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\title{\vspace{-2.5em}
  {\color{ink}\bfseries Interior barriers for robust positivity programs\\
  toward the Riemann hypothesis}}
\author{Author information withheld until publication}
\date{Submission-neutral version, July 19, 2026}

\begin{document}
\maketitle
\thispagestyle{titlepage}

\begin{abstract}
Rodgers and Tao's theorem that the de Bruijn--Newman constant is
nonnegative supplies an explicit sequence of real entire functions
\(H_{-1/n}\), each with a nonreal zero, converging analytically to the
Riemann function \(H_0\).  We isolate the exact topological consequence:
if a certificate map sends those counterexamples to the target
certificate, if its accepted region is open at the target, and if the
same acceptance implication forces real-rootedness on the target and on
every counterexample, then the target cannot be accepted.  The proof is
elementary; the substance is the operational membership test and its
application to complete positivity architectures.

We prove the corresponding cone-interior dichotomies for infinite scalar
hierarchies and self-adjoint operators.  Weak coordinate topologies have
empty positive interior, whereas weighted sup and operator-norm
topologies have interior exactly at a uniform positive floor or a
coercive bound.  We then audit seven primary-source programs: de
Bruijn--Newman heat flow, Jensen polynomials, generalized Laguerre and
Turan inequalities, Weil forms, de Branges spaces, canonical systems,
and Polya-frequency criteria.  The Li and Nyman--Beurling criteria serve
as boundary controls.  None of the cited papers is itself a member
of the obstruction class.  What is excluded is a precisely specified
future completion combining a fixed topology, regular transport along
an explicit false family, an open uniform reserve, and common
sufficiency.  Semidefinite, zero-infimum, singular, changing-domain,
endpoint-arithmetic, and exact boundary mechanisms remain outside the
barrier.  The abstract theorem and its coercive and finite-test
specializations are checked in Lean.
\end{abstract}

\noindent\textbf{Keywords.}
Riemann hypothesis; de Bruijn--Newman constant; positivity criteria;
Laguerre--Polya class; Weil quadratic form; total positivity; natural
barriers; formal verification.

\noindent\textbf{MSC 2020.}
11M26; 30D15; 46C05; 47A63.

\section{Introduction}

Many criteria equivalent to the Riemann hypothesis are positivity
statements: hyperbolicity of every Jensen polynomial, nonnegativity of
every generalized Laguerre expression, positivity of the Weil
quadratic form, positive-semidefinite canonical Hamiltonians, or total
positivity of a reciprocal transform.  Their finite or localized
instances often admit strict estimates.  It is therefore natural to ask
whether a complete proof can be obtained by making those estimates
uniform and stable.

This paper identifies a precise obstruction to that final move.  It does
\emph{not} say that positivity methods are impossible, nor that the
papers audited below made an invalid argument.  It says that a complete
certificate cannot simultaneously have all four properties:

\begin{enumerate}
\item an explicit sequence of false objects approaching the target;
\item convergence of their certificate values in one fixed topology;
\item an open accepted neighborhood at the target; and
\item one acceptance theorem valid for both target and counterexamples.
\end{enumerate}

The topological proof is one line.  The RH-specific force comes from a
canonical, unconditional false sequence.  Writing
\begin{equation}\label{eq:heat-family}
 H_t(z)=\int_0^\infty e^{t u^2}\Phi(u)\cos(zu)\,du,
\end{equation}
we normalize the real Riemann function as \(\Xi(z)=H_0(z)\).
There is a finite de Bruijn--Newman constant \(\Lambda\) such that
\(H_t\) has only real zeros exactly when \(t\geq\Lambda\).
Rodgers and Tao proved \(\Lambda\geq0\)
\cite[Theorem 1]{RodgersTao2020}.  Consequently,
\begin{equation}\label{eq:false-family}
 F_0=H_0,\qquad F_n=H_{-1/n}\quad(n\geq1)
\end{equation}
is an explicit sequence in which every \(F_n\), \(n\geq1\), has a
nonreal zero.

The contribution has three parts.  First, Section~\ref{sec:abstract}
states the exact obstruction in topological and metric form.  Second,
Section~\ref{sec:cones} identifies what ``open positivity'' means for
the complete scalar and operator cones that occur in RH programs.
Third, Sections~\ref{sec:protocol}--\ref{sec:dossiers} give a
primary-source membership protocol and apply it program by program.
The audit deliberately separates an actual paper's theorem from a
hypothetical robust completion of its architecture.

The conclusion is prospective and falsifiable.  In the audited
architectures, a successful proof must do at least one of the following:
remain on a sharp boundary, lose uniform regularity along the canonical
false family, use a target-specific mechanism unavailable on that
family, change domain or normalization essentially, or quantitatively
separate the false family from \(\Xi\).  Seeking a uniform interior
reserve while preserving common backward-heat regularity is not the
missing step.

\section{The accumulating-counterexample barrier}
\label{sec:abstract}

Let \(X\) be a class of objects, \(\good\subseteq X\) the desired
property, \(x_0\in X\) the target, and \(x_n\in X\) explicit
counterexamples.  A certificate is a map \(\cert:X\to Y\) into a
topological space \(Y\), together with an accepted set \(U\subseteq Y\).
Only the restriction of these data to
\[
 \mathcal S=\{x_0\}\cup\{x_n:n\geq1\}
\]
matters.

\begin{definition}[AIC membership]\label{def:aic}
An argument is an \emph{accumulating-counterexample interior
certificate} argument, abbreviated \(\AIC\), if an outside reader can
verify:
\begin{description}
\item[M1: explicit false family.] \(x_n\notin\good\) for every \(n\geq1\).
\item[M2: certificate accumulation.]
  \(\cert(x_n)\to\cert(x_0)\) in \(Y\).
\item[M3: open acceptance.]
  \(U\) is open and \(\cert(x_0)\in U\).
\item[M4: common sufficiency.]
  For every \(x\in\mathcal S\), \(\cert(x)\in U\) implies
  \(x\in\good\).
\end{description}
The carrier \(Y\), topology, normalization, index set, and map
\(\cert\) must be fixed as part of the argument before the terminal
estimate at \(x_0\).  An endpoint-tailored or discrete topology is a
nonmember rather than a counterexample to the theorem.
\end{definition}

\begin{theorem}[Interior barrier]\label{thm:barrier}
Conditions M1--M4 are inconsistent.  More precisely, under M1, M2, and
M4, the point \(\cert(x_0)\) does not lie in the interior of any accepted
set \(U\).
\end{theorem}

\begin{proof}
If \(\cert(x_0)\in\interior U\), then \(\interior U\) is an open
neighborhood of \(\cert(x_0)\).  Convergence gives \(N\) such that
\(\cert(x_n)\in\interior U\subseteq U\) for every \(n\geq N\).
M4 then gives \(x_n\in\good\), contradicting M1.  Taking \(U\) itself
open proves the M1--M4 formulation.
\end{proof}

\begin{remark}[Why state an elementary lemma?]\label{rem:elementary}
Theorem~\ref{thm:barrier} is elementary and should not be advertised as
deep topology.  Its use is diagnostic.  Rodgers--Tao supplies a
canonical false family; complete positivity programs supply recognizable
certificate cones; and the M1--M4 checklist prevents pointwise
continuity, finite strictness, or target-only sufficiency from being
silently upgraded to a robust RH proof.
\end{remark}

\begin{corollary}[Metric form]\label{cor:metric}
Let \((Y,d)\) be a metric space.  Suppose \(d(\cert(x_n),\cert(x_0))\to0\)
and acceptance is sufficient on \(\mathcal S\), while every \(x_n\) is
bad.  Then there is no \(r>0\) for which
\(B(\cert(x_0),r)\subseteq U\).
\end{corollary}

\begin{proof}
Choose \(n\) with \(d(\cert(x_n),\cert(x_0))<r\) and apply M4.
\end{proof}

\begin{corollary}[Canonical RH specialization]\label{cor:rh}
Take \(x_0=H_0\), \(x_n=H_{-1/n}\), and let \(\good\) mean ``has only
real zeros.''  Under M2 and M4, \(H_0\) cannot satisfy M3.
\end{corollary}

\begin{proof}
Rodgers--Tao's \(\Lambda\geq0\) theorem makes every \(H_{-1/n}\) bad,
so Theorem~\ref{thm:barrier} applies.
\end{proof}

The same statement can be phrased directly on the heat path.  If a
certificate at \(t=0\) persists on some interval
\((-\eps,0]\), and if the certificate implies real-rootedness for each
\(H_t\) there, then it contradicts \(\Lambda\geq0\).  The sequence form
is stronger because it applies to any explicit false family and any
certificate topology, not only to interval persistence.

\section{Interiors of positivity cones}
\label{sec:cones}

The abstract theorem becomes informative only after the relevant
notion of interior is made explicit.  Complete positivity criteria
usually face a dichotomy: weak topologies make the positive cone have
empty interior, while strong uniform topologies make interior
equivalent to a positive reserve.

\subsection{Infinite scalar hierarchies}

Let \(A\) be an infinite index set and
\[
 C_+=\{q\in\R^A:q_\alpha\geq0\text{ for every }\alpha\in A\}.
\]

\begin{proposition}[Product and weighted-sup dichotomy]
\label{prop:scalar-cones}
\leavevmode
\begin{enumerate}
\item In the product topology on \(\R^A\), \(C_+\) has empty interior.
\item Fix weights \(w_\alpha>0\) and let
\[
 \ell^\infty_w(A)=
 \left\{q:\|q\|_{\infty,w}:=
 \sup_{\alpha\in A}\frac{|q_\alpha|}{w_\alpha}<\infty\right\}.
\]
Write \(C_{+,w}=C_+\cap\ell^\infty_w(A)\).  Then
\[
 \interior_{\ell^\infty_w(A)} C_{+,w}
 =
 \left\{q\in\ell^\infty_w(A):
 \inf_{\alpha\in A}\frac{q_\alpha}{w_\alpha}>0\right\}.
\]
\end{enumerate}
\end{proposition}

\begin{proof}
A basic product neighborhood controls only finitely many coordinates.
Choose an uncontrolled coordinate and make it negative; hence no point
of \(C_+\) is interior.

For the weighted sup norm, if
\(\delta=\inf_\alpha q_\alpha/w_\alpha>0\), then every \(p\) with
\(\|p-q\|_{\infty,w}<\delta\) is positive coordinatewise.  Conversely,
if the infimum is zero, then for every \(r>0\) choose \(\alpha\) with
\(q_\alpha/w_\alpha<r/2\) and alter only that coordinate by
\(-3r w_\alpha/4\).  The perturbation has norm \(3r/4<r\) and leaves
the cone.
\end{proof}

\begin{corollary}[Uniform hierarchy barrier]\label{cor:hierarchy}
Suppose \(q_\alpha(t)\) is a complete sufficient hierarchy for
real-rootedness of \(H_t\), with
\((q_\alpha(t))_{\alpha\in A}\in\ell^\infty_w(A)\) for \(t\) near
zero.  If, for fixed a-priori weights,
\[
 \sup_\alpha
 \frac{|q_\alpha(t)-q_\alpha(0)|}{w_\alpha}\longrightarrow0
 \quad(t\uparrow0)
\]
and \(q_\alpha(0)\geq\delta w_\alpha\) for every \(\alpha\) and some
\(\delta>0\), then a contradiction follows.
\end{corollary}

\begin{proof}
The endpoint lies in the weighted-sup interior of the positive cone,
and the negative-heat certificates converge to it.  Apply
Corollary~\ref{cor:rh}.
\end{proof}

This also exposes a normalization tradeoff.  If positive endpoint
coordinates are rescaled after inspection by
\(w_\alpha=q_\alpha(0)\), the margin becomes one, but M2 becomes the
relative uniform estimate
\[
 \sup_\alpha
 \frac{|q_\alpha(t)-q_\alpha(0)|}{q_\alpha(0)}
 \longrightarrow0.
\]
The barrier forbids that estimate together with common sufficiency.
Moreover, endpoint-dependent rescaling violates the a-priori
normalization clause in Definition~\ref{def:aic}.

\subsection{Quadratic forms and operators}

Let \(\mathcal H\) be an infinite-dimensional Hilbert space and
\(B(\mathcal H)_{\rm sa}\) the bounded self-adjoint operators.

\begin{proposition}[Operator-cone dichotomy]\label{prop:operator-cones}
For the positive cone
\[
 \mathcal P=\{A\in B(\mathcal H)_{\rm sa}:A\geq0\},
\]
one has:
\begin{enumerate}
\item in operator norm,
\[
 \interior\mathcal P
 =\{A:A\geq\delta I\text{ for some }\delta>0\};
\]
\item in the strong or weak operator topology, \(\mathcal P\) has empty
interior.
\end{enumerate}
\end{proposition}

\begin{proof}
If \(A\geq\delta I\), every self-adjoint \(B\) with
\(\|B-A\|<\delta\) is positive.  Conversely, if no such \(\delta\)
exists, choose unit vectors \(u_n\) with
\(\langle Au_n,u_n\rangle\to0\); arbitrarily small negative rank-one
perturbations leave the cone.

A basic strong neighborhood controls \(A\) on finitely many vectors.
Choose a unit vector \(u\) orthogonal to their span and replace \(A\) by
\(A-M(u\otimes u)\), with \(M>\langle Au,u\rangle\).  The controlled
vectors are unchanged but positivity fails.  The same perturbation
works for a basic weak neighborhood after choosing \(u\) orthogonal to
both finite lists of test vectors appearing in its matrix coefficients.
\end{proof}

\begin{corollary}[Coercive-form barrier]\label{cor:coercive}
Let \(Q_t\) be quadratic forms on one normed space \(V\), and suppose
nonnegativity of \(Q_t\) implies real-rootedness of \(H_t\).  If
\[
 Q_0(v)\geq\delta\|v\|^2
\]
for some \(\delta>0\), and for every \(\eta>0\) there is
\(\eps>0\) such that
\[
 |Q_t(v)-Q_0(v)|\leq\eta\|v\|^2
 \quad(-\eps<t\leq0,\ v\in V),
\]
then a contradiction follows.
\end{corollary}

\begin{proof}
Use \(\eta=\delta/2\).  Then \(Q_t(v)\geq
(\delta/2)\|v\|^2\) for all sufficiently small negative \(t\), contrary
to Corollary~\ref{cor:rh}.
\end{proof}

The norm scaling in Corollary~\ref{cor:coercive} is essential.  A bound
\(Q_0(v)\geq\delta\) for every \(v\) is vacuous at \(v=0\); a continuity
estimate independent of \(\|v\|\) does not describe form-norm
continuity.

\subsection{Finite strict tests}

\begin{corollary}[Finite-test barrier]\label{cor:finite}
Let \(I\) be finite.  Suppose every \(q_i(t)\) is continuous at \(0\),
\(q_i(0)>0\) for all \(i\in I\), and simultaneous nonnegativity of the
\(q_i(t)\) implies real-rootedness of \(H_t\).  Then these assumptions
are inconsistent.
\end{corollary}

\begin{proof}
Finiteness and strict positivity give a common negative-time
neighborhood on which every \(q_i(t)>0\), contradicting
\(\Lambda\geq0\).
\end{proof}

The corollary does not reject finite computations stated at finite
scope.  It rejects only the additional assertion that finitely many
strict continuous tests are sufficient for the entire function.

\section{Membership protocol}
\label{sec:protocol}

The source audit uses two separate questions.

\begin{quote}
\textbf{Source question.}  Does the cited paper itself supply M1--M4?

\textbf{Completion question.}  What exact additional statements would
turn a recognizable certificate architecture into an \(\AIC\) member?
\end{quote}

\paragraph{Outside-reader worksheet.}
For any paper or proposed proof, record the following data without
substituting words such as ``stable,'' ``strict,'' or ``uniform'' for
an estimate.
\begin{enumerate}
\item State the target \(x_0\), the desired property \(\good\), and an
explicit sequence \(x_n\) with a cited proof that every \(x_n\) is bad
(M1).
\item Write the certificate map \(\cert\), its domain, index set,
normalization, carrier \(Y\), and topology.  These must be fixed
independently of the terminal margin claimed at \(x_0\).
\item Prove \(\cert(x_n)\to\cert(x_0)\) in that exact topology, not only
coordinatewise in each fixed finite truncation (M2).
\item Identify the accepted set \(U\) and exhibit an open neighborhood
of \(\cert(x_0)\) contained in \(U\), equivalently an explicit reserve
in a metric model (M3).
\item Quote or prove one same-object implication
\(\cert(x)\in U\Rightarrow x\in\good\) for \(x_0\) and every named
\(x_n\), checking all domain and transform hypotheses there (M4).
\item Stop at the first failed item and report ``nonmember'' together
with that escape hatch.  If all four premises are verified, the
purported completion is inconsistent by Theorem~\ref{thm:barrier};
one of its claimed premises must be false.
\end{enumerate}

No paper in Table~\ref{tab:source-verdicts} supplies all four premises.
Thus none is refuted by Theorem~\ref{thm:barrier}.  Each ``ruled out''
verdict below has the conditional form: \emph{if a future argument adds
every listed admission clause, then the theorem applies}.

The seven rows are architecture representatives, not a claim that the
bibliography exhausts every equivalent formulation of RH.  An omitted
paper receives no verdict by analogy: an outside reader must run the
same worksheet.  Any omitted argument that passes all four items is
covered by the abstract theorem; one that does not pass is outside the
class.

\begin{center}
\refstepcounter{table}
\label{tab:source-verdicts}
\begin{minipage}{\textwidth}
\small
\textbf{Table \thetable.}
Premises supplied by the primary sources.  ``General M4'' means that a
source theorem gives sufficiency for a general real-entire or transform
class once its hypotheses are verified.

\vspace{0.45em}
\begin{tabularx}{\textwidth}{
  >{\raggedright\arraybackslash}p{0.24\textwidth}
  >{\raggedright\arraybackslash}X
  >{\raggedright\arraybackslash}p{0.25\textwidth}}
\toprule
Program & Source-supplied M1--M4 & Actual-source verdict\\
\midrule
de Bruijn--Newman heat
& M1 yes; M2--M4 absent for an argument-chosen certificate
& Theorem input, not a member\\

Jensen polynomials
& General M4 via Polya--Schur; M1--M3 absent
& Nonmember\\

Generalized Laguerre--Turan
& General M4 via Theorem 2.3; M1--M3 absent
& Nonmember\\

Weil--Bombieri forms
& Target-only M4; no false family or heat form; no norm-open reserve
& Nonmember\\

de Branges positivity
& Proposed M3 fails at the target; no false-family transport
& Refuted in its own source\\

Canonical systems
& No M1--M4 heat package; \(\omega\) is not heat time
& Nonmember\\

Polya frequency
& General M4 is conditional on transform hypotheses; M1--M3 absent
& Nonmember\\
\bottomrule
\end{tabularx}
\end{minipage}
\end{center}

Three rules prevent false membership.

\paragraph{Topology and normalization are argument data.}
One may not choose a metric after seeing the endpoint estimates, isolate
\(\Xi\) by decree, or divide each coordinate by its observed endpoint
value and then call the result robust.  Such constructions either fail
the a-priori clause or move the burden to M2.

\paragraph{Pointwise continuity is not M2.}
Continuity of each fixed Jensen polynomial, Laguerre expression, minor,
or matrix entry does not imply convergence in the complete uniform
topology that gives the positive cone an interior.

\paragraph{Sufficiency must follow the same objects.}
A zeta-specific prime identity at \(t=0\) does not imply
real-rootedness of \(H_t\).  A reciprocal transform or canonical system
defined only at the endpoint cannot be transported formally.  Failure
of a common domain, transform, or sufficiency theorem is a genuine
escape hatch, not a technical omission.

\section{Primary-source dossiers}
\label{sec:dossiers}

\subsection{de Bruijn--Newman heat flow and universal factors}

De Bruijn's Theorem 13 contracts a strip of zeros under multiplication
of the Fourier kernel by a Gaussian factor, and Theorem 14 places this
in the universal-factor framework \cite[pp.~204--205]{deBruijn1950}.
Rodgers and Tao state the threshold characterization and prove
\(\Lambda\geq0\) \cite[pp.~2--3]{RodgersTao2020}.

\paragraph{Actual source.}
M1 is supplied by \eqref{eq:false-family}.  The papers do not choose an
arbitrary certificate \(\cert\), prove \(\cert(H_{-1/n})\to\cert(H_0)\),
or claim an open RH-sufficient endpoint certificate.  They are theorem
inputs, not \(\AIC\) members.

\paragraph{Precisely excluded completion.}
Any argument in this deformation architecture is ruled out if it defines
one fixed certificate topology on \(H_0,H_{-1},H_{-1/2},\ldots\), proves
certificate convergence, accepts \(H_0\) in an open set, and proves
acceptance sufficient for real-rootedness on the same family.

\paragraph{What survives.}
Positive-time upper bounds for \(\Lambda\), exact identities at \(t=0\),
one-sided limits that yield no negative interval, and certificates
singular under backward heat.

\subsection{Jensen-polynomial hyperbolicity}

O'Sullivan's Theorem 1.1 states that RH is equivalent to hyperbolicity
of every \(J^{d,n}\), \(d\geq1,n\geq0\)
\cite[p.~2]{OSullivan2021}.  His Theorem 3.1 records the general
Polya--Schur theorem: all Jensen polynomials of a real formal power
series are hyperbolic exactly when it converges to an entire function
in the Laguerre--Polya class \cite[p.~6]{OSullivan2021}.  Thus the
complete hierarchy has a general sufficiency direction, not merely a
zeta-specific one.

The heat embedding can be made exact.  Since \(H_t\) is even, the
power-series definition
\[
 \Theta_t(w)=H_t(2i\sqrt w),\qquad
 H_t(z)=\Theta_t(-z^2/4),
\]
is single-valued and entire.  The displayed Riemann kernel
\(\Phi(u)\) is positive for \(u\geq0\), so \(\Theta_t\) has positive
Taylor coefficients for \(t\leq0\).  If every Jensen polynomial of
\(\Theta_t\) is hyperbolic, Polya--Schur gives
\(\Theta_t\in\LP\); its positive coefficients exclude positive real
zeros.  Hence all zeros of \(\Theta_t\) are nonpositive, and the second
identity forces every zero of \(H_t\) to be real.  This verifies M4 on
the same heat family rather than borrowing a target-only zeta
implication.

Let \(\mathcal P_d\) be the coefficient space of monic degree-\(d\)
polynomials and \(\mathcal H_d\subseteq\mathcal P_d\) its closed
hyperbolic set.  The a-priori normalization is assumed to place each
Jensen polynomial in this fixed chart and to preserve hyperbolicity.
For \(d\geq2\) and an accepted polynomial \(p\), the robust reserve is
\[
 r_d(p)=\dist(p,\mathcal P_d\setminus\mathcal H_d).
\]
The complete certificate consists of a-priori normalized polynomials
\(N_{d,n}J_F^{d,n}\).  In a weighted sup metric, open acceptance means a
positive infimum of the corresponding distances to the complements.
The distance is a reserve on accepted points; it should not be confused
with a globally signed certificate, since distance to a set is always
nonnegative.

\paragraph{Actual source.}
The paper supplies neither the negative-heat family nor a uniform
coefficient modulus over all \((d,n)\), and it proves no positive
complete reserve.  Griffin, Ono, Rolen, and Zagier prove in Theorem 1
that for every fixed \(d\), \(J^{d,n}\) is hyperbolic for all sufficiently
large \(n\) \cite[p.~2]{GriffinOnoRolenZagier2019}.  This is a
fixed-degree tail theorem, not M3.  Farmer further shows that
fixed-degree hyperbolicity can persist for off-axis entire functions and
therefore loses the information needed for RH \cite{Farmer2022}.

\paragraph{Precisely excluded completion.}
Fix \(N_{d,n}\) and a complete coefficient metric independently of the
endpoint; apply the construction to every \(H_{-1/k}\); prove convergence
in that complete metric; and prove a positive uniform distance from the
nonhyperbolic complements.  These additions make the program an
\(\AIC\) member and are inconsistent.

\paragraph{What survives.}
Every fixed degree, every finite index range, and complete
hyperbolicity whose normalized distances have infimum zero.

\subsection{Generalized Laguerre and Turan inequalities}

Csordas, Norfolk, and Varga prove the classical Turan inequalities in
Theorem 2.5 \cite[p.~528]{CsordasNorfolkVarga1986}.  Their introduction
identifies these inequalities as necessary for the relevant
real-rootedness statement; failure would disprove RH, but validity does
not prove it.  Csordas and Vishnyakova define
\begin{equation}\label{eq:laguerre}
 L_m(x;F)=
 \sum_{j=0}^{2m}
 \frac{(-1)^{j+m}}{(2m)!}\binom{2m}{j}
 F^{(j)}(x)F^{(2m-j)}(x)
\end{equation}
and prove in Theorem 2.3 that \(L_m(x;F)\geq0\) for all \(m\geq0\) and
\(x\in\R\) forces a nonzero real entire \(F\) into the
Laguerre--Polya class \cite[pp.~1645--1647]{CsordasVishnyakova2013}.
Their Remarks 1.3 give the direct real-zero implication through
\[
 |F(x+iy)|^2=\sum_{m=0}^{\infty}L_m(x;F)y^{2m}.
\]

\paragraph{Certificate discipline.}
The automatic coordinate \(L_0(x;F)=F(x)^2\) vanishes at every real
zero and cannot carry a strict margin.  It may be omitted from a
nonnegativity certificate without weakening sufficiency.  A robust
completion would instead fix weights \(w_{m,x}>0\) for \(m\geq1\) and
use a topology controlling \(L_m(x;F)/w_{m,x}\) uniformly over order
and point.

\paragraph{Actual source.}
The general sufficiency theorem supplies M4 once the hierarchy is
defined, but no source supplies the heat false family, a uniform
backward modulus, or a positive normalized all-order/all-point floor.
Unnormalized expressions may also decay as \(|x|\to\infty\).

\paragraph{Precisely excluded completion.}
A fixed normalized hierarchy on all \(H_{-1/k}\), a uniform heat
modulus over \(m\) and \(x\), and a positive complete floor at \(H_0\).

\paragraph{What survives.}
The proved Turan inequalities, fixed Laguerre order, compact \(x\)
ranges, pointwise strictness, and complete hierarchies with zero
normalized infimum.

\subsection{Weil and Bombieri quadratic forms}

Bombieri states Weil's criterion as positivity of a quadratic
functional and gives a necessary-and-sufficient formulation in
Theorem 1 \cite[pp.~191--192]{Bombieri2000}.  He studies variational
problems on functions with prescribed support and finite matrix
truncations.  His theorem is pointwise strict for each nonzero test
function under RH, but it does not give a positive lower bound for the
Rayleigh quotient in any fixed complete norm.  Pointwise positive
definiteness is therefore not coercive interior.  Suzuki restates
\[
 Q_W(v)\geq0\quad\text{for every }v\in C_c^\infty(\R)
 \quad\Longleftrightarrow\quad\RH
\]
and proves continuity of the lowest localized eigenvalue
\(a\mapsto\lambda_a\) in Theorem 1.3
\cite[pp.~1,5]{SuzukiWeil2026}.  If RH fails, \(\lambda_a<0\) for some
\(a\), whereas it is positive for small \(a\); continuity therefore
produces a zero crossing.

\paragraph{Actual source.}
The forms are arithmetic zeta forms.  Neither paper defines \(Q_t\) for
the functions \(H_t\), proves a common real-rootedness implication, or
asserts a global coercive floor.  Pointwise strict positivity,
semidefinite positivity, and support-dependent lowest eigenvalues are
boundary data rather than open operator-norm acceptance unless a
uniform Rayleigh floor is proved.

\paragraph{Precisely excluded completion.}
Forms \(Q_t\) on one fixed form domain \(V\), with positivity of \(Q_t\)
implying real-rootedness of the same \(H_t\), convergence in form norm,
and \(Q_0(v)\geq\delta\|v\|_V^2\).  Corollary~\ref{cor:coercive} rules
out exactly this completion.

\paragraph{What survives.}
Semidefinite forms, explicit null sequences, support-dependent floors,
changing domains, distributional limits, and prime-side identities
available only at \(t=0\).  Suzuki's zero crossing is an exemplar of a
surviving boundary mechanism.

\subsection{de Branges-space positivity}

Conrey and Li's 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 \cite[p.~930]{ConreyLi2000}.  They then test
\(E(z)=\xi(1-iz)\).  Equation (3.2), evaluated at the 34th zeta zero,
has the wrong sign, and the authors conclude that the zeta-associated
space does not satisfy condition (3.1)
\cite[p.~934]{ConreyLi2000}.

\paragraph{Actual source.}
The proposed endpoint positivity already fails.  It is not an
\(\AIC\) member and is not refuted by Theorem~\ref{thm:barrier}; it is
refuted inside its own primary source.

\paragraph{Precisely excluded replacement.}
A different de Branges condition would enter the class only after it is
true at \(H_0\), is defined on \(H_{-1/n}\) using one common domain,
has same-object real-rootedness sufficiency, converges in form norm, and
has a coercive interior reserve.

\paragraph{What survives.}
Different de Branges spaces, semidefinite kernels, changing domains,
and singular endpoint constructions.

\subsection{Canonical-system Hamiltonians}

Suzuki defines shifted functions \(A^\omega\) and proves in Proposition
1.1 that RH is equivalent to real-rootedness of \(A^\omega\) for every
\(\omega>0\) \cite[p.~1]{SuzukiCanonical}.  Theorem 2.3 constructs the
associated canonical system unconditionally for \(\omega>1\); the
paper explains that an extension to all \(\omega>0\) would yield a
criterion through positive-semidefinite Hamiltonian matrices
\cite[pp.~2,23--24]{SuzukiCanonical}.

\paragraph{Actual source.}
The parameter \(\omega\) is a shift in the completed zeta function, not
de Bruijn--Newman heat time.  The construction is semidefinite, has
parameter-dependent analytic difficulties, and is not defined on
\(H_{-1/n}\).  M1--M4 are therefore absent.

\paragraph{Precisely excluded completion.}
A backward-heat canonical system on one common measurable domain and
normalization, same-object real-rootedness sufficiency, convergence in
a positivity-preserving norm, and uniform positive definiteness.

\paragraph{What survives.}
Positive-semidefinite Hamiltonians, \(\omega\)-singular limits,
changing domains, and constructions with no heat extension.

\subsection{Polya-frequency and total positivity}

Gr\"ochenig's Theorem 4 states that RH holds exactly when
\begin{equation}\label{eq:pf-transform}
 \Lambda(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
 \frac{e^{-ix\tau}}{\xi(1/2+\tau)}\,d\tau
\end{equation}
is a Polya-frequency function \cite[p.~4]{Grochenig2023}.  This follows
from Schoenberg's general transform characterization.  The paper also
states that even the order-one positivity, equivalently positive
definiteness of the reciprocal function, is not known
\cite[p.~5]{Grochenig2023}.

\paragraph{Certificate discipline.}
Raw translation minors vanish when nodes coalesce.  Thus a purported
positive floor over all raw node configurations is impossible.  Any
robust all-minor certificate must specify in advance either separated
configurations or a confluent normalization and must include every
limiting configuration needed for PF sufficiency.

\paragraph{Actual source.}
The reciprocal transform in \eqref{eq:pf-transform} is proved
well-defined for the endpoint used in the paper.  The source does not
define corresponding transforms for \(H_{-1/n}\), verify
integrability there, prove a uniform minor modulus, or produce an open
PF reserve.

\paragraph{Precisely excluded completion.}
Define source-compatible reciprocal transforms throughout the canonical
false family; verify Schoenberg's hypotheses there; fix a complete
minor normalization and topology; and prove both certificate
accumulation and open PF acceptance.

\paragraph{What survives.}
Fixed PF order, rank-dependent reserves tending to zero,
coalescing-node boundary behavior, and singular or nonexistent
reciprocal transforms.

\section{Boundary controls and escape hatches}
\label{sec:controls}

\subsection{Li coefficients}

Bombieri and Lagarias prove a general multiset criterion in Theorem 1
and its critical-line corollary in Corollary 1
\cite[pp.~276--278]{BombieriLagarias1999}.  Under explicit summability
and symmetry hypotheses, critical-line location is equivalent to
nonnegativity of every generalized Li sum
\[
 \lambda_n(R)=\sum_{\rho\in R}
 \left[1-\left(1-\frac1\rho\right)^n\right].
\]
This is an all-index boundary criterion.  The source contains no
canonical heat deformation of \(R\), no fixed normalization producing
an open reserve, and no uniform modulus.  A future source-matched false
multiset family satisfying those additions would fall under
Theorem~\ref{thm:barrier}; endpoint arithmetic formulas and zero
complete infimum do not.

\subsection{Nyman--Beurling}

B\'aez-Duarte's Theorem 1.1 states RH as membership of a fixed vector in
the closure of the natural Beurling span
\cite[p.~1]{BaezDuarte2003}.  The terminal condition is distance zero
to a closure.  It is therefore an exact boundary statement, not an open
positivity certificate.  This control prevents ``Hilbert space'' or
``stable approximation'' from being used as proxy tests for \(\AIC\)
membership.

\subsection{Recorded surviving mechanisms}

At the theorem level the alternatives are exhaustive: an argument
evades the contradiction only by failing M1, M2, M3, or M4.  With the
Rodgers--Tao false family fixing M1, the recurrent concrete failures of
M2--M4 in the audited sources include:
\begin{enumerate}
\item all-index positivity with zero complete infimum;
\item semidefinite forms, null sequences, and zero eigenvalue crossings;
\item rank-, degree-, point-, support-, or height-dependent margins;
\item heat-singular transforms and changing spaces, domains, or
normalizations;
\item arithmetic identities defined only for the zeta endpoint;
\item exact closure, density, and limiting criteria;
\item certificates that quantitatively separate every named false
family from \(\Xi\);
\item positive-time de Bruijn--Newman estimates; or
\item finite results stated only at finite scope.
\end{enumerate}

Conversely, saying that a program is ``robust'' does not place it in the
class.  The outside reader must exhibit the exact false family,
certificate topology, open accepted set, and common sufficiency theorem.
At the first missing premise, the verdict narrows to nonmembership.

\section{Hostile checks}
\label{sec:hostile}

The following objections are load-bearing.

\paragraph{The theorem is tautological.}
Yes: openness plus convergence eventually accepts the sequence.  The
paper's claim is the exact RH specialization, cone-interior
identification, and source classification, not a difficult topological
lemma.

\paragraph{A topology can be chosen to isolate the target.}
Yes, but that makes M2 false or violates the a-priori topology clause.
The theorem does not outlaw unnatural certificates; it classifies them
as nonmembers.

\paragraph{Every fixed coordinate is continuous.}
That proves only product-topology convergence.  By
Proposition~\ref{prop:scalar-cones}, the positive cone then has empty
interior.  A topology with positive interior requires the complete
uniform estimate that triggers the barrier.

\paragraph{Strict positivity at every index should be enough.}
No.  The sequence \(q_n=1/n\) is strictly positive but lies on the
boundary of the positive cone in \(\ell^\infty\).  The operator
\(\operatorname{diag}(1,1/2,1/3,\ldots)\) is positive and injective but
not coercive.

\paragraph{The false family may leave the certificate domain.}
Correct.  This defeats M2 or M4.  It is a central surviving mechanism
for reciprocal transforms, arithmetic forms, de Branges spaces, and
canonical systems.

\paragraph{A criterion may imply RH only through zeta arithmetic.}
Correct.  Endpoint-only prime identities are outside M4 and therefore
survive.

\paragraph{Finite strict computations are useful.}
Correct.  Corollary~\ref{cor:finite} applies only if those finite tests
are also claimed sufficient for real-rootedness of the complete
function.

\paragraph{No cited paper is a member.}
Correct.  The result is prospective: it excludes a common proposed
completion move, not the papers' valid finite and boundary theorems.

\section{Formal verification and reproducibility}
\label{sec:formal}

The supplementary Lean file \texttt{HeatBoundary.lean} checks:
\begin{enumerate}
\item direct backward-persistence and metric heat-path barriers;
\item metric, open-set, and arbitrary-interior
accumulating-counterexample theorems;
\item a uniform scalar-hierarchy specialization;
\item the norm-scaled common-domain coercive-form specialization; and
\item the finite strict continuous-test corollary.
\end{enumerate}

The topological theorem's sufficiency premise is restricted exactly to
\[
 \{x_0\}\cup\operatorname{range}(x_n),
\]
not to every object of the ambient type.  In Lean this scope is
\texttt{Set.insert target (Set.range counterexample)}, matching M4
premise-for-premise.

The reproducibility bundle also contains:
\begin{itemize}
\item SHA-256 hashes for every primary-source PDF;
\item OCR or text extraction with theorem anchors;
\item a machine-readable dossier giving source and completion verdicts;
\item a verifier that compiles Lean, validates source hashes and
anchors, checks the exact seven claimed program dossiers and two
boundary controls, builds this manuscript, and inspects the resulting
PDF text.
\end{itemize}

The analytic statement \(\Lambda\geq0\) is cited rather than formalized.
The Lean development proves the complete deduction from the explicit
bad-family premise.

\section{Conclusion}

The de Bruijn--Newman theorem turns backward heat into a universal
stress test for robust RH certificates.  If a complete positivity
architecture gives \(\Xi\) an open reserve and transports that reserve
continuously to the canonical negative-heat functions, common
sufficiency certifies a known false object.  The obstruction is
topological, but its practical content is quantitative: product
topologies have no positive interior, while the natural strong
topologies demand exactly the uniform floor or coercivity that cannot
persist.

The source audit sets a narrow boundary around this claim.  The cited
heat, Jensen, Laguerre--Turan, Weil, de Branges, canonical, and
total-positivity sources are not members of the obstruction class.
Their semidefinite, finite, localized, singular, and exact endpoint
results remain valid.  What future work should abandon is the search
for a terminal \emph{uniform interior reserve} that is regular on an
explicit false family and sufficient there.

The resulting fork is constructive.  A viable program must reach RH on
a boundary, explain a controlled singularity, retain genuinely
zeta-specific arithmetic, change its domain essentially, or build a
certificate that separates the canonical counterexamples.  Those are
not loopholes around the theorem; they are the remaining mathematical
directions.

\begin{thebibliography}{99}
\small
\setlength{\itemsep}{0.15em}
\setlength{\parsep}{0pt}
\vspace{0.35em}

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\newblock \emph{Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.}
\textbf{14} (2003), 5--11.

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E. Bombieri,
\newblock Remarks on Weil's quadratic functional in the theory of
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\newblock \emph{Cent. Eur. J. Math.} \textbf{11} (2013), 1643--1650.

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N. G. de Bruijn,
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\newblock \emph{Duke Math. J.} \textbf{17} (1950), 197--226.

\bibitem{Farmer2022}
D. W. Farmer,
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\newblock \emph{Res. Number Theory} \textbf{8} (2022), article 12.

\bibitem{Grochenig2023}
K. Gr\"ochenig,
\newblock Schoenberg's theory of totally positive functions and the
Riemann zeta function,
\newblock in \emph{Sampling, Approximation, and Signal Analysis},
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pp.~193--210.

\bibitem{GriffinOnoRolenZagier2019}
M. J. Griffin, K. Ono, L. Rolen, and D. Zagier,
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11103--11110.

\bibitem{OSullivan2021}
C. O'Sullivan,
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\bibitem{RodgersTao2020}
B. Rodgers and T. Tao,
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\bibitem{SuzukiCanonical}
M. Suzuki,
\newblock A canonical system of differential equations arising from
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\bibitem{SuzukiWeil2026}
M. Suzuki,
\newblock Weil's quadratic form via the screw function,
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\end{thebibliography}

\end{document}
