# Reviewer 12 — independent final admission decision

**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.

## The target has not become RH or an easier surrogate

Keep the complete original function and every real height:

```text
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.

## Three apparent opportunities do not yet supply an estimating step

**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

```text
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.

## The scoped rejections survive, without broader closures

- **01:** the original complete theta tail gives `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.
- **06:** smooth even C has derivative zero at the origin, positive value there and decay to zero, so it cannot be convex on the whole positive half-line. Its double-exponential upper tail also excludes nonzero positive Gaussian scale mixtures. Neither argument excludes general autocorrelations or positive definiteness. The exact convolution decomposition retains its negative term.
- **02/03/05:** rescaling all translates makes every positive separation parameter equivalent to the final transport target; lattice reindexing preserves signed phases and mixed terms; phase monotonicity restates J, while a Herglotz logarithmic derivative already demands exclusion of nonreal zeros. None supplies a missing generator, arithmetic regrouping or Hamiltonian.

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.

## Controls, stop and exact reopening evidence

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.