Proposal: Bounded admission audit of three concrete original-theta mechanisms. Principal candidate: for the complete original phi, M(t)=int phi(s+t)phi(s-t)ds, C(t)=int s^2 phi(s+t)phi(s-t)ds and k=C/M, assert for every real t k(t)=int_[0,infinity) sech(2*t)^a dnu(a), nu finite nonnegative. Assess its original-lattice tail before any computation. Also assess exact divisor-variance and Hardy/Riemann-Siegel arithmetic decompositions for an independent signed estimating step. This PR records paper audit outcomes.
Shared-input group: ZERO-GEOMETRY (NS100/101, PR76/77). Status: frozen. Thaw requires specific original arithmetic absent from the matched controls, with an estimating mechanism and complete dependency edge. No thaw or research-row claim is made.
Arithmetic input beyond the functional equation: Original complete integer-square theta series, all coefficients one, n=1 leading term and n>=2 exponent gap. These determine the quotient’s two tail coefficients and reject the proposed positive mixture. The divisor identity also uses exact Euler coefficients and unique factorization. Hardy sums retain weights n^(-1/2), frequencies log n, original gamma phase and original complete remainder. These identities alone supply no sign.
Dependency edge and arithmetic use (before any new candidate computation): Positive sech powers are positive definite. Their positive mixture gives k positive definite; M is an autocorrelation, hence C=Mk is positive definite. The complete associated-kernel identity then gives L1X=4 int C(t)cos(2rt)dt>=0 and J(r)=16 L1X>=0 for all real r. The explicit centered Volterra correction retains both flux boundaries. This is only first Laguerre, not RH. No zero division or omitted polynomial correction. The candidate’s sole additional sign hypothesis is rejected: Q(z)=k(arcosh(exp z)/2)=exp(-z)/(16pi)-exp(-2z)/(128*pi^2)+O(exp(-3z)) cannot be a positive Laplace mixture. The other two proposals leave a signed completed-block estimate, compatible remainder bound and (if eventual) compact-range proof unproved. No such unspecified bridge is computed.
Control screen (actual candidate, before proof/certification): Davenport-Heilbronn: not-applicable to literal original-lattice hypothesis; different coefficients, conductor and completion. Generic contour/transform manipulations are not original arithmetic estimates. No sampled pass. NS100 log-concave order-2: actual negative original slice retained as a known restriction; candidate integrates the complete pair, assumes no slice sign. Thus no matched exclusion of its full implication and no sampled pass. NS101 reciprocal dilation: generic centering, autocorrelation and complete transform identities are shared; exact original lattice/IVP are changed. Their mismatch is not a pass. Its established negative-Laguerre range forces its quotient k_beta not to be positive definite. The original input itself rejects the proposed mixture. Additional control: NS74/83 concern NB target sensitivity/dyadic rates; no NB projected Gram, target or relative-gain sequence occurs here. PR77 Gaussian-mixture rejection concerns C, not the new quotient k. Command/artifact, precision, sample domain, values/residuals: Paper only; no numerical command, precision or sampled domain applies. reviews/01-nonlocal.md and independent reviews/04-algebra.md,05-controls.md retain the complete tail and prove the necessary-condition rejection. reviews/02-poisson.md,03-riemann-siegel.md state missing-method stops. Full-scope obstruction: the specified positive sech-power mixture for the original quotient is incompatible with its second tail coefficient. No obstruction to general positive-definite k, nonlocal certificates, signed Poisson techniques, or the original J target is established.
Wall check: Same open gap. Closest result: NS100/101, PR76/77. What changes: a concrete nonlocal sufficient class is now explicitly assessed and rejected. No signed arithmetic lower estimate for the target is added. The other two attempts give exact decompositions without an estimating step.
What success changes: the principal mixture would supply a sufficient complete first-Laguerre certificate. A genuine signed divisor/Hardy estimate would need its listed complementary inputs. No RH/G2 inference follows from first Laguerre. What failure changes: stop the exact mixture class and the two missing-method attempts as specified. Preserve broader original-arithmetic routes as open. Budget: six paper reviews, one audit PR, zero numerical scans/certification runs. Disposition before claim: reject principal mixture; defer the two arithmetic expressions because no independent estimating mechanism was supplied. Stop this round, no automatic variant or route switch. No research row is claimed. Version bump expected: no. Audit/failure-only work; manuscript unchanged.