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% Addendum 360 (extends Paper 24, the Riemann/Hilbert-Polya construction).
% Number-theory-in-TOE lane. Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P360.py (11/11 pass).
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\title{Addendum 360: The Prime Number Theorem and Goldbach in TOE Context\\
\large PNT as the leading pole of the renormalisation face; the Goldbach\\
$2$-vs-$3$ parity as dimension, not metric\\[0.3em]
\normalsize (Addendum to Paper 24)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Two number-theory statements read through the master operator. \textbf{Part~A (solid).} The
prime number theorem $\pi(x)\sim x/\ln x$ is the leading-order spectral shadow of the
renormalisation face $\zR=r\partial_r$ of $\hat{O}$. The dilation operator $x\partial_x$ has
Mellin eigenfunctions $x^s$ (eigenvalue $s$); Paper~24's $H=-i(x\partial_x+\tfrac12)$ is its
self-adjoint symmetrisation, whose real spectrum is the critical line $\mathrm{Re}(s)=\tfrac12$
(the Riemann boundary spectrum). PNT is the weaker statement that $\zeta$ has no zero on
$\mathrm{Re}(s)=1$; its main term $\pi(x)\sim\mathrm{Li}(x)\sim x/\ln x$ is the $s=1$ pole of
$\zeta$ (residue $1$) read back through the inverse Mellin transform. So the $1/\ln x$ prime
density is the leading pole of the $\zR$ face --- a corpus identification, not an analogy
(verified: $\pi(x)/(x/\ln x)\to1$, $\pi(x)/\mathrm{Li}(x)\to1$). \textbf{Part~B (a bounded
resonance).} The binary (strong, even $n=p+q$) and ternary (weak, odd $n=p+q+r$) Goldbach
conjectures --- ternary proven (Helfgott~2013), binary open --- map onto the TOE by
\emph{parity}, not metric. The Hardy--Littlewood count is $r_k(n)\sim\mathfrak{S}_k(n)\,
n^{k-1}/(\ln n)^k$: the power of $\ln$ equals the number of primes $k$ (one PNT density per
prime, on the $(k{-}1)$-simplex), so binary gives $(\ln n)^2$ and ternary $(\ln n)^3$
(verified). The reason ternary closes and binary is marginal --- the $k=3$ main term dominates
the minor-arc error, $k=2$ does not --- is the same dimensional fact as the corpus's
``composition closes at three'' ($\alpha^{-1}=4\pi^3+\pi^2+\pi$ is a three-term sum; the $\mathbb{Z}_3$
lens, the three layers) versus the two-fold dual-observer (Papers~27/28) that keeps an
irreducible residual. \textbf{But the layer weights $4\pi^3\!:\!\pi^2\!:\!\pi$ do not enter the
representation counts}: $\mathfrak{S}_k(n)$ is the arithmetic singular series
$\prod_{p\mid n}(\cdots)$, set by $n$'s prime divisors, not by any $\pi$-power (verified:
$r_2(99990)=1855\gg r_2(100000)=810$, $r_2(99991)=1$). The connection is dimensional, not
metric --- explicitly \emph{not} a claim that the TOE proves or quantitatively predicts
Goldbach. All claims pinned by \texttt{verify\_P360.py} (11/11).
\end{abstract}

\section{Part A: PNT is the leading pole of the renormalisation face}
\label{sec:pnt}

\begin{proposition}[The dilation operator and the $s=1$ pole]
\label{prop:pnt}
The renormalisation face of the master operator is $\zR=r\partial_r$ (Paper~18; the dilation
generator). On the multiplicative half-line it is $x\partial_x$, with eigenfunctions $x^s$ and
eigenvalue $s$: the Mellin transform diagonalises it. Paper~24 takes its self-adjoint
symmetrisation $H=-i(x\partial_x+\tfrac12)$, whose real spectrum forces $\mathrm{Re}(s)=\tfrac12$
--- the Riemann critical line (the boundary spectrum). The prime number theorem is the weaker
statement that $\zeta$ has no zero on $\mathrm{Re}(s)=1$, equivalently that the inverse Mellin
transform of $\zeta'/\zeta$ is dominated by the simple pole at $s=1$ (residue $1$), giving the
main term $\pi(x)\sim\mathrm{Li}(x)\sim x/\ln x$. Numerically $\pi(x)/(x/\ln x)\to1$ and
$\pi(x)/\mathrm{Li}(x)\to1$ (the latter to $5$ figures by $x=2\times10^6$).
\end{proposition}

\begin{remark}[Why this is a real identification]
\label{rmk:pnt}
The $1/\ln x$ density is not imposed; it is the residue of the $s=1$ pole transported by the
$\zR$ face. The corpus already builds this operator (Paper~24) and is careful to claim only the
Hilbert--P\'olya framework, not a proof of RH. The PNT statement is strictly the leading pole
and is unconditional; it sits one layer below RH on the same operator.
\end{remark}

\section{Part B: the Goldbach $2$-vs-$3$ parity is dimensional}
\label{sec:goldbach}

\begin{proposition}[The $\ln$-exponent counts the primes]
\label{prop:exponent}
For $n$ a sum of $k$ primes the Hardy--Littlewood asymptotic is
$r_k(n)\sim\mathfrak{S}_k(n)\,n^{k-1}/(\ln n)^k$: each prime contributes one PNT density factor
$1/\ln n$, and the compositions fill the $(k{-}1)$-simplex (volume $\sim n^{k-1}$). Hence binary
($k=2$) carries $(\ln n)^2$ and ternary ($k=3$) carries $(\ln n)^3$. Verified:
$r_2(n)(\ln n)^2/n$ stays $\approx1.05$--$1.08$ and $r_3(n)(\ln n)^3/n^2$ stays bounded across
$n$. The proof status follows the same dimension: the $k=3$ main term dominates the minor-arc
error (ternary proven, Helfgott~2013), while $k=2$ is marginal (binary open).
\end{proposition}

\begin{remark}[The corpus parity]
\label{rmk:parity}
The same ``three closes, two leaves a residual'' is the corpus's own signature. Composition
closes at three: $\alpha^{-1}=4\pi^3+\pi^2+\pi$ is a three-term layer sum, the $\mathbb{Z}_3$
lens $L(3,1)$ is the rigid family structure, and the three-layer ontology is complete. The
two-fold structure --- the dual observer of Papers~27/28, ``why one reader is not enough'' ---
carries a permanent residual (the standing wave's minus sign). Ternary Goldbach closing and
binary staying open echoes this parity exactly.
\end{remark}

\begin{theorem}[The limit: weights do not transfer]
\label{thm:limit}
The geometric layer weights $4\pi^3:\pi^2:\pi$ do \emph{not} appear in the Goldbach
representation counts. The constant $\mathfrak{S}_k(n)$ is the arithmetic singular series
$\prod_{p\mid n}(\cdots)$, governed entirely by the small-prime divisors of $n$: numerically
$r_2(99990)=1855$ (with $99990$ divisible by $2,3,5,11$) against $r_2(100000)=810$ (divisible
by $2,5$) and $r_2(99991)=1$ ($99991$ prime). The Goldbach comet's scatter is arithmetic, not
geometric. Hence the TOE--Goldbach connection is one of \emph{dimension} (the count $k$, the
$(k{-}1)$-simplex, the $(\ln)^k$ parity), not of \emph{metric} (the $\pi$-weights).
\end{theorem}

\section*{Status}
\textbf{Part~A is solid.} PNT is the leading-order spectral shadow of the renormalisation face
$\zR=r\partial_r$: the dilation operator whose Mellin eigenfunctions are $x^s$, whose self-adjoint
symmetrisation (Paper~24) has the critical line as spectrum, and whose $s=1$ pole is the
$x/\ln x$ main term. This is a corpus identification, verified numerically, and sits one layer
below the (unproven) RH on the same operator. \textbf{Part~B is a bounded structural resonance.}
The Goldbach binary/ternary split maps onto the TOE by parity: the $\ln$-exponent equals the
number of primes $k$ (binary $2$, ternary $3$), the $(k{-}1)$-simplex picture gives ``three
closes, two marginal,'' and this matches the corpus's three-fold closure ($\alpha=4\pi^3+\pi^2+\pi$,
$\mathbb{Z}_3$) versus the two-fold dual-observer residual. The honest limit, proven here: the
geometric weights $4\pi^3:\pi^2:\pi$ do \emph{not} enter the representation counts, which are
fixed by the arithmetic singular series. So the connection is dimensional, not metric, and is
explicitly \emph{not} a claim that the TOE proves or quantitatively predicts Goldbach --- a
discipline the PRVE prime-resonance programme earned, where most family-resonance claims were
retracted as artifacts. No new objects; no published number changes. Verifier:
\texttt{verify\_P360.py}, 11/11 pass.

\paragraph{What a genuine quantitative probe would require.}
To upgrade Part~B from resonance to result, the three-layer weights would have to predict
something the singular series does not --- e.g.\ a geometric correction to the Goldbach comet's
envelope that survives a residue-matched null. Absent such a prediction, the parity is the
content, and it is named as such.

\end{document}
