# Hostile referee disposition

Date: 2026-07-19

## Recommendation

Accept the mathematical barrier and the source classifications as a
draft suitable for expert circulation after author identification and
ordinary journal formatting. Reject any stronger description saying
that all positivity methods, or any cited paper, has been disproved.

The defensible contribution is exact: an open sufficient certificate
cannot accept a target while its values are the limit of certificates
for an explicitly proved bad sequence. For RH, Rodgers--Tao supplies
the canonical sequence \(H_{-1/n}\to H_0\). The field-level consequence
is prospective: a uniform interior reserve with regular transport and
same-object sufficiency is not the missing final step in the audited
architectures.

## Theorem audit

- **Abstract theorem: pass.** M1--M4 are inconsistent by the defining
  property of convergence.
- **Scope: pass.** M4 is required only on the target and named
  counterexamples, not an ambient function class.
- **Metric and arbitrary-interior forms: pass.** The manuscript and
  Lean core cover both an open accepted set and the interior of an
  arbitrary accepted set.
- **Scalar hierarchy: pass after repair.** The positive cone is
  intersected with its weighted carrier, and the corollary assumes
  certificate values belong to that carrier.
- **Coercive form: pass after repair.** Both the margin and continuity
  estimate scale by \(\|v\|^2\); the statement is not vacuous at zero.
- **Finite tests: pass.** The corollary rejects only finite strict tests
  additionally claimed sufficient for the complete function.
- **Formal synchronization: pass.** Shared and pinned Lean files are
  byte-identical and compile in the frozen Mathlib environment.

## Membership dispositions

| Program | Primary-source verdict | First failed or adverse premise | Conditional completion excluded |
|---|---|---|---|
| de Bruijn--Newman heat flow | Theorem input, not a member | M2: no argument-chosen certificate topology or convergence | Fixed certificate, M2, open M3, and common M4 |
| Jensen polynomials | Nonmember | M2 and M3: no complete heat modulus or uniform hyperbolicity reserve | Same heat transform, fixed normalization, complete convergence, positive complete distance |
| Generalized Laguerre--Turan | Nonmember | M2 and M3: no all-order/all-point heat modulus or floor | Fixed \(m\geq1\) hierarchy, uniform modulus, positive normalized floor |
| Weil--Bombieri forms | Nonmember | M2 and M4: no common heat form/domain or same-object theorem; M3 has no Rayleigh floor | Common-domain heat forms, form-norm convergence, same-object sufficiency, coercivity |
| de Branges positivity | Fails at target | M3: the primary source computes the wrong sign for the zeta-associated space | Only a replacement true at \(H_0\), transported on one domain with M4 and coercivity |
| Canonical systems | Nonmember | M2: \(\omega\) is a zeta shift, not heat time; no backward-heat system | Common heat systems, fixed domain, convergence, uniform definiteness, same-object M4 |
| Pólya frequency / total positivity | Nonmember | M2: no reciprocal transform or regularity on negative heat; M3 has no nondegenerate complete reserve | Source-compatible transforms, common Schoenberg hypotheses, normalized minors, convergence, open acceptance |

The common Rodgers--Tao false family can supply M1 to a future
completion even when a program's own source does not mention it. The
table therefore reports the first architecture-specific failure rather
than using source-local M1 absence as a rhetorical shortcut.

## Boundary controls

- **Li coefficients:** nonmember. The published multiset criterion has
  no canonical heat multiset, fixed complete normalization, uniform
  accumulation, or open complete floor.
- **Nyman--Beurling:** outside-class control. Its terminal statement is
  distance zero to a closure, exactly a boundary mechanism.

## Escape-hatch audit

Every genuine escape fails at least one of M1--M4. The manuscript
records the recurrent concrete forms: zero complete infimum,
semidefinite or null-sequence behavior, rank/degree/support dependence,
singular transforms, changing domains, endpoint-only arithmetic,
closure criteria, finite scope, and quantitative separation from the
named false family.

Three false admissions are specifically blocked:

1. Jensen distance to a complement is only a local reserve on an
   already accepted polynomial, not a globally signed criterion.
2. \(L_0=F^2\) is automatic and vanishes at real zeros, so strict
   Laguerre margins begin at \(m\geq1\).
3. Raw total-positivity minors vanish as nodes coalesce, requiring
   separated configurations or a sufficiency-preserving confluent
   normalization.

## Residual limits

- The paper does not claim an exhaustive bibliography of RH
  equivalences. Omitted papers must be tested independently with the
  worksheet.
- Rodgers--Tao's analytic theorem is cited, not formalized in Lean.
- The Lean core checks the barrier and quantitative persistence
  deductions; the scalar and operator cone-interior propositions remain
  conventional handwritten proofs.
- Text anchors and hashes prove source identity and location, while the
  semantic membership judgments remain mathematical arguments exposed
  in the dossiers.
- Suzuki's 2026 Weil paper is a current preprint supplement; the
  published Weil architecture is independently anchored in Bombieri's
  2000 paper.

## Final referee verdict

A hostile referee should concede the theorem, the exact M1--M4
worksheet, and every stated actual-paper verdict. The referee need not
concede, and the manuscript does not claim, that positivity methods as a
whole are dead. What is dead is the precisely defined robust-interior
completion move whenever all admission clauses are genuinely present.
