# Primary-source membership dossiers

Date: 2026-07-19

## Scope

This audit covers every published certificate architecture named in
O-0176:

1. de Bruijn--Newman heat flow and universal factors;
2. Jensen-polynomial hyperbolicity;
3. generalized Laguerre and Turan inequalities;
4. Weil and Bombieri quadratic forms;
5. de Branges-space positivity;
6. canonical-system Hamiltonians; and
7. Polya-frequency and total-positivity criteria.

Li coefficients and Nyman--Beurling are included as boundary controls.
The source PDFs, extraction/OCR aids, hashes, and theorem anchors are
pinned in `source_manifest.json` and `sources/SHA256SUMS`.

These seven dossiers are architecture representatives, not a claim that
the bibliography exhausts every RH equivalence. An omitted paper
receives no verdict by analogy: an outside reader must run M1--M4
against its own displayed certificate, topology, accepted set, and
sufficiency theorem.

## Exact class

Let \(x_0\) be the target, let \(x_n\) be an explicitly named bad
sequence, let \(D\) map the target and counterexamples into a fixed
topological certificate space \(Y\), and let \(U\subseteq Y\) be
accepted. An argument is in the accumulating-counterexample interior
certificate class only when all four premises hold:

- **M1, false family:** every \(x_n\), \(n\geq1\), is proved bad.
- **M2, certificate accumulation:** \(D(x_n)\to D(x_0)\) in a topology
  fixed before the terminal estimate.
- **M3, open acceptance:** \(U\) is open and \(D(x_0)\in U\).
- **M4, common sufficiency:** for every
  \(x\in\{x_0,x_1,\ldots\}\), \(D(x)\in U\) implies that \(x\) is good.

The theorem says that M1--M4 are inconsistent. The RH specialization is
\[
x_0=H_0,\qquad x_n=H_{-1/n},
\]
with goodness equal to real-rootedness. Rodgers--Tao prove every
\(H_{-1/n}\) has a nonreal zero.

## Verdict discipline

- **Theorem input, not a member:** the source supplies the false family
  or endpoint criterion but no AIC terminal certificate.
- **Actual-paper nonmember:** at least one of M1--M4 is absent.
- **Fails at target:** the source itself shows its proposed positivity
  condition is false for the zeta-associated object.
- **Conditionally ruled out completion:** a future argument is ruled out
  only after every listed admission clause is proved.
- **Outside-class control:** the terminal criterion is intrinsically a
  closure or boundary statement.

No cited paper is classified as an actual AIC member.

## Source summary

| Program | M1 supplied by source | M2 | M3 | M4 supplied by source | Actual-source verdict |
|---|---:|---:|---:|---:|---|
| de Bruijn--Newman heat | yes | no certificate topology | no | no | theorem input, not a member |
| Jensen polynomials | no | no complete modulus | no | general theorem | nonmember |
| generalized Laguerre--Turan | no | no complete modulus | no | general theorem | nonmember |
| Weil--Bombieri forms | no | no heat form | no norm-open reserve | endpoint only | nonmember |
| de Branges positivity | no | no | fails at target | target architecture only | refuted in source |
| canonical systems | no | \(\omega\) is not heat time | semidefinite | no heat theorem | nonmember |
| Polya frequency | no | no reciprocal-heat modulus | no | conditional general theorem | nonmember |

## Dossier 1: de Bruijn--Newman heat flow

### Primary anchors

- de Bruijn, *The roots of trigonometric integrals*, Theorems 13--14,
  journal pp. 204--205.
- Rodgers--Tao, *The de Bruijn--Newman constant is non-negative*,
  threshold discussion on PDF p. 2 and Theorem 1 on PDF p. 3.

### Source statement

The heat family is
\[
H_t(z)=\int_0^\infty e^{t u^2}\Phi(u)\cos(zu)\,du.
\]
There is a finite \(\Lambda\) for which \(H_t\) is real-rooted exactly
when \(t\geq\Lambda\). Rodgers--Tao prove \(\Lambda\geq0\). De Bruijn's
Theorem 13 supplies the classical Gaussian strip-contraction mechanism,
and Theorem 14 places it in the universal-factor framework.

### Actual-source M1--M4

- **M1: supplied.** \(H_{-1/n}\) is bad for every \(n\geq1\).
- **M2: not supplied for an argument-chosen certificate.** Analytic
  convergence of \(H_t\) does not imply convergence of an arbitrary
  \(D(H_t)\).
- **M3: not supplied.** Neither source proposes an open endpoint
  certificate accepting \(H_0\).
- **M4: not supplied.** It must come from the proposed certificate.

### Verdict

The papers are theorem inputs, not AIC members.

### Exact admission clauses for a future completion

1. define one certificate \(D\), carrier, normalization, and topology
   on \(H_0,H_{-1},H_{-1/2},\ldots\);
2. prove \(D(H_{-1/n})\to D(H_0)\);
3. prove that the accepted set is open at \(D(H_0)\); and
4. prove acceptance implies real-rootedness of \(H_0\) and every
   \(H_{-1/n}\).

If all four are present, O-0176 rules out the completion.

### Surviving mechanisms

Positive-time bounds, exact endpoint identities, one-sided limits with
no negative interval, and backward-heat-singular certificates.

## Dossier 2: Jensen-polynomial hyperbolicity

### Primary anchors

- O'Sullivan, *Zeros of Jensen polynomials and asymptotics for the
  Riemann xi function*, Theorem 1.1 on PDF p. 2 and Polya--Schur
  Theorem 3.1 on PDF p. 6.
- Griffin--Ono--Rolen--Zagier, *Jensen polynomials for the Riemann zeta
  function and other sequences*, Theorem 1 on PDF p. 2.
- Farmer, *Jensen polynomials are not a plausible route to proving the
  Riemann hypothesis*, hostile fixed-degree controls.

### Source statement

O'Sullivan's Theorem 1.1 states
\[
\mathrm{RH}
\quad\Longleftrightarrow\quad
J^{d,n}\text{ is hyperbolic for every }d\geq1,\ n\geq0.
\]
Theorem 3.1 gives the general Polya--Schur sufficiency theorem.
Griffin--Ono--Rolen--Zagier prove that for each fixed \(d\),
\(J^{d,n}\) is hyperbolic for all sufficiently large \(n\). Farmer
shows why such fixed-degree tails can lose the zero information needed
for RH.

For the same heat objects, set
\[
\Theta_t(w)=H_t(2i\sqrt w).
\]
Evenness of \(H_t\) makes this a single-valued entire function, and its
Taylor coefficients are positive for \(t\leq0\) because the displayed
Riemann kernel is positive on \([0,\infty)\). The identity
\[
H_t(z)=\Theta_t(-z^2/4)
\]
then shows that complete Jensen hyperbolicity for \(\Theta_t\) implies
real-rootedness of the same \(H_t\): Polya--Schur puts \(\Theta_t\) in
the Laguerre--Polya class, positive coefficients exclude positive real
zeros, and every remaining zero pulls back to a real zero of \(H_t\).
Thus M4 is genuinely same-family sufficiency rather than a
zeta-endpoint implication.

### Candidate certificate

Fix, before the endpoint estimate, a monic normalization
\[
N_{d,n}J_F^{d,n}\in\mathcal P_d
\]
that preserves hyperbolicity, and a complete coefficient metric. Let
\(\mathcal H_d\subset\mathcal P_d\) be the closed hyperbolic set.
For an accepted polynomial, the reserve is
\[
r_{d,n}(F)=
\operatorname{dist}\!\left(
N_{d,n}J_F^{d,n},
\mathcal P_d\setminus\mathcal H_d
\right).
\]
This is a distance-to-complement reserve on accepted points, not a
globally signed certificate.

### Actual-source M1--M4

- **M1: not supplied.** The sources do not formulate the criterion on
  the canonical negative-heat sequence.
- **M2: not supplied.** Fixed-\((d,n)\) coefficient continuity is not a
  complete uniform modulus.
- **M3: not supplied.** It would require a positive infimum of all
  normalized distances.
- **M4: supplied for the complete family.** Polya--Schur gives a general
  real-entire sufficiency theorem; the \(n=0\) subfamily already forces
  Laguerre--Polya membership.

### Verdict

All cited Jensen papers are nonmembers. Fixed-degree, large-shift
results are tail results, not open complete acceptance.

### Exact admission clauses for a future completion

1. use the same complete Jensen construction on \(H_0\) and every
   \(H_{-1/n}\);
2. fix all normalizations and the complete metric in advance;
3. prove convergence in that metric, not just coordinatewise; and
4. prove a positive uniform distance from every nonhyperbolic
   complement.

If all four are present, O-0176 rules out the completion.

### Surviving mechanisms

Fixed degree, finite index ranges, and complete hyperbolicity whose
normalized distance infimum is zero.

## Dossier 3: generalized Laguerre and Turan inequalities

### Primary anchors

- Csordas--Norfolk--Varga, *The Riemann hypothesis and the Turan
  inequalities*, introduction on journal p. 522 and Theorem 2.5 on
  journal p. 528.
- Csordas--Vishnyakova, *The generalized Laguerre inequalities and
  functions in the Laguerre--Polya class*, Remarks 1.3 on pp.
  1644--1645 and Theorem 2.3 on pp. 1645--1647.

### Source statement

The generalized expressions are
\[
L_m(x;F)=
\sum_{j=0}^{2m}
\frac{(-1)^{j+m}}{(2m)!}\binom{2m}{j}
F^{(j)}(x)F^{(2m-j)}(x).
\]
Theorem 2.3 proves that \(L_m(x;F)\geq0\) for every \(m\geq0\) and
\(x\in\mathbb R\) forces a nonzero real entire \(F\) into the
Laguerre--Polya class. The 1986 paper proves the classical zeta Turan
inequalities and explicitly treats them as necessary, not sufficient,
conditions.

### Candidate certificate

The automatic coordinate \(L_0(x;F)=F(x)^2\) is nonnegative and
vanishes at real zeros, so it cannot carry a strict margin. Use
\(m\geq1\), fix weights \(w_{m,x}>0\) in advance, and equip
\[
\left(L_m(x;F)/w_{m,x}\right)_{m\geq1,\ x\in\mathbb R}
\]
with a complete weighted sup metric.

### Actual-source M1--M4

- **M1: not supplied.**
- **M2: not supplied.** Derivative continuity at each fixed \((m,x)\)
  is not a uniform all-order/all-point estimate.
- **M3: not supplied.** It would require a positive normalized complete
  floor.
- **M4: supplied.** Theorem 2.3 is a general real-entire sufficiency
  theorem.

### Verdict

The published all-order characterization and classical Turan theorem
are nonmembers.

### Exact admission clauses for a future completion

1. evaluate the same \(m\geq1\) hierarchy on the canonical heat family;
2. fix weights and topology before inspecting \(H_0\);
3. prove a uniform backward-heat modulus over \(m\) and \(x\); and
4. prove a positive normalized complete floor at \(H_0\).

If all four are present, O-0176 rules out the completion.

### Surviving mechanisms

The classical Turan inequalities, fixed order, compact \(x\)-ranges,
pointwise strictness, and complete zero-infimum hierarchies.

## Dossier 4: Weil and Bombieri quadratic forms

### Primary anchors

- Bombieri, *Remarks on Weil's quadratic functional in the theory of
  prime numbers, I*, abstract and Theorem 1 on journal pp. 183 and
  191--192.
- Suzuki, *Weil's quadratic form via the screw function*, Weil
  criterion on PDF p. 1 and Theorem 1.3 on PDF p. 5.

### Source statement

The global criterion is
\[
Q_W(v)\geq0\quad\text{for every }v\in C_c^\infty(\mathbb R)
\quad\Longleftrightarrow\quad\mathrm{RH}.
\]
Bombieri's Theorem 1 is pointwise strict for every nonzero test
function under RH, but it supplies no positive lower Rayleigh bound in
a fixed complete norm. Bombieri also studies support-localized
variational problems and finite matrix truncations. Suzuki proves
continuity of the lowest localized eigenvalue \(\lambda_a\). Under
failure of RH, \(\lambda_a\) becomes negative for some support radius
while remaining positive for small radius, so continuity produces a
zero crossing.

### Candidate certificate

A robust completion would require forms \(Q_t\) on one fixed normed form
domain \(V\), with
\[
Q_0(v)\geq\delta\|v\|_V^2,\qquad
|Q_t(v)-Q_0(v)|\leq\omega(t)\|v\|_V^2,\qquad
\omega(t)\to0.
\]

### Actual-source M1--M4

- **M1: not supplied.**
- **M2: not supplied.** No heat-deformed \(Q_t\) on a common domain is
  constructed.
- **M3: not supplied.** Pointwise strict positivity and semidefinite
  positivity do not give norm interior; localized floors depend on
  support and can cross zero.
- **M4: endpoint only.** The arithmetic form is a zeta criterion, not a
  common real-rootedness theorem for \(H_t\).

### Verdict

The actual Weil, Bombieri, localized, and screw-function programs are
nonmembers.

### Exact admission clauses for a future completion

1. construct \(Q_t\) for \(H_0\) and \(H_{-1/n}\) on one fixed domain;
2. prove positivity of \(Q_t\) implies real-rootedness of the same
   \(H_t\);
3. prove form-norm convergence to \(Q_0\); and
4. prove a common coercive floor at \(t=0\).

If all four are present, the formal coercive corollary of O-0176 rules
out the completion.

### Surviving mechanisms

Pointwise strict forms with zero Rayleigh infimum, semidefinite forms,
null sequences, support-dependent eigenvalues, zero crossings, changing
domains, distributional limits, and arithmetic identities available
only at \(t=0\).

## Dossier 5: de Branges-space positivity

### Primary anchors

- Conrey--Li, *A note on some positivity conditions related to zeta- and
  L-functions*, Theorem 1 on journal p. 930 and equation (3.2) with its
  conclusion on journal p. 934.

### Source statement

Theorem 1 gives hypotheses under which
\[
\operatorname{Re}\langle F(z+i),F(z)\rangle_{H(E)}\geq0
\]
for every admissible \(F\) forces the zeros of \(E\) onto the target
line. The paper then tests \(E(z)=\xi(1-iz)\), obtains a negative value
in equation (3.2), and concludes that the zeta-associated space does
not satisfy condition (3.1).

### Actual-source M1--M4

- **M1: not supplied.**
- **M2: not supplied.**
- **M3: fails at target.**
- **M4: not a heat-family theorem.**

### Verdict

The proposed positivity condition is refuted by its own source,
independently of O-0176. It must not be described as an AIC member.

### Exact admission clauses for a replacement

1. replace the failed condition by one true at \(H_0\);
2. define it on every \(H_{-1/n}\) using one common domain;
3. prove same-object real-rootedness sufficiency;
4. prove form-norm accumulation; and
5. prove a coercive open reserve.

Only such a replacement is conditionally ruled out.

### Surviving mechanisms

Different de Branges spaces, semidefinite kernels, changing domains,
and singular endpoint constructions.

## Dossier 6: canonical-system Hamiltonians

### Primary anchors

- Suzuki, *A canonical system of differential equations arising from
  the Riemann zeta-function*, Proposition 1.1 on PDF p. 1, Theorem 2.3,
  and the extension discussion on PDF pp. 23--24.

### Source statement

Proposition 1.1 uses shifted functions \(A^\omega\) and states that RH is
equivalent to real-rootedness of \(A^\omega\) for every \(\omega>0\).
Theorem 2.3 constructs the associated canonical system
unconditionally for \(\omega>1\). Extending the construction to all
\(\omega>0\) would produce a criterion through
positive-semidefinite Hamiltonians.

### Actual-source M1--M4

- **M1: not supplied.**
- **M2: not supplied.** The shift \(\omega\) is not
  de Bruijn--Newman heat time.
- **M3: not supplied.** The source uses positive semidefiniteness, not a
  common uniform positive-definiteness reserve.
- **M4: not supplied for heat-deformed objects.**

### Verdict

The published canonical-system program is a nonmember.

### Exact admission clauses for a future completion

1. construct canonical systems for \(H_0\) and every \(H_{-1/n}\);
2. use one common measurable domain and normalization;
3. prove Hamiltonian convergence in a positivity-preserving topology;
4. prove uniform positive definiteness; and
5. prove same-object real-rootedness sufficiency.

Only that new heat completion is ruled out.

### Surviving mechanisms

Positive-semidefinite Hamiltonians, \(\omega\)-dependent singularities,
changing domains, and constructions with no heat extension.

## Dossier 7: Polya-frequency and total positivity

### Primary anchors

- Gröchenig, *Schoenberg's theory of totally positive functions and the
  Riemann zeta function*, Theorems 3--4 on PDF p. 4 and the hostile
  feasibility discussion on PDF p. 5.

### Source statement

Theorem 4 states that RH holds exactly when
\[
\Lambda(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
\frac{e^{-ix\tau}}{\xi(1/2+\tau)}\,d\tau
\]
is a Polya-frequency function. The paper explicitly notes that even
order-one positivity is not known.

### Candidate certificate

Raw translation minors vanish when nodes coalesce. Therefore a positive
floor over all raw configurations is impossible. A robust candidate
must fix in advance:

1. a reciprocal transform for each object;
2. separated node configurations or a confluent normalization;
3. a complete topology controlling every normalized minor; and
4. the limiting configurations required by PF sufficiency.

### Actual-source M1--M4

- **M1: not supplied.**
- **M2: not supplied.** Reciprocal transforms are not defined and proved
  regular on \(H_{-1/n}\).
- **M3: not supplied.**
- **M4: conditional general theorem.** Schoenberg supplies sufficiency
  only after the transform, integrability, and analytic hypotheses are
  verified for each family member.

### Verdict

The published equivalence is a boundary all-minor criterion and is not
an AIC member.

### Exact admission clauses for a future completion

1. define source-compatible reciprocal transforms throughout the
   canonical false family;
2. verify Schoenberg's hypotheses for every member;
3. fix a complete nondegenerate minor normalization and topology;
4. prove certificate accumulation; and
5. prove open PF acceptance.

Only that regularized completion is ruled out.

### Surviving mechanisms

Fixed PF order, zero complete reserve, coalescing-node boundary behavior,
and singular or nonexistent reciprocal transforms.

## Boundary control 1: Li coefficients

### Primary anchors

- Bombieri--Lagarias, *Complements to Li's criterion for the Riemann
  hypothesis*, Theorem 1 on journal p. 276 and Corollary 1 on pp.
  277--278.

Under explicit summability and symmetry hypotheses, critical-line
location is equivalent to nonnegativity of every generalized Li sum
\[
\lambda_n(R)=
\sum_{\rho\in R}
\left[1-\left(1-\frac1\rho\right)^n\right].
\]
The source supplies no canonical heat deformation of \(R\), no fixed
normalization yielding an open reserve, and no uniform coefficient
modulus. The actual paper is a nonmember.

A source-matched false multiset family with all source hypotheses,
fixed normalization, uniform accumulation, positive complete floor, and
common sufficiency would satisfy AIC and be ruled out. Endpoint
arithmetic formulas, raw all-index nonnegativity, and zero complete
infimum survive.

## Boundary control 2: Nyman--Beurling

### Primary anchor

- Baez-Duarte, *A strengthening of the Nyman--Beurling criterion for the
  Riemann hypothesis*, Theorem 1.1 on PDF p. 1.

The theorem states RH as membership of a fixed vector in the closure of
the natural Beurling span. The terminal condition is distance zero to a
closure, not membership in an open positive set. It is an
outside-class control, and the published boundary criterion survives
unchanged.

## Cross-family topology conclusions

For an infinite scalar hierarchy:

- in the product topology, the nonnegative cone has empty interior;
- in a weighted sup topology, its interior is exactly the positive
  complete-floor set.

For bounded self-adjoint operators:

- in operator norm, positive interior is exactly uniform coercivity;
- in strong or weak operator topology on an infinite-dimensional space,
  the positive cone has empty interior.

Thus weak topologies do not provide robust acceptance. Strong
topologies do so exactly through the reserve forbidden by O-0176 when
M1, M2, and M4 also hold.

## Audit conclusion

No published paper in the named families is an AIC member. The
field-level conclusion is prospective:

> A uniform interior reserve with regular transport along an explicit
> false family and common sufficiency is not the missing step.

The viable directions are boundary, zero-infimum, semidefinite,
singular, changing-domain, endpoint-arithmetic, or quantitatively
counterexample-separating.
