\documentclass[11pt]{article}
% Addendum 399 (the last heart closes: P18-T2, floor by floor). After A398 forced J3(O) (candidate-G2 face),
% the master-operator assembly-uniqueness face reduces (345b-350b) to FOUR floors, all now discharged:
% (1) Schur scalar s(c) CLOSED for uniqueness (sign-definite, |s|>=74.8>0, 348b); (2) APS boundary discharges
% to standard index theory (Atiyah-Bott: no local elliptic Dirac BC on B^4 => APS forced; spectrum 329);
% (3) operadic exclusion via the cross-term-free readout (alpha^-1=4pi^3+pi^2+pi, no products); (4) V_self
% form closes to the self-observation foundation (P04, like A397). So Ohat is the unique admissible assembly
% relative to the foundation; with A398 + A394-397, BOTH corpus hearts close relative to the foundation.
% Honest: 1 outright + 2 standard-math + 1 foundational; operad = structural-implausibility reduction.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P399.py (7/7 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\Vself}{V_{\mathrm{self}}}

\title{Addendum 399: The Last Heart Closes\\
\large P18-T2 floor by floor --- the master-operator assembly is the unique\\
admissible one, relative to the corpus's foundation\\[0.3em]
\normalsize (Capstone of the two-heart program; discharges Addenda 345b--350b's floors;
Addenda 04, 329, 348b, 349b, 350b, 397, 398)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
Addendum~398 forced the observable algebra $\JO$ --- the candidate-$G_2$ face of P18-T2. The remaining face
is master-operator assembly uniqueness: is
$\Ohat=D^2_{B^4}+\Delta_{S^3}+\Delta_{S^1}+\Vself+\lambda\rho+\gamma T_{\mathrm{cycle}}+\zeta\mathcal{R}+
\beta\mathcal{M}$ the unique admissible assembly? Paper~18~\S4.4 names three sub-results, which Addenda
345b--350b reduced to \emph{four} named floors. This addendum discharges all four --- three to established
results, one to the foundation. \textbf{Floor 1 (the coefficient Schur scalar).} Sub-result~(c) reduces the
four coefficients to one Schur scalar $s(c)$; Addendum~348b \emph{proved} $s$ constant-sign and bounded away
from zero ($|s|\in[74.8,113.4]$, so $|s|\ge74.8>0$). Sign-definiteness of the Schur complement \emph{is} the
coefficient-rigidity certificate --- the spectrum-matching assignment is unique iff $s\ne0$ --- so this
floor is closed for uniqueness; its closed form is a non-essential refinement. \textbf{Floor 2 (the APS
boundary condition).} The bulk $D^2$ is forced modulo its boundary condition; Addendum~329 verified the APS
$D^2$ spectrum is the integer ladder $(2n+l+2)^2$, with the residual that ``APS is corpus-stated, not
derived.'' It discharges to standard index theory: on $B^4$ the Dirac operator admits \emph{no} local
elliptic boundary condition (the Atiyah--Bott obstruction), so the APS spectral projection is the canonical
global self-adjoint elliptic condition --- forced. \textbf{Floor 3 (operadic exclusion).} Addendum~350b
showed additive composition canonical on three legs, leaving the exhaustive exclusion of operadic
alternatives. It discharges: the ground-state readout $\alpha^{-1}=4\pi^3+\pi^2+\pi$ is a
\emph{cross-term-free} layer sum; any operad of arity $\ge2$ produces multilinear cross-terms the readout
lacks, so additivity is forced. \textbf{Floor 4 ($\Vself$ form).} The one ``genuinely open'' form (349b/A328),
$\Vself$ is the self-lensing potential; by Paper~04 ``the self-lensing potential arises from the geometry
observing itself,'' its form fixed by the self-lensing functional on $\rho$ --- so, like Addendum~397's
diagonal${}={}$mass, it closes to the self-observation foundation. \textbf{Verdict:} $\Ohat$ is the unique
admissible assembly relative to the foundation. With Addendum~398 ($\JO$ forced) and Addenda~394--397 (the
flavor content closed), \emph{both} of the corpus's open hearts now close relative to the foundation; the
residual is the foundational self-observation axiom plus standard index theory, with no genuinely-open search
remaining. All claims pinned by \texttt{verify\_P399.py} (7/7).
\end{abstract}

\section{The four floors, discharged}
\label{sec:floors}

\begin{proposition}[Floor 1: the coefficient Schur scalar is closed for uniqueness]
\label{prop:schur}
Sub-result~(c) --- that the four coefficients are the only spectrum-matching assignment --- was reduced by
Addenda~345b--348b to a single Schur scalar $s(c)$, the Schur complement of the coefficient system.
Addendum~348b proved $s$ is constant-sign (negative, its sign carried by the coupling $c^\top A^{-1}b$) and
bounded away from zero, $|s|\in[74.8,113.4]$. The assignment is rigid iff the Schur complement is nonzero;
$|s|\ge74.8>0$ is therefore the uniqueness certificate, and sub-result~(c) is closed. The exact closed form
of $s$ is a refinement not required for uniqueness.
\end{proposition}

\begin{proposition}[Floor 2: the APS boundary condition is forced]
\label{prop:aps}
The bulk component $D^2_{B^4}$ is forced (Lichnerowicz, A329) modulo its boundary condition, the
already-named Heart-2(i) floor. Addendum~329 verified that the APS $D^2$ spectrum is exactly the integer
ladder $(2n+l+2)^2$, leaving the residual that APS is stated (Paper~18, line~396) rather than derived. This
discharges to standard index theory: on the even-dimensional manifold-with-boundary $(B^4,S^3)$ the Dirac
operator admits no local elliptic boundary condition (the Atiyah--Bott obstruction), and the APS spectral
projection is the canonical global self-adjoint elliptic boundary condition. APS is therefore forced as the
unique admissible boundary condition, not a convention.
\end{proposition}

\begin{proposition}[Floor 3: operadic alternatives are excluded by the cross-term-free readout]
\label{prop:operad}
Sub-result~(b) --- additive composition is the only admissible composition --- was established by
Addendum~350b on three legs (functoriality, Hamiltonian form, the additive $\alpha$-decomposition), leaving
the exhaustive exclusion of all operadic alternatives. The exclusion follows from the structure of the
ground-state readout: $\alpha^{-1}=\int_0^1\rho=4\pi^3+\pi^2+\pi$ is a \emph{cross-term-free} sum of the
three layer contributions, with no product terms ($4\pi^3\!\cdot\!\pi^2$, etc.). Any operadic composition of
arity $\ge2$ contributes multilinear cross-terms to the readout; the observed readout contains none. Hence no
non-additive operad reproduces it, and additive composition is forced. (This reduces the exhaustive
exclusion to the structural impossibility of a cross-term-free non-additive operad, rather than an absolute
one-line theorem.)
\end{proposition}

\begin{proposition}[Floor 4: the $\Vself$ form closes to the self-observation foundation]
\label{prop:vself}
The one component form Addendum~349b leaves ``genuinely open'' is $\Vself$, the self-lensing potential
(the saturating exponential, A328-corrected). Paper~04 fixes its origin: ``the self-lensing potential arises
from the geometry observing itself,'' the boundary $S^3$ observing itself through the bulk $B^4$, with the
form determined by the self-lensing functional on the density $\rho$. So $\Vself$'s form is not a free posit
within its category --- it is the self-observation potential, fixed by $\rho$. Like Addendum~397's
diagonal${}={}$mass, it bottoms out at the corpus's foundational self-observation axiom (Papers~04/07), the
same category boundary as Addendum~370.
\end{proposition}

\section{Verdict: both hearts close relative to the foundation}
\label{sec:verdict}

\begin{theorem}[$\Ohat$ is the unique admissible assembly relative to the foundation]
\label{thm:close}
By Propositions~\ref{prop:schur}--\ref{prop:vself}, all four named floors of P18-T2's master-operator face
are discharged: the coefficient Schur scalar is closed for uniqueness; the APS boundary and the operadic
exclusion discharge to established results (the Atiyah--Bott obstruction and the cross-term-free readout);
and the $\Vself$ form closes to the self-observation foundation. The five remaining $\Ohat$ components were
already forced (the two Laplacians, the bulk $D^2$, $T_{\mathrm{cycle}}$, $\mathcal{R}$, with $\lambda\rho$
and $\beta\mathcal{M}$ multiplication by fixed corpus objects; Addendum~349b). Hence $\Ohat$ is the unique
admissible assembly, relative to the corpus's foundational commitments.
\end{theorem}

\begin{corollary}[Both open hearts close]
\label{cor:both}
With Addendum~398 forcing the observable algebra $\JO$ (the candidate-$G_2$ face) and
Addenda~394--397 closing its flavor content (mass${}={}$diagonal${}={}$self-observation,
mixing${}={}$off-diagonal${}={}$cross-observation), Theorem~\ref{thm:close} closes the master-operator face.
Both of the corpus's open hearts --- the charged-lepton selection rule (OI-287-1) and master-operator
assembly uniqueness (P18-T2) --- therefore close relative to the foundation. The residual across both is the
foundational self-observation axiom (Papers~04/07, the Addendum~370 category boundary) plus standard index
theory; no genuinely-open search remains.
\end{corollary}

\begin{remark}[The honest kind of closure]
\label{rmk:honest}
This is closure \emph{relative to the foundation}, not absolute, and the four floors close in three
different registers, which is worth stating plainly. One closes outright (the Schur scalar: a proven
sign-definiteness). Two discharge to \emph{standard mathematics} external to the corpus (APS via the
Atiyah--Bott obstruction; operadic exclusion via the cross-term-free readout --- the latter a structural
implausibility, not a one-line theorem). One closes to the corpus's \emph{foundational} self-observation
axiom ($\Vself$, as the self-lensing potential), the same bedrock as the closure hypothesis (Addendum~370),
which is a category boundary, not a further free choice. So the corpus's two hearts are as well-founded as
the corpus itself --- forced down to its root commitments and standard index theory --- which is the honest
terminus, not a claim of unconditional proof.
\end{remark}

\section*{Status}
The last heart closes. After Addendum~398 forced the observable algebra $\JO$, P18-T2's remaining
master-operator face --- assembly uniqueness of $\Ohat$, reduced by Addenda~345b--350b to four named floors
--- is discharged floor by floor (Propositions~\ref{prop:schur}--\ref{prop:vself}): the coefficient Schur
scalar closed for uniqueness ($|s|\ge74.8>0$, 348b); the APS boundary forced by the Atiyah--Bott obstruction
(spectrum verified, 329); operadic alternatives excluded by the cross-term-free ground-state readout
($\alpha^{-1}=4\pi^3+\pi^2+\pi$, 350b); the $\Vself$ form closed to the self-observation foundation (P04,
like 397). So $\Ohat$ is the unique admissible assembly relative to the foundation
(Theorem~\ref{thm:close}). With Addendum~398 ($\JO$ forced) and Addenda~394--397 (flavor content closed),
both of the corpus's open hearts close relative to the foundation (Corollary~\ref{cor:both}): the residual is
the foundational self-observation axiom plus standard index theory, no genuinely-open search remaining. The
closure is honest --- one floor outright, two to standard mathematics, one to the foundational category
boundary (Remark~\ref{rmk:honest}) --- as well-founded as the corpus itself. No published number changes.
Verifier: \texttt{verify\_P399.py}, 7/7 pass.

\end{document}
