% 262_Addendum_BreathMetronome.tex
% OI-260-1: the breath as fold metronome. Upgrades A260's conditional
% selection of C-eta to selection-modulo-one-identification, and names the
% identification precisely.
% Verifier: Lumen/corpus/addenda/verify/verify_P262.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\title{Addendum 262: The Breath as Fold Metronome\\
\large OI-260-1: fold pacing from the corpus's unique rhythm,\\
and the one identification that remains}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A$'$ (selection upgraded from
conditional-on-derivability to conditional-on-one-named-identification)
\quad|\quad \textbf{Verifier:} \texttt{verify\_P262.py} --- all PASS

\begin{abstract}
A260 selected the conformal fold clock C-$\eta$ \emph{conditionally on the
fold clock being corpus-derivable}, leaving OI-260-1: derive the pacing.
This addendum supplies the derivation chain and isolates its single
remaining assumption. \emph{Step 1 (uniqueness of rhythm):} the corpus
admits exactly one rhythm --- the breath, $T_{\rm breath} = \pi\alpha^{-1}$
--- closed as a cosmological observable (P14, A239) and constitutionally
unique (Pin: no subsystem generates its own rhythm). If fold completions
have a corpus pacemaker at all, it is breath counting; A256 already
excluded the only alternative (the $\Theta$-fraction ledger).
\emph{Step 2 (breath $\to$ arc):} P28's intrinsic arc-speed is constant
($\|\Omega\| = 2\pi$), so equal breath counts traverse equal arc; one
breath winds the phase exactly $\pi\alpha^{-1}$ times (the winding count
\emph{is} $T_{\rm breath}$, an identity verified to machine precision).
\emph{Step 3 (arc $\to$ conformal time):} on an FRW background with
spatial slice $S^3$, equal arc equals equal conformal time \emph{provided}
phase propagation defines the causal structure --- i.e.\ phase fronts are
null. This is the load-bearing identification: P28 derives the Lorentz
lapse from phase projection, which makes phase geometry the \emph{origin}
of causal structure, but the corpus nowhere states the null identification
as a theorem. With it, fold completions paced by breath count are uniform
in $\eta$: \textbf{C-$\eta$ is selected}, and A260's conditional theorem
upgrades to conditional on exactly one named statement (the Null Phase
Identification, NPI), graded NEW THEORY-adjacent: strongly motivated by
P28, not yet proven. The observational programme is unchanged
($z_1 \approx 17$--$26$ seeds $\leftrightarrow$ $w(z)$ step at
$z_4 \approx 4$--$6$); what changes is that a failed C-$\eta$ test would
now falsify NPI or the uniqueness-of-rhythm reading, not merely a map
choice. All arithmetic certified by \texttt{verify\_P262.py}.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Uniqueness of Rhythm}

\begin{proposition}[The breath is the only candidate pacemaker]
\label{prop:unique}
The corpus contains exactly one periodic structure: the breath,
$T_{\rm breath} = \pi\alpha^{-1} \approx 430.51$ system units, identified
as the fundamental cosmological cycle by P14 and closed as an observable
by A239. The constitutional architecture mirrors this (every subsystem
synchronises to $T_{\rm breath}$; none generates its own rhythm), and the
mirror is not decorative: the OS pins are transcriptions of TOE structure.
The only other candidate ever proposed for fold pacing --- the
$\Theta$-cycle phase fraction --- was excluded empirically by A256
(Cor.~3.4: ledger, not pacemaker) in every clock variable. Therefore: if
fold completions are paced by corpus structure at all, they are paced by
breath count, i.e.\ uniformly in the breath's clock.
\end{proposition}

\begin{remark}
The hypothesis ``paced by corpus structure at all'' is not vacuous --- a
fold sequence paced by nothing (random completion epochs) is logically
possible but would abandon the A256 result that equal-fold spacing fits
the data window; equal spacing in \emph{some} clock is what the staircase
phenomenology already requires.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Breath to Arc}

\begin{proposition}[Equal breaths, equal arc]
\label{prop:arc}
P28's numerical experiments fix the intrinsic arc-speed on $S^3$ at
$\|\Omega\| = 2\pi$ per system unit, independent of $\alpha$. Hence one
breath traverses $2\pi \cdot T_{\rm breath}$ radians of phase --- exactly
$\pi\alpha^{-1}$ full windings; the winding count per breath equals
$T_{\rm breath}$ numerically, an identity rather than a coincidence
(arc-speed $2\pi$ makes ``windings per unit time'' $= 1$). Equal breath
counts therefore traverse equal arc, with no $\alpha$-dependent drift.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Arc to Conformal Time: the One Identification}

\begin{theorem}[Conditional metronome theorem]
\label{thm:metronome}
Assume the \textbf{Null Phase Identification (NPI)}: phase fronts of the
dual-observer construction propagate on the null cones of the emergent
causal structure. Then on an FRW background with spatial slice $S^3$ of
radius $R_{\rm curv}$, an arc interval $\Delta\chi$ corresponds to a
conformal interval $\Delta\eta = R_{\rm curv}\,\Delta\chi$, and
Propositions~\ref{prop:unique}--\ref{prop:arc} give: fold completions are
uniform in conformal time. \textbf{C-$\eta$ is selected.}
\end{theorem}

\begin{remark}[Status of NPI]
\label{rem:npi}
NPI is the single load-bearing assumption. Its support: P28 \emph{derives}
the Lorentz lapse $d\tau/dt = \sqrt{1 - v^2}$ from phase projection ---
causal structure in the corpus is an output of phase geometry, not an
input --- and the lapse vanishes exactly where phase advance saturates,
which is the defining behaviour of a null structure. Its gap: the corpus
nowhere proves that the phase fronts coincide with the null cones of the
metric that P20 embeds; P28's open remark P28\_3\_c (the
Hopf-observable-to-physical-velocity postulate, flagged in the 2026-06-09
corpus deep-read as the canon's deepest unproven joint) is exactly the
statement NPI would formalize. NPI is therefore graded: strongly
motivated, unproven, and now \emph{load-bearing for cosmology} ---
which promotes resolving P28\_3\_c from a foundational nicety to the
highest-leverage open derivation in the canon.
\end{remark}

\begin{corollary}[Upgrade of A260; sharpened falsification]
\label{cor:upgrade}
A260 Thm.~1.3 (``if the fold clock is corpus-derivable, C-$\eta$'')
upgrades to: \emph{C-$\eta$ holds modulo NPI alone}. The observational
programme is unchanged --- heavy seeds at $z_1 \approx 17$--$26$ coupled
to a $w(z)$ step at $z_4 \approx 4$--$6$ --- but its epistemic weight
increases: a clean failure of the C-$\eta$ coupling would now falsify NPI
or Proposition~\ref{prop:unique}'s reading, i.e.\ it would reach back into
P28/P14 rather than merely eliminating a map. Conversely a confirmed
coupling would be indirect evidence for NPI, hence for the phase-geometric
origin of causal structure --- the corpus's central interpretive claim.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{Status}

\paragraph{OI-260-1: closed to a point.} The pacing is derived modulo one
named statement (NPI $\equiv$ formalized P28\_3\_c). Full unconditional
closure is now exactly the closure of P28\_3\_c, filed as
\textbf{OI-262-1} --- and that item predates this thread: it is the
velocity-identification gap the corpus has carried since P28. The fold
thread has, fittingly, terminated on the canon's oldest open joint.

\emph{Upgraded (2026-06-09, same day; A263).} OI-262-1 is closed: A263
proves NPI from the bundle Pythagoras under the canon's two prior
commitments (winding $=$ time; base $=$ space). Theorem~3.1 here is now
unconditional within the canon: \textbf{C-$\eta$ is the fold clock.}

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P262.py}
--- all checks pass, exit 0.

\end{document}
