete transfer, but its proposed positive sech-power representation fails a necessary condition on the original kernel. The divisor and Riemann–Siegel attempts retain correct arithmetic identities but supply no separate signed estimating step. Their failure to qualify is not a demand that an admitted conjecture already be proved.
Reviewed baseline: 670864385024041d24612cf620054c1975473928, local scope commit b914ac47e8006951a22d030e920d592243e7e0fe. Read AGENTS, the admission brief and review record, all three proposer notes, PR77’s factorization/control findings, the relevant continuation and missing-evidence records, and the two control scripts as source only. The coordinator’s full-register review is inherited; historical proofs and numerical certificates were not replayed. Both original-evidence recovery groups remain OPEN.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–77. What changes: a specific sufficient nonlocal mixture class is assessed on the original coefficients, and two complete arithmetic regroupings are checked for genuine estimating content. The narrow mixture rejection does not settle general positive definiteness or the complete first-Laguerre inequality.
The distinction between the complete pair C and its normalized second moment is essential:
w(s,t)=phi(s+t)*phi(s-t),
M(t)=int_R w(s,t)ds, C(t)=int_R s^2*w(s,t)ds,
k(t)=C(t)/M(t), M(t)>0.
The positivity of the denominator follows from the original strictly positive real kernel, not from Fourier positivity. The centering construction gives partial_s V=(s^2-k(t))*w and zero flux at both infinite endpoints for each fixed t. It therefore supplies the claimed residual k((u-v)/2) exactly. It does not prove that residual positive definite.
The conditional transfer is nevertheless complete: M is a scaled autocorrelation of phi; if k is positive definite, their product C is positive definite. C is integrable and its Fourier transform is continuous, so nonnegative transform density gives L1[X](r)>=0 at every real r, including zeros. This uses the full C and automatically retains PR76’s low-height correction in J. No additional sign estimate is hidden in the transfer.
I independently checked the proposed sufficient class and its rejection. For a>0, elementary beta integration gives
int_R sech(2t)^a*exp(i*r*t)dt
=2^(a-2)*Gamma(a/2+i*r/4)*Gamma(a/2-i*r/4)/Gamma(a).
This is positive; a=0 is the constant positive-definite member. Thus a finite positive mixture would indeed suffice. PR77’s positive-Gaussian-mixture obstruction concerned C, whose decay is double exponential; it does not decide this sech-power mixture of the different quotient k.
For the original n=1 tail, write A=2piexp(2t). Factoring both summands gives exactly
w(s,t) ~ 4*pi^4*exp(9t)*exp(-A*cosh(2s))
*[1-6*cosh(2s)/A+9/A^2].
The coefficients 6 and 9 are correct. On the proposer’s growing central interval the other lattice terms are uniformly exponentially smaller; outside it the complete tails are negligible relative to the central integral. With y=2s=x/sqrt(A), the first Gaussian correction is -6-x^4/24. The zeroth and second moments therefore have corrections -49/8 and -53/8, giving
k(t)=1/(4A)-1/(8A^2)+O(A^-3).
Q(z)=k(arcosh(exp(z))/2)
=exp(-z)/(16*pi)-exp(-2z)/(128*pi^2)+O(exp(-3z)).
A finite nonnegative measure with Q(z)=int exp(-a*z)dnu(a) and the displayed leading asymptotic must have no mass below 1 and mass exactly 1/(16*pi) at 1. It follows that exp(z)*Q(z)>=1/(16*pi). The negative next coefficient contradicts this. No numerical onset or finite sample is involved. This rejects precisely the nonnegative sech-power measure, not arbitrary positive-definite k, signed mixtures, or every nonlocal correction.
For ordinary absolutely convergent Dirichlet products, Re(s)>1, symmetrizing the ordered divisors gives
D(s)=zeta*zeta''-zeta'^2=sum c(n)*n^(-s),
c(n)=1/2*sum_(d|n) [2*log(d)-log(n)]^2.
For n=prod p^a, divisor exponents are independently uniform on 0,...,a. Their centered second moments are a*(a+2)/3. Thus
c(n)=d(n)/6*sum_(p^a || n) a*(a+2)*(log p)^2.
In particular c(p)=log(p)^2. This is a valid Euler-coefficient identity, not a positivity theorem for a critical-line transform. It is not necessary to assume c itself multiplicative.
Let g=pi^(-s/2)*Gamma(s/2), A=s*(s-1)*g/2, and C_A=(log A)''. Product differentiation gives
C_A=-1/s^2-1/(s-1)^2+psi1(s/2)/4,
xi*xi''-xi'^2=A^2*(D+C_A*zeta^2).
Since X(r)=xi(1/2+i*r)/2, each r derivative contributes i, and consequently J=4*(xi*xi''-xi'^2)=p^2*g^2*(D+C_A*zeta^2). Both the sign and factor are correct. The original gamma and polynomial contributions cannot be dropped. The expression is undivided at zeta zeros. Absolute convergence in Re(s)>1 does not authorize termwise evaluation on Re(s)=1/2. The proposed two-shift completed product is a legitimate starting object, but no completed Voronoi identity or sign-producing remainder theorem is actually supplied or relied upon.
The fixed-N quadratic identity retains both ratio and product phases, the 2*theta''*sin(sigma_nm) term, and both -2*d terms. Direct differentiation confirms these signs. The complete remainder correction also retains all cross products and orders zero through two. Taking derivatives at fixed N before selecting the usual square-root cutoff avoids falsely differentiating a floor function.
Here R_N=Z-P_N is an exact definition. No asymptotic Riemann–Siegel error theorem is imported, and no admissible uniform derivative bounds or positive margin follow merely from naming that expansion. The balanced dual scale explains why automatic smallness is unjustified; it does not prove every Poisson method fails. None of the three notes imports a new external result claimed to close an estimate. The beta integral, finite variance identity and product differentiation are derived in the notes. Any future specialized summation or derivative theorem must be checked against an actual arXiv statement, including its range and remainder; there is currently no source-based admission to endorse.
Gamma((s+1)/2) differ from the conductor-one expressions. Its full positivity premises for the k construction have not been established here. Generic completion or differentiation identities may match, but the full literal arithmetic hypotheses do not. Moreover off-axis zeros alone do not establish failure of the weaker first-Laguerre inequality. The existing example script samples a different horizontal-growth expression; running it unchanged would not screen these proposals.Disposition and budget: no control script or sign scan was run, because no surviving estimating candidate is ready for computation. The bounded paper review is complete. The mixture class is stopped; the other two lack a signed arithmetic estimating mechanism and a compatible remainder comparison. No row, thaw, manuscript version, commit or outreach. First-Laguerre positivity, RH, G2 and the separate cofinal arithmetic input remain open.
Final wall check: Same open gap. The reviewed narrow failure is sound; it supplies no new bound on the original complete target.