---
id: O-0176
type: obstruction
title: "Accumulating false-RH certificates cannot contain Xi in their interior"
status: binding
family: proof-system
statement: >
  Let x_0 be the target and x_n an explicit sequence of bad objects. Let
  D map exactly the target and named counterexamples into a topological
  certificate space Y, and let U be an open accepted set. If
  D(x_n) converges to D(x_0), while acceptance implies goodness for
  every object in {x_0,x_1,...}, then D(x_0) is not in U. For RH, take
  x_0=H_0=Xi and x_n=H_{-1/n}; Rodgers--Tao prove every x_n is not
  real-rooted.
scope: >
  An outside reader admits an argument to the accumulating-counterexample
  interior-certificate class only after verifying all four clauses:
  (M1) every named counterexample is proved bad; (M2) its certificate
  converges to the target certificate in a topology and normalization
  fixed before the terminal estimate; (M3) the target certificate lies
  in an open accepted set; and (M4) the same acceptance implication is
  sufficient for the target and every named counterexample. The
  sufficiency domain is exactly the target plus the named sequence, not
  an unspecified ambient class. No cited paper currently supplies all
  four clauses. The obstruction applies only to future completions that
  do.
witness: >
  Since U is open and D(x_n) converges to D(x_0) in U, eventually
  D(x_n) is in U. Common sufficiency then makes x_n good, contradicting
  M1. Metric open-ball, uniform hierarchy, norm-coercive form/operator,
  and finite strict-test corollaries are formal specializations.
verifier: canon/witnesses/O-0176/verify.py
depends: []
supersedes: [O-0113]
sources:
  - archive/notes/530-debruijn-newman-backward-heat-openness-obstruction.md
  - literature/2026-07-19-uniform-interior-certificate-barrier-primary-sources.md
tags: [proof-system, de-bruijn-newman, argument-class, accumulating-counterfeit, open-certificate, uniform-margin, coercivity, natural-barrier]
aliases: [backward-open certificate barrier, robust positivity barrier, heat-stable reserve obstruction, accumulating-counterexample interior barrier]
gates: [counterfeit, same-space, scaling]
---

# Accumulating false-RH certificates cannot contain Xi in their interior

## Exact theorem

Let \(x_0\) be a target, let \(x_n\) be an explicitly named sequence,
and let `Good` be the desired property. An argument belongs to the
**accumulating-counterexample interior-certificate class** (AIC) only if
an outside reader can verify:

1. **M1, false family.** \(\neg\operatorname{Good}(x_n)\) for every
   \(n\geq 1\).
2. **M2, certificate accumulation.** A map \(D\) into one topological
   space \(Y\) satisfies \(D(x_n)\to D(x_0)\). Its carrier,
   normalization, and topology are fixed before the terminal estimate.
3. **M3, open acceptance.** There is an open set \(U\subseteq Y\) with
   \(D(x_0)\in U\).
4. **M4, common sufficiency.** For every
   \(x\in\{x_0,x_1,\ldots\}\), \(D(x)\in U\) implies
   \(\operatorname{Good}(x)\).

**Theorem.** M1--M4 are inconsistent.

The proof is the defining property of convergence: openness of \(U\)
and \(D(x_0)\in U\) imply \(D(x_n)\in U\) eventually. M4 then makes
that \(x_n\) good, contradicting M1.

The formal scope of M4 is exactly

```text
Set.insert target (Set.range counterexample)
```

and not all objects in an ambient function space.

## RH specialization

Set

\[
H_t(z)=\int_0^\infty e^{t u^2}\Phi(u)\cos(zu)\,du,\qquad
x_0=H_0=\Xi,\qquad x_n=H_{-1/n}.
\]

Rodgers--Tao prove the de Bruijn--Newman constant is nonnegative.
Consequently every \(H_{-1/n}\) has a nonreal zero, supplying M1.
Any proposed certificate must still supply M2, M3, and M4; analytic
convergence of the heat family does not automatically transport an
argument-chosen certificate.

The metric form is immediate: if
\(B_d(D(x_0),r)\subseteq U\) and
\(d(D(x_n),D(x_0))\to0\), some bad \(x_n\) enters the ball.
This strictly contains the heat-specific metric obstruction O-0113
because the sequence and topology are abstract.

## Quantitative specializations

- For a complete scalar hierarchy in a weighted sup norm, interior of
  the positive cone intersected with the weighted carrier is equivalent
  to a positive normalized complete infimum. Coordinatewise strict
  positivity is not enough.
- In the product topology, the positive cone of an infinite hierarchy
  has empty interior.
- In operator norm, interior of the positive self-adjoint cone is the
  uniformly coercive cone \(A\succeq\delta I\). In the strong and weak
  operator topologies on an infinite-dimensional Hilbert space, that
  cone has empty interior.
- A common-domain form estimate
  \(Q_0(v)\geq\delta\lVert v\rVert^2\), together with
  \(\lvert Q_n(v)-Q_0(v)\rvert\leq\omega_n\lVert v\rVert^2\) and
  \(\omega_n\to0\), would persist to a bad counterexample and is
  impossible when positivity is sufficient there.
- A finite family of continuous strict sufficient tests is an open
  certificate and is therefore impossible on an accumulating bad
  sequence.

## Published-program membership audit

No cited paper is an AIC member. The primary sources provide endpoint
criteria, false-family input, finite theorems, or boundary
characterizations. The table records what a **future completion** must
add before O-0176 applies.

An outside reader runs the test in this order:

1. name the target, desired property, and a proved bad sequence;
2. write the certificate map, domain, indices, normalization, carrier,
   and topology;
3. prove convergence in that exact topology;
4. exhibit an open accepted neighborhood or metric reserve at the
   target;
5. prove the same acceptance implication for the target and every
   named counterexample; and
6. stop at the first failed item and record that precise escape hatch.

If every item is present, the proposed completion is inconsistent. No
paper is admitted by resemblance or by words such as "stable,"
"strict," or "uniform."

| Program | What the primary source supplies | Missing admission clauses | Actual-paper verdict |
|---|---|---|---|
| de Bruijn--Newman heat flow and universal factors | Canonical heat path and M1 | Certificate topology and accumulation, open endpoint acceptance, common sufficiency | Theorem input, not a member |
| Jensen-polynomial hyperbolicity | General complete-family M4 via Polya--Schur; fixed-degree tails | Heat-family construction, fixed complete normalization, complete-metric convergence, positive complete reserve | Nonmember |
| Generalized Laguerre and Turan inequalities | General complete-hierarchy M4; classical necessary inequalities | Heat-family hierarchy, fixed weights, all-order/all-point modulus, positive complete floor | Nonmember |
| Weil and Bombieri quadratic forms | Endpoint criterion and pointwise strict positivity under RH | Common heat forms/domain, same-object M4, form-norm convergence, coercive floor | Nonmember |
| de Branges-space positivity | A sufficient zero-line condition which the source shows fails for zeta | A replacement true at the target, common heat domain, same-object M4, accumulation and open reserve | Fails at target |
| Canonical-system Hamiltonians | An arithmetic \(\omega\)-family and semidefinite criterion | Heat extension, common domain/normalization, positivity-preserving convergence, uniform definiteness and M4 | Nonmember |
| Polya-frequency and total positivity | Conditional reciprocal-transform criterion | Heat-family transforms, common Schoenberg hypotheses, nondegenerate complete minor topology, convergence and open reserve | Nonmember |

For each row, a proposed completion is ruled out only if every missing
clause is proved. Calling a method "positive," "stable," or "uniform"
does not establish membership.

The seven rows are architecture representatives, not an exhaustive
bibliography of RH equivalences. An omitted paper receives no verdict
by analogy. The abstract theorem covers it only if an outside reader can
verify M1--M4 from that paper's actual statements and any explicitly
cited inputs.

Two controls sharpen the boundary:

- **Li--Keiper coefficients** are not admitted without a heat-deformed
  multiset satisfying the Bombieri--Lagarias hypotheses, one fixed
  normalization, complete convergence, and a positive complete floor.
- **Nyman--Beurling** is intrinsically a closure/distance-zero
  criterion, so its published formulation is outside the open-acceptance
  class.

The frozen source-by-source verdicts and admission clauses are in
`canon/witnesses/O-0176/membership.json`.

## Hostile checks

At the premise level the alternatives are exhaustive: an argument
escapes only by failing M1, M2, M3, or M4. The following are recurrent
concrete escape hatches in the audited sources:

- the proposed certificate is undefined or discontinuous on the named
  counterexamples;
- its space, domain, or normalization changes along the sequence;
- sufficiency uses arithmetic identities available only at \(H_0\);
- the terminal statement is semidefinite, closure-based, or otherwise
  on the boundary rather than in an open set;
- all coordinates are positive but their normalized infimum is zero;
- margins depend on degree, order, rank, support, height, point, or test;
- a finite result lacks sufficiency for the entire function;
- the certificate quantitatively separates the target from every known
  bad family.

Three recurring false admissions are explicitly rejected:

1. Distance to the complement of the hyperbolic set is a local Jensen
   reserve only after hyperbolicity is known; unsigned distance is not a
   globally signed criterion.
2. The automatic Laguerre coordinate \(L_0=F^2\) vanishes at real zeros
   and cannot participate in a strict complete floor; strict-margin
   tests begin at \(m\geq1\).
3. Raw total-positivity minors vanish at coalescing nodes; any robust
   floor must specify separated configurations or a confluent
   normalization without losing sufficiency.

## What is redirected

O-0176 does not invalidate any cited paper. It excludes one prospective
proof move across seven published architectures: transport the complete
criterion regularly to the canonical negative-heat sequence and then
prove that the endpoint lies a positive uniform distance inside the
sufficient region. If the transport and common sufficiency are real,
the reserve cannot be.

Surviving work must therefore exploit a sharp boundary, a zero complete
infimum, singular or nonuniform heat behavior, a changing domain, an
endpoint-only arithmetic identity, quantitative separation from the
false sequence, or a finite result honestly kept finite.

```text
exact_map: D(x) in U -> Good(x), evaluated only on the target and named counterexamples
counterfeit: an explicit bad sequence x_n with D(x_n) -> D(x_0); canonically H_{-1/n} -> H_0
counterfeit_failure: openness eventually accepts a proved bad member
success_delta: a topology-level obstruction with exact source-specific admission tests for seven robust-positivity architectures
failure_delta: a row missing M1, M2, M3, or M4 remains a nonmember and is not rhetorically absorbed
stop_condition: topology and normalization predate the endpoint estimate, and M4 holds on the exact target-plus-sequence scope
```

## Witness and provenance

Run:

```bash
python3 canon/witnesses/O-0176/verify.py
```

The verifier compiles the synchronized Lean theorem, checks the exact
M4 scope, validates seven program dossiers and two controls against the
manuscript route, and rejects any actual-paper membership claim.

O-0176 strengthens and supersedes O-0113 and archive Note 530. The
primary-source audit was completed on 2026-07-19. Formal proof:
`canon/witnesses/O-0176/HeatBoundary.lean`.
