# Hostile expert audit

Date: 2026-07-19

## Executive verdict

The abstract theorem is correct and elementary. A journal paper is
defensible only if its contribution is stated as:

1. an exact operational definition of robust interior certification;
2. a machine-checked no-go theorem using the canonical negative-heat
   counterexamples;
3. topology dichotomies for scalar hierarchies and operator cones; and
4. a primary-source classification that never labels a paper a member
   when it supplies only an endpoint criterion.

The title and abstract must not say that all positivity approaches, or
the cited papers themselves, are false.

## Attack 1: the theorem is a tautology

**Objection.** An open set containing a limit point eventually contains
the sequence. The proof is one line.

**Finding.** Correct. Hiding this would damage the paper.

**Required response.** State openly that the topological lemma is
elementary. The mathematical content specific to RH is Rodgers--Tao's
canonical false sequence, the exact identification of uniform margins
with cone interiors, and the source-level classification of proposed
completion moves. The paper is a barrier/taxonomy paper, not a deep
topology paper.

## Attack 2: the class is gerrymandered by topology

**Objection.** One can choose a metric after seeing \(\Xi\), make the
target isolated, or make the false sequence fail to converge.

**Finding.** The theorem cannot prevent pathological certificate design.

**Required response.** Membership requires the certificate carrier,
normalization, index set, and topology to be part of the proposed
argument before its terminal estimate. Every dossier identifies that
data. A discrete or endpoint-tailored topology is an explicit nonmember,
not a rebuttal.

## Attack 3: rescaling manufactures a margin

**Objection.** If every endpoint coordinate is positive, divide each
coordinate by its endpoint value and obtain margin one.

**Finding.** This does not evade the theorem. The rescaling is forbidden
if chosen after the endpoint estimate. Even if fixed legitimately, the
uniform backward modulus becomes
\[
\sup_\alpha
\frac{|q_\alpha(t)-q_\alpha(0)|}{q_\alpha(0)},
\]
which cannot tend to zero together with common sufficiency.

**Required response.** State this normalization tradeoff explicitly.

## Attack 4: pointwise continuity is mistaken for robustness

**Objection.** Every fixed Jensen polynomial, Laguerre expression, or
finite matrix varies continuously under heat.

**Finding.** Pointwise continuity over an infinite index set does not
give convergence in a uniform certificate topology.

**Required response.** Separate finite-coordinate continuity from the
complete sup modulus in every dossier. Product topology has empty cone
interior; sup topology requires the forbidden floor.

## Attack 5: strict positivity at every index looks interior

**Objection.** A sequence \(q_n>0\) for all \(n\) should count as robust.

**Counterexample.** \(q_n=1/n\) is strictly positive but lies on the
boundary of the positive cone in \(\ell^\infty\).

**Operator analogue.** The diagonal operator
\(\operatorname{diag}(1,1/2,1/3,\ldots)\) is positive and injective but is
on the norm boundary of the positive cone.

**Required response.** Use complete infimum/coercivity, not
fixed-index strictness.

## Attack 6: weak topology avoids the uniform floor

**Objection.** Use the product topology for scalar hierarchies or the
strong operator topology for operators.

**Finding.** The positive cone then has empty interior in the relevant
infinite-dimensional setting. The argument is not robust in that
topology.

**Required response.** Present the weak/strong topology dichotomy as a
proposition.

## Attack 7: the false family leaves the certificate domain

**Objection.** A reciprocal transform, form domain, or canonical system
defined for \(\Xi\) may not exist for \(H_t\), \(t<0\).

**Finding.** This is a valid escape hatch and defeats M2 or M4.

**Required response.** The PF, Weil, de Branges, and canonical-system
dossiers are conditional at exactly this point. Do not infer a heat
extension from formal similarity.

## Attack 8: endpoint arithmetic is not transportable

**Objection.** Weil's prime-side form and Li's arithmetic formulas are
specific to zeta; \(H_t\) has no Euler product.

**Finding.** Valid. O-0176 kills only a common sufficient certificate
defined on the false family. Arithmetic identities unique to \(t=0\)
survive.

**Required response.** Label the actual Weil and endpoint Li programs
nonmembers. Kill only the explicitly added heat-deformed common
criterion.

## Attack 9: the same-path sufficiency theorem is absent

**Objection.** Positivity may imply RH only because of zeta-specific
functional identities; it need not imply real-rootedness for \(H_t\).

**Finding.** Valid and load-bearing.

**Required response.** M4 is a separate checklist line in every dossier.
For generalized Laguerre inequalities it follows from a general
real-entire theorem. For Weil and canonical systems it is absent.

## Attack 10: Jensen sufficiency is misstated

**Objection.** The dossier either treats O'Sullivan's zeta criterion as a
general heat-family theorem or burdens the full Pólya--Schur theorem
with genus assumptions that occur only in the paper's proof sketch.

**Finding.** O'Sullivan Theorem 3.1 is stated for a real formal power
series with no genus assumption: all Jensen polynomials hyperbolic is
equivalent to convergence to an entire function in the
Laguerre--Pólya class. Thus its sufficiency direction is general. For
the heat family, the exact transform
\(\Theta_t(w)=H_t(2i\sqrt w)\) has positive Taylor coefficients and
satisfies \(H_t(z)=\Theta_t(-z^2/4)\), so complete Jensen
hyperbolicity implies real-rootedness of the same \(H_t\). The heat
completion still must define one normalized complete family and prove
M2 and M3.

**Required response.** Mark M4 as source-supplied via the full
Pólya--Schur theorem, display the heat transform proving same-object
sufficiency, and keep M1--M3 absent from the actual Jensen papers.

## Attack 11: Li's criterion has no canonical heat normalization

**Objection.** Mapping zeros of \(H_t\) to a Bombieri--Lagarias multiset
requires a fixed affine convention, summability, symmetry, and a
normalization for \(\lambda_n\).

**Finding.** Valid.

**Required response.** The Li dossier is conditional. It becomes an
exact killed completion only after these source hypotheses and uniform
normalization are verified.

## Attack 12: reciprocal-\(\xi\) PF may be singular under heat

**Objection.** The reciprocal transform may fail to exist or deform
regularly for \(H_t\).

**Finding.** Valid.

**Required response.** Existence, integrability, and a uniform minor
modulus are explicit membership premises. Their failure is a surviving
route, not a technicality to suppress.

## Attack 13: Conrey--Li is misclassified

**Objection.** Their de Branges positivity condition is not an open
certificate killed by O-0176; their paper directly shows it fails.

**Finding.** Correct.

**Required response.** Classify it as independently refuted. Use it as
evidence that overstrong positivity had already failed, not as an AIC
member.

## Attack 14: Suzuki's canonical-system parameter is not heat

**Objection.** The parameter \(\omega\) shifts the completed zeta
function and controls a canonical system; it is not \(t\) in the
de Bruijn--Newman flow.

**Finding.** Correct.

**Required response.** The source is a nonmember. Only a new
backward-heat extension with common sufficiency and uniform positivity
would be killed.

## Attack 15: localized Weil continuity looks like a contradiction

**Objection.** Suzuki proves continuity of the lowest localized Weil
eigenvalue and positivity for small support. Why does O-0176 not reject
it?

**Finding.** The varying parameter is support radius, the complete RH
criterion is nondegeneracy for every radius, and no common positive floor
is asserted. Under failure of RH, continuity produces a zero crossing,
which is a boundary mechanism.

**Required response.** Present this as a surviving exemplar.

## Attack 16: Nyman--Beurling is robust Hilbert-space geometry

**Objection.** It is a major Hilbert-space approach and should be covered.

**Finding.** Its terminal condition is membership in a closure, or
distance zero. It is not open acceptance.

**Required response.** Include it as a nonmember control to prevent
“positivity” or “Hilbert space” from becoming proxy membership tests.

## Attack 17: finite computations are overkilled

**Objection.** Fixed degree, rank, support, or verified height often has a
strict margin and continuous deformation.

**Finding.** Such a finite result does not imply real-rootedness of the
entire function, so M4 fails.

**Required response.** State this in the theorem and in every program
section. Farmer's off-axis examples make the Jensen case concrete.

## Attack 18: the formal coercive theorem is vacuous

**Objection.** The previous Lean theorem asked for
\(Q_0(v)\geq\delta\) for every \(v\), impossible at \(v=0\), and used an
unscaled continuity estimate.

**Finding.** Confirmed and repaired.

**Repair.** The synchronized theorem now assumes
\[
Q_0(v)\geq\delta\|v\|^2,\qquad
|Q_t(v)-Q_0(v)|\leq\omega(t)\|v\|^2.
\]
The pinned Lean verifier compiles this nonvacuous statement.

## Attack 19: the formal theorem does not match the manuscript

**Objection.** The old formal theorem encoded balls manually rather than
the paper's open-set and convergence language.

**Finding.** Repaired.

**Repair.** `no_open_certificate_at_accumulation` now takes
`Filter.Tendsto` and `IsOpen` directly, and
`no_certificate_interior_at_accumulation` performs the exact reduction
from an arbitrary accepted set to its interior. The earlier metric
theorem remains as a quantitative specialization. All abstract
accumulating-sequence theorems restrict sufficiency to
`Set.insert target (Set.range counterexample)`, exactly the target and
named counterexamples in manuscript M4.

## Attack 20: “field-redirecting” overstates publication status

**Objection.** No cited paper proves all AIC premises, so the theorem
does not refute the papers.

**Finding.** Correct.

**Required response.** The field redirection is prospective and exact:
uniform interior reserve plus regular false-family transport is not the
missing step in any of these programs. The papers' finite and boundary
theorems remain valid. The title must name “robust interior positivity,”
not “all positivity approaches.”

## Attack 21: Jensen distance is not a signed certificate

**Objection.** A formula such as
\(\operatorname{dist}(p,\{\text{nonhyperbolic polynomials}\})\geq0\)
accepts every polynomial because distance is always nonnegative.

**Finding.** Correct.

**Required response.** Work in a fixed monic coefficient chart, define
the accepted set as the closed hyperbolic set itself, and use distance to
its complement only as the local reserve at an accepted point.

## Attack 22: the Laguerre floor is trivially impossible at \(L_0\)

**Objection.** The complete hierarchy includes
\(L_0(x;F)=F(x)^2\), which vanishes at every real zero. A positive
all-coordinate floor therefore fails before O-0176 is invoked.

**Finding.** Correct.

**Required response.** Omit the automatic \(m=0\) coordinate from the
strict-margin certificate. Its nonnegativity is automatic, and the
general sufficiency theorem remains available once every \(m\geq1\)
coordinate is nonnegative.

## Attack 23: raw total-positivity minors have forced zeros

**Objection.** Translation minors vanish when nodes coalesce, so a
positive floor over all raw node configurations is impossible.

**Finding.** Correct.

**Required response.** Any proposed robust PF completion must specify
separated configurations or a confluent normalization in advance and
must retain every limiting configuration needed for PF sufficiency.
Otherwise M3 fails trivially or M4 is lost.

## Attack 24: fixed-degree Jensen tails are presented as complete

**Objection.** Griffin--Ono--Rolen--Zagier prove fixed-degree,
large-shift hyperbolicity, not a complete uniform interior theorem.

**Finding.** Correct.

**Required response.** Cite their Theorem 1 as a surviving finite-tail
result. Do not infer a complete metric modulus or a positive infimum over
degree and shift.

## Attack 25: the weighted-sup boundary perturbation misses the open ball

**Objection.** Perturbing one coordinate by exactly \(r w_\alpha\)
has weighted-sup norm \(r\), so it does not prove that every open
radius-\(r\) ball leaves the positive cone.

**Finding.** Correct and repaired.

**Repair.** Choose \(q_\alpha/w_\alpha<r/2\) and perturb by
\(-3r w_\alpha/4\). The new coordinate is negative and the perturbation
has norm \(3r/4<r\).

## Attack 26: Bombieri strict positivity is confused with semidefiniteness

**Objection.** Bombieri's Theorem 1 is strict for every nonzero test
function under RH, so calling the whole published criterion merely
semidefinite is inaccurate.

**Finding.** Correct. Pointwise strict positive definiteness still need
not be interior in any fixed complete form norm; the Rayleigh infimum may
be zero.

**Required response.** State Bombieri's pointwise strict theorem and
separate it from the additional coercive estimate
\(Q_0(v)\geq\delta\|v\|^2\) required for M3.

## Attack 27: the weighted cone is not a subset of its stated carrier

**Objection.** The manuscript first defines \(C_+\subseteq\mathbb R^A\)
and then takes its interior in \(\ell^\infty_w(A)\), although an
arbitrary positive family need not belong to that weighted space.

**Finding.** Correct as a notation defect. The intended subset is
\[
C_{+,w}=C_+\cap\ell^\infty_w(A).
\]
The hierarchy corollary must also assume that its certificate values
belong to the weighted carrier.

**Repair.** State the intersection explicitly and include carrier
membership in the corollary hypotheses.

## Attack 28: M1--M4 is not yet a runnable literature test

**Objection.** A definition alone leaves room to classify papers by
analogy or by adjectives such as "stable" and "uniform."

**Finding.** Correct. Field redirection requires a worksheet that forces
the reader to write the exact false family, certificate map, domain,
normalization, topology, convergence estimate, open reserve, and
same-object sufficiency theorem.

**Repair.** Add an outside-reader worksheet with a stop-at-first-failure
rule. No omitted paper receives a verdict merely because it resembles
one of the seven audited architectures.

## Attack 29: the escape-hatch list claims unsupported completeness

**Objection.** A finite list of examples cannot be advertised as every
possible mechanism outside the class.

**Finding.** The premise-level alternatives are exhaustive: a proposed
argument avoids the contradiction only by failing M1, M2, M3, or M4.
The concrete mechanisms observed in the sources are representative
ways M2--M4 fail, not a theorem that no other implementation exists.

**Repair.** Rename the section "Recorded surviving mechanisms" and
state the exhaustive conclusion only at the M1--M4 level.

## Surviving research directions

1. all-index positivity with zero complete infimum;
2. semidefinite forms and explicit null sequences;
3. rank-, height-, support-, or test-dependent margins;
4. heat-singular or domain-changing constructions;
5. arithmetic identities available only for \(\Xi\);
6. certificates that quantitatively separate every explicit false family;
7. exact closure/density criteria;
8. positive-time de Bruijn--Newman bounds;
9. finite theorems stated only at finite scope.

## Referee threshold

The manuscript is not acceptable until:

- every bibliography claim has a primary-source theorem/page anchor;
- actual-paper and robust-completion verdicts are visibly separate;
- no table row silently assumes a heat extension;
- the topology dichotomies are proved;
- the Lean theorem and coercive corollary compile;
- an independent build verifies all source keys and dossier fields; and
- the title, abstract, and conclusion state the surviving boundary
  programs with the same prominence as the conditionally ruled-out
  completions.
