# Local model pilot: proposal and critic calibration

2026-09-29. Reviewed three first-model proposals, three critic replies, and three second-model audits as untrusted data. Scope: candidate admission and pipeline calibration, not an RH result, capability benchmark, or numerical scan. Baseline: parent-reviewed `b5bd5fa`.

**Wall check: Same open gap.** The pilot supplies no OPERATOR-BRIDGES construction or WEIL-FLOOR estimate. Its proposed ordinary smooth reduction additionally fails a concrete geometric prerequisite. No research row or thaw is justified.

All three proposal lanes returned variations of the supplied 4D contact-block template. This demonstrates inadequate diversity in this batch, without establishing its cause or ranking models generally.

| Proposal | Admission disposition | Decisive issue |
|---|---|---|
| Symplectic | Reject the ordinary smooth reduction as stated; a differently specified quotient would require fresh admission | Incorrect energy-surface description and non-free, nonproper action at the orbit being encoded |
| Arithmetic geometry | Reject current bridge; retain only the already supplied local coefficient | Trace distribution, Euler-factor value, positivity and zero location are conflated |
| Spectral | Reject current candidate and proposed test | Undefined operator/remainder and orbit weights misidentified as spectral eigenvalues |

## Concrete proposal errors

**Symplectic.** `H=τ+xy=1` is a regular three-dimensional energy hypersurface: `∂H/∂τ=1`. It cannot be a symplectic leaf of the ambient symplectic manifold; its restricted two-form has the flow as kernel. The usual dimension count, restricting by one equation then quotienting by one direction, *could* yield dimension two under the required hypotheses. The count itself is not the error.

The stated R-action has stabilizer `log(p) Z` on the central periodic orbit. It is not free there, and is not proper: the preimage of a fixed orbit-point pair contains an unbounded sequence of return times. Consequently the standard smooth symplectic-reduction conclusion is unavailable precisely at the encoded orbit. Deleting that orbit changes the proposed arithmetic input. A groupoid or other singular replacement is a separate construction, not an automatic repair. The reduced spectral measure and retained trace are also unspecified. Matching one local coefficient would not complete either parity-sector transfer.

**Arithmetic geometry.** The coefficient `−log(p)p^(−m/2)` is a signed weight at return time `m log p`, not the value of `(1−p^(−s))^(−1)` at `s=m/2`. The former is negative and the latter positive for positive m. The repetition index and transform variable are different. This single Euler factor already has a convergent geometric series for `Re s>0`; analytic continuation is not the missing operation at `s=1/2`. A signed supertrace need not be positive.

The proposal's nonvanishing claim is qualified by “t corresponding to the flow periods,” so it should not be reported as a universal assertion of critical-line nonvanishing. Nevertheless that correspondence is undefined, the asserted implication is unsupported, and such nonvanishing is not the RH target. No complete trace identity or positivity mechanism is specified.

**Spectral.** The displayed λ is a one-form; the symplectic form is dλ. More substantially, periodic-orbit trace weights are not thereby eigenvalues. No Hamiltonian quantization, transport realization, Hilbert domain or trace-to-spectrum theorem supplies the proposed identification. The remainder in its first test is never defined. Even for a defined quadratic remainder, growth under rescaling f is compatible with a bound `C||f||²`; the relevant quantity must be normalized. The title's assertion of explicit cofinal positivity failure is unsupported.

## The critics require correction too

The first critics correctly identify missing global constructions and the undefined remainder, but their rejection labels are not certificates:

- Infinite geometric measure does **not** preclude a spectral measure. Multiplication by x on `L²(R,dx)` is a standard self-adjoint example: its spectral projections multiply by indicator functions. Trace-classness, finite total volume and the spectral theorem are separate questions.
- Assuming an explicitly labeled missing hypothesis in a conditional implication is not inherently circular. The useful objection is that these replies give neither a sound complete implication nor an independent route to their assumptions.
- Comparing a reduced model's coefficient with the established parent model can be a legitimate preservation test. It is not automatically circular. Replacing that test with a fitted zeta zero or isolated L-value would not repair the missing transfer.
- The spectral critic repeats the category error by recommending a search for the orbit coefficient as an eigenvalue. An arbitrary operator could be manufactured with that eigenvalue; this would establish no desired trace identity.
- The arithmetic critic's suggested single-prime gamma completion has no specified global target. NS47/100/101 cannot be invoked merely because positivity or geometry appears: their actual hypotheses must match. No ordinary-L² closable factorization is established in these drafts.

The second-model audits repeat substantial portions of the first critics. Their `critic_errors` fields often describe proposal defects instead of errors in the critics; the spectral entry duplicates the same list in both fields. They also repeat the orbit-coefficient-as-eigenvalue recommendation. Agreement therefore did not provide independent mathematical validation in this batch.

## Process recommendation

Use separate construction briefs, then check object type, dimension, action/domain and transfer target before requesting calculations. Require each critic finding to identify the disputed statement and its reason; require reviews of critics to address the critic's assertion explicitly. Include matched counterexamples and legitimate conditional arguments so that blanket rejection is itself detectable. Deduplicate outputs before expanding the batch. Preserve unsuccessful generations as calibration evidence, without promoting votes or schema compliance into mathematical evidence.

**Final wall: Same open gap.** None of these three generated proposals is admitted for an RH-directed computation. The supplied local constructions remain available with their original limited scope.