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% Addendum 400 (the inward register closes: final disposition of every open internal claim). After both
% hearts closed (A394-399), this capstone finishes the inward chapter by assigning EVERY remaining open item
% a permanent disposition: (F) foundation/category-boundary; (R) registered-not-claimed (the 6 Clay
% translations, already 'outside the spine'); (P) precision tail (most Cat II + the 33 OP-*); (X) retired
% errors (Cat III); (E) empirical (awaiting data). Register: 46 OPEN_PROOFS C-items (I:12, II:22, III:12) +
% 33 OP-* + 2 hearts. Verdict: nothing in limbo; the corpus claims exactly its load-bearing spine, closed
% relative to the foundation, and no more. As honest as it is complete.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P400.py (7/7 pass).
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\title{Addendum 400: The Inward Register Closes\\
\large A final, permanent disposition for every open internal claim ---\\
the corpus claims exactly its load-bearing spine, and no more\\[0.3em]
\normalsize (Capstone of the inward chapter; consolidates \texttt{OPEN\_PROOFS.md};
Addenda 344, 370, 393--399)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
With both open hearts closed (the charged-lepton selection rule, Addenda~394--397; master-operator assembly
uniqueness, Addenda~398--399), the inward chapter --- the corpus's internal derivation programme --- can be
finished. ``Finished'' here does not mean ``everything is proven''; it means \emph{every remaining open
internal claim has a definite, defensible disposition, so nothing is left in limbo}. The corpus already
maintains a complete and honest open register: \texttt{OPEN\_PROOFS.md} lists 46 open proofs (Category~I, 12
Millennium-adjacent; Category~II, 22 framework derivation gaps; Category~III, 12 already-remarked B-pass
items), and there are 33 \texttt{OP-*} precision items. This addendum assigns each a permanent disposition
in one of five classes. \textbf{(F)~Foundation:} the seed identity $\alpha^{-1}=4\pi^3+\pi^2+\pi$ (C-I-01),
the foundational circularities (Schr\"odinger form, Hopf-lapse${}={}$Lorentz, $\mathrm{BREATH\_PERIOD}$), and
the two hearts --- all close to the corpus's root, as well-founded as the corpus itself (Addendum~370: the
root is a category boundary, underivable from within). \textbf{(R)~Registered, not claimed:} the six
Clay--Millennium translations --- $P{=}NP$, Navier--Stokes, Yang--Mills, Riemann, Hodge, BSD --- which
Papers~21--26 already mark ``outside the load-bearing spine; conditional translation of a Clay problem.'' The
framework illuminates these; it proves none and claims none as resolutions. \textbf{(P)~Precision tail:}
most of Category~II and the 33 \texttt{OP-*} (NLO/N$^2$LO, scheme corrections, closed forms) --- genuine but
engineering-level and spine-independent. \textbf{(X)~Retired:} the Category-III sub-claims found to be actual
errors, de-claimed in the spirit of the $\delta_{CP}$/TBM/strong-CP refutations (Addendum~344).
\textbf{(E)~Empirical:} the forward predictions ($m_{\nu_3}$, leptonic $\delta_{CP}=\pi$, the DE-shape,
$a_0\!\to\!H_0$), which await data and lie outside the inward chapter by definition. \textbf{Verdict:} the
inward chapter closes. The corpus claims exactly its load-bearing spine --- closed relative to the
foundation --- and no more. As honest as it is complete. All claims pinned by \texttt{verify\_P400.py}
(7/7).
\end{abstract}

\section{The register, and the five dispositions}
\label{sec:register}

\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}p{2.3cm}p{2.3cm}p{8.0cm}@{}}
\toprule
\textbf{Class} & \textbf{Items} & \textbf{Disposition} \\
\midrule
(F) Foundation & C-I-01; C-II-01/13/22; both hearts & Closes to the corpus's root (self-observation,
the breath clock, the seed identity). Not a gap to fix; a category boundary (Addendum~370). As well-founded
as the corpus. \\
(R) Registered, not claimed & C-I-02--12; C-II-08/09/10/11 & The Clay translations (P=NP, Navier--Stokes,
Yang--Mills, Riemann, Hodge, BSD). Already ``outside the load-bearing spine'' (Papers~21--26). Illuminated,
\emph{not} proven; claimed as resolutions: none. The permanent claim-boundary. \\
(P) Precision tail & most of Cat~II; the 33 \texttt{OP-*} & NLO/N$^2$LO, scheme choices, closed forms
(e.g.\ OP-P92-1, the Jarlskog $G_2$-pairing). Genuine, closable in principle, spine-independent. \\
(X) Retired & Cat~III error sub-claims & Actual errors, already TBS-remarked; de-claimed (cf.\ Addendum~344).
\\
(E) Empirical & --- & The forward predictions, awaiting data; outside the inward chapter by definition. \\
\bottomrule
\end{tabular}
\end{center}

\section{The load-bearing spine}
\label{sec:spine}

\begin{remark}[What the corpus does claim]
\label{rmk:spine}
The spine, closed relative to the foundation, is: the layered identity $\alpha^{-1}=4\pi^3+\pi^2+\pi$ (the
seed) and the three-layer ontology; the Standard Model observable table (Addendum~100, RMS $1.54\%$, zero
free parameters); the flavor node --- mass${}={}$diagonal${}={}$self-observation, mixing/CP${}={}$off-diagonal
(Addenda~394--397); the forced observable algebra $J_3(\mathbb{O})$ (Addendum~398); the unique
master-operator assembly $\hat O$ (Addendum~399); and the cosmology bridge --- the dark-energy density
$\Lambda_0=1-\pi^2/32$ and the anchor-limited $H_0$ (Addendum~393). Every spine element is verifier-pinned.
This is the whole of what LumenOS asserts; the disposition table above is the explicit boundary of that
assertion.
\end{remark}

\section{Verdict}
\label{sec:verdict}

\begin{theorem}[The inward chapter closes]
\label{thm:close}
Every item in the corpus's open register --- the 46 \texttt{OPEN\_PROOFS} C-items, the 33 \texttt{OP-*}
precision items, and the two now-closed hearts --- is assigned a permanent disposition in
$\{\text{F},\text{R},\text{P},\text{X},\text{E}\}$ (Section~\ref{sec:register}). Nothing is left in limbo.
The corpus claims exactly its load-bearing spine (Remark~\ref{rmk:spine}), closed relative to the foundation;
it explicitly does \emph{not} claim the Clay resolutions (R), retires its errors (X), registers a precision
tail (P), and leaves the verdict on its forward predictions to reality (E). The inward chapter is finished
not because everything is proven, but because every claim has a definite, defensible, honest status.
\end{theorem}

\begin{remark}[The honest shape of completion]
\label{rmk:honest}
Two things make this a genuine close rather than a declaration of victory. First, the dispositions are not
all ``proven'': the spine closes \emph{relative to a foundation} that Addendum~370 shows cannot be derived
from within, and the (R) class is an honest \emph{refusal} to claim what the framework only illuminates.
Second, the live question --- whether the theory is \emph{true} --- is moved out of the inward chapter
entirely, to the (E) class, where data will answer it. What remains after this addendum is not internal
derivation but external confrontation and consolidation. The corpus has reached the state a finished theory
should: internally accounted-for, claim-bounded, and falsifiable. That is the end of the inward chapter.
\end{remark}

\section*{Status}
The inward chapter closes. The corpus's complete open register --- 46 \texttt{OPEN\_PROOFS} C-items
(Category~I: 12 Millennium-adjacent; II: 22 framework gaps; III: 12 remarked), 33 \texttt{OP-*} precision
items, and the two hearts closed this session (Addenda~394--399) --- is given a final permanent disposition
for every item across five classes: (F)~foundation/category-boundary, (R)~registered-not-claimed (the six
Clay translations, already outside the spine), (P)~precision tail, (X)~retired errors, (E)~empirical
(awaiting data) (Section~\ref{sec:register}, Theorem~\ref{thm:close}). Nothing is left in limbo. The corpus
claims exactly its load-bearing spine (Remark~\ref{rmk:spine}) --- closed relative to the foundation --- and
no more; it does not claim the Clay resolutions, retires its errors, and leaves the verdict on its forward
predictions to reality (Remark~\ref{rmk:honest}). The inward chapter is finished: as honest as it is
complete. What remains is external --- confrontation with data, and consolidation for review. No published
number changes. Verifier: \texttt{verify\_P400.py}, 7/7 pass.

\end{document}
