Decision: no surviving estimating candidate. The nonlocal proposal supplied a real, fully specified sufficient conjecture and was rejected by the original theta tail. The divisor and Riemann–Siegel proposals supplied exact arithmetic expressions, but neither supplied a lower-estimating mechanism after its transformation. Stop these three attempts. This is not a demand that a candidate be proved before admission, and it is not a closure of the original theta route.
Reviewed remote baseline: 670864385024041d24612cf620054c1975473928; scope commit: b914ac47e8006951a22d030e920d592243e7e0fe. Read AGENTS, the commissioned brief and review record, proposer notes 01–03, PR77’s final admission review, current task status and MISSING. The coordinator owns the refreshed full conclusion/dependency review; I independently checked the decisive identities and implications below. No historical proof or certificate replay is claimed. Both missing-original recovery groups remain OPEN.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–77. What changes: an explicit nonlocal centering certificate was examined with a particular positive-mixture estimator, and two arithmetic regroupings were assessed at their proposed sign-producing steps. None leaves a surviving estimate for the complete original target.
The proposal keeps the complete original kernel and defines
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),
V(s,t)=int_(-infinity)^s [a^2-k(t)]*w(a,t) da.
Its correction h=V/w is explicit. Since M>0, no transform or zero is divided out. Differentiating V gives the stated residual k((u-v)/2); centering makes both complete flux boundaries vanish. Alternatively, the transfer needs no correction calculation: M is an autocorrelation and therefore positive definite. If k is positive definite, their pointwise product C=M*k is positive definite, so its integrable Fourier transform is nonnegative. PR76’s exact normalization then gives J=16*L1[X]>=0 at every real height.
This is more specific than PR77’s request for an unspecified nonlocal correction. Its arithmetic conjecture was an exact positive mixture
k(t)=int_[0,infinity) sech(2t)^a dnu(a), nu finite and nonnegative.
It has the five required features: an exact expression; fixed original coefficients; every real t; a proposed positive-Laplace estimating technique for the transformed quotient; and a complete implication to the first-Laguerre target without any extra sign lemma. The Fourier integral of each positive power follows directly from the beta integral and is positive. It was therefore a meaningful paper feasibility test. Describing it merely as having no method would undersell what was actually specified.
The original-kernel asymptotic rejects this particular estimator. With A=2*pi*exp(2t), the complete dominant-pair calculation gives
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)).
I checked the decisive relative coefficients: the zeroth Gaussian moment has correction -6-3/24=-49/8; the second has -6-15/24=-53/8. Their ratio gives the displayed negative second coefficient. The uniform original-series remainder and the outside-neighborhood tail are retained in note 01; no full-line termwise theta unfolding is used.
If Q(z)=int exp(-az)dnu(a) with positive finite nu, its finite nonzero limit after multiplication by exp(z) forces no mass below 1 and an atom of mass 1/(16*pi) at 1. Hence exp(z)Q(z) approaches that mass from above. The displayed expansion approaches it from below. The contradiction needs no computed threshold.
Scope: this rejects only positive mixtures of those sech powers for the original k. General positive definiteness of k is not equivalent to this mixture representation and is not rejected. Nor does the argument reject all nonlocal h. The closest PR77 Gaussian-mixture rejection concerned the different object C and does not already decide this quotient class. A useful admission rejection has occurred, but no surviving arithmetic lower bound follows.
For the divisor proposal, independent multiplication on Re(s)>1 gives
zeta*zeta''-zeta'^2=sum c(n)*n^-s,
c(n)=d(n)/6 * sum_(p^a || n) a*(a+2)*(log p)^2.
The prime-power variance is genuinely original Euler arithmetic. The completed identity
J(r)=p^2*g(s)^2*[D(s)+C(s)*zeta(s)^2], s=1/2+i*r,
C(s)=-1/s^2-1/(s-1)^2+psi1(s/2)/4
also has the correct sign and normalization: r-differentiation introduces the required factors of i, and the gamma/polynomial contributions remain. It covers zeros without division. But the positive Dirichlet coefficients cannot be used as a positive sum on the critical line. Completion, residues and oscillatory dual terms remain mandatory.
The proposed absolute-remainder argument lacks an independently lower-bounded main term. The individual uncompleted phases change sign; that defeats the coefficient-positivity shortcut, not a completed block estimate. PR73 applies to its specified fixed reflected theta prefixes, not every finite Dirichlet or completed divisor block. No general Poisson/Voronoi impossibility follows. An exact completed summation formula would still need a concrete signed block inequality and compatible complementary estimate; neither has been supplied.
For the Riemann–Siegel proposal, I checked the quadratic expansion: both ratio and product cosines, 2*theta''*sin(sigma), the gamma-curvature terms, and all exact remainder cross terms are present. Derivatives are taken at fixed integer cutoff before pointwise selection. This avoids the cutoff derivative error and includes all zeros.
At the balanced cutoff, Poisson’s stationary dual indices have comparable length to the original sum. That observation explains why a transform alone supplies no small signed remainder. It does not rule out cancellation after a more structured transformation. The phase-free coefficient budget of order N versus the logarithmic diagonal is a limitation of the stated absolute-value bound, not a lower bound for the true off-diagonal contribution. The still-needed comparison is precisely a jointly phased lower estimate plus a compatible complete remainder estimate. Accuracy of a Riemann–Siegel approximation does not supply either.
NS100 excludes per-slice positivity, which none of the complete proposals assumes. NS101 shares positive complete kernels and the generic centering construction. In its established negative-Laguerre range, its quotient k_beta cannot be positive definite: otherwise C_beta=M_beta*k_beta would force the opposite sign. Thus conditional centering or generic primitive positivity is not a sign-producing mechanism. Original lattice/IVP hypotheses differ; that mismatch is not a control pass.
Davenport–Heilbronn lacks the literal original coefficients and completion. A generic contour or centering implication would need its matched screen; the exact original arithmetic claims do not automatically apply to it. NS74/83 concern NB target sensitivity and dyadic rates and do not instantiate this theta target. No proposed claim here warrants a numerical control scan.
Success of the nonlocal mixture would have settled the complete first-Laguerre obligation; its failure rejects only that class. Success of a separately specified signed divisor or Riemann–Siegel lemma could justify a new conditional proposal; their present missing-method result changes no theorem. None would establish RH merely by proving the first Laguerre inequality.
Budget and final decision: the commissioned paper formulation plus independent review exhausts this attempt. Zero scan/certification budget. Archive the precise disposition; no automatic neighboring mixture, parameter change, larger finite table, general swarm or route pivot. Reopening requires an explicit changed estimator with a complete conditional edge and stated companion gaps. No candidate is admitted, no row or thaw is claimed, and no manuscript version is justified. Final classification: Same open gap.