Decision: no estimating method in reviews 01–11 meets the gate. Finish this commissioned assessment and stop the automatic research cycle. No additional computation, adjacent trial, general next-step round, substitute brief, or automatic pivot to another frozen group is justified by these findings. This is a disposition of the supplied proposals, not an impossibility theorem.
Reviewed remote baseline: 18d324703969b5bcc9aa94043cfd90a6e8211f34 (PR76), refreshed in the coordinator’s isolated checkout. Read AGENTS’s proposal/relevance gates, current README/checkpoints, NEXT_STEPS, the conclusion register’s relevant dependencies and all continuation demands, MISSING, reviews 01–11, and the prior complete smooth-completion comparison. The coordinator owns the full-register review and primary-source refresh; I independently checked the principal algebraic and asymptotic implications below. This is not a historical proof/certificate replay. Both original-recovery groups remain OPEN.
Wall check: Same open gap. Closest: ZERO-GEOMETRY, NS100/101, PR73–76. What changes: stronger admission and scope assessment; no independent estimate for the complete original sign is supplied. Scoped failed constructions remain failed, while the original arithmetic input remains open.
Keep the complete original function and every real height:
p(r)=r²+1/4, X=pY/4,
J=p²(Y'²−YY'')+2(r²−1/4)Y²=16(X'²−XX'') >= 0.
The identity includes zeros without division. For |r|<1/2, the polynomial correction is negative: even L1[Y]>=0 would still require p²L1[Y]>=2(1/4−r²)Y². The first-Laguerre inequality alone is not RH. A stronger RH-equivalent premise may suffice, but calling it a weaker moment or phase estimate would misstate the obligation.
Prime smoothing (04). The reviewed exact formula retains the pole, trivial zeros and every nontrivial zero. Repeated integration by parts with a fixed smooth cutoff supports the stated remote-zero bound: the summand’s leading allowance is sqrt(x)L^(1−K)|s−rho|^(−K−1), and local zero counting yields the displayed U^(−K)log(2+|r|+U) tail. This is a genuine supporting upper estimate.
It leaves the local signed expression unchanged. Near a known critical-line zero the quotient contains a positive inverse-square pole, precluding a uniformly bounded absolute residue error there. A hypothetical symmetric off-line pair contributes −2/a²+L²mu_2+O(L⁴a²) when aL is small. That calculation tests the proposed geometry/counting bound; it neither exhibits such a zeta pair nor rules out compensation by the other arithmetic terms. Consequently this is same open gap, not a general Euler-product wall. Enlarging a cutoff cannot replace the missing signed local estimate. NS47’s different all-test undamping obstruction does not settle this question.
Jensen backward transfer (07). The derivative recurrence loses the constant coefficient. The displayed alternating critical-value signs correctly express the additional information needed to integrate back when the critical points are distinct; multiplicities need their own retained conditions. The compact limit P_(d,0)(−r²/d) -> X(r), together with convergence of two derivatives, is a valid transfer from unshifted all-degree hyperbolicity to J. The cited large-shift region does not estimate those unshifted critical values. No bound for their alternating areas is supplied. Ordinary moment positivity or a degree-two Turan inequality is insufficient. This conditional theorem is useful bookkeeping, not an admitted backward-estimating method.
Bessel margin plus smooth completion (08–09). The Bessel theorem estimates L1[H]; the completion theorem estimates X−X_M. These concern different differences. Writing U_M=X_M−H leaves
L1[X_M]=L1[H]+2H'U_M'−H U_M''−H''U_M
+U_M'^2−U_M U_M''.
The missing signed comparison is precisely this complete mixed expression against the completion error. Increasing M sends U_M to X−H, not to zero. The error bound is not a bridge to H’s positivity.
There is also a rate limitation for the displayed bounds: the proved crude Bessel allowance has scale r^(−2)exp(−r²), whereas the explicit completion budget with linear M decays exponentially in |r|. That comparison fails for these allowances; it says nothing about the sharp Bessel margin or the true signed error. Quadratic M would address only that rate comparison, not the object mismatch. No cutoff search is warranted. Strict margins would also impose simplicity consequences absent from the non-strict target.
b=−2z+9/2+O(1/z), b'=−4z+O(1/z). Its score divided-difference determinant at nodes 0,u is eventually negative. This rejects that all-node positive-semidefinite kernel, not every two-point invariant. It was only a proposed supporting building block; success of two-node checks would not have proved J.Existing known walls remain restricted to their matching premises: NS100 per-slice positivity, PR73 fixed reflected prefixes, and PR74’s particular drift residual. The rejected new building blocks settle bounded feasibility questions. They do not constitute a surviving distinct estimating test for the original sign.
NS101 directly violates the first-Laguerre target in its proved parameter range while sharing the primitive’s structural properties. It changes the original coefficients/Jacobi IVP. This excludes the shared-property inference; its mismatch is neither a counterexample to original arithmetic nor a control pass. Davenport–Heilbronn likewise does not instantiate the original theta/Euler premises. Generic supporting lemmas remain permissible if their separate arithmetic premises and complete transfer are supplied.
Review 11 correctly keeps NB, Möbius, Weil-floor and operator gaps distinct. No new supported arithmetic method there warrants an automatic switch. The already commissioned synthesis is now the final useful action; another synthesis would duplicate it.
The stop changes only upon receiving one concrete estimating mechanism with its exact expression, domain and rate; an identifiable use of original arithmetic; a complete conditional transfer retaining tails, mixed terms and zeros; and every remaining companion estimate stated. A partial lemma can qualify if it improves a named bound or settles an independent feasibility decision. Merely requesting the missing sign, a positive factorization, critical-value inequalities or a generator is not such a mechanism. No present research allocation follows from this conditional reopening rule.
Final classification: Same open gap. No major mathematical defect found in the reviewed admission conclusions. No candidate, row, thaw, new theorem node or manuscript version is admitted. RH, G2, the actual cofinal signed-arithmetic lower bound and the complete original first-Laguerre target remain open.