% 260_Addendum_FoldClock.tex
% TASK_031 / OI-256-1: deriving the fold clock. Two corpus readings, the
% conditional selection of C-eta, and the unit-anchor reduction to the
% B4-scale problem with A259's conditional closure.
% Verifier: Lumen/corpus/addenda/verify/verify_P260.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\newtheorem{theorem}{Theorem}[section]
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\newcommand{\zfour}{z_4}
\newcommand{\Rc}{R_{\rm curv}}

\title{Addendum 260: The Fold Clock\\
\large Two corpus readings, conditional selection of C-$\eta$,\\
and the reduction of the unit problem}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (clock selected conditionally; unit problem
reduced, not closed) \quad|\quad \textbf{Task:} TASK\_031 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P260.py} --- all PASS

\begin{abstract}
A256 established empirically that fold completions are equally spaced in
their clock and left the clock's identity open (OI-256-1). This addendum
derives what the corpus can and cannot say about it. \emph{First}, the
corpus supports exactly two readings of fold pacing: the P32 fold-rate
remark (interior observer counts completed folds as proper time) selects
cosmic time --- the C-$t$ map, viable only in A256's untestable
$\zfour \ge 8$ tail --- while the P28 dual-observer construction, in which
all phase propagation on $S^3$ is angular and hence uniform in conformal
time, selects C-$\eta$. The $\ln a$ clock has no corpus reading at all:
e-fold counting appears nowhere in the canon as a dynamical variable.
\emph{Hence}: if the fold clock is derivable from existing corpus
structure, \textbf{C-$\eta$ is selected}, and the C-$\ln a$ survivor's
PBH/SIGW predictions become tests of new structure rather than of the
canon. \emph{Second}, the absolute calibration of any fold clock reduces to
a single unknown length: the curvature radius of the spatial $S^3$, which
is the same unknown as the $B^4$ physical scale (P04, open). One unit
problem, not two. \emph{Third}, A259's $\alpha^6$ candidate conditionally
closes it: $\Rc = 7.5\times10^{4}$ Mpc gives the universe's present
conformal age as $\chi_0 = \eta_0/\Rc \approx 0.19$ rad of $S^3$ arc ---
the fold history is then a measured angular traversal, and each fold
quarter subtends a computable arc. All numerics certified by
\texttt{verify\_P260.py}.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Two Readings, One Selection}

\begin{proposition}[Reading 1: fold-rate proper time $\Rightarrow$ C-$t$]
\label{prop:r1}
P32's fold-rate remark (\S Octonionic Correspondence) assigns the interior
observer a proper time that counts completed folds. For a comoving
cosmological observer, proper time is cosmic time $t$; uniform fold ticks in
interior proper time give $t(J_1) = \tfrac12 t(J_2)$ --- the C-$t$ map.
A256's grid (checks 27, 33b of \texttt{verify\_P256.py}) shows C-$t$ places
the seeding epoch at $z_1 \in [9,13]$ for $\zfour \in [4,6]$ --- too late to
grow UHZ1-class objects --- and becomes viable only in the untestable
$\zfour \ge 8$ tail. Reading 1 is therefore either wrong or pushes the
framework into unfalsifiability.
\end{proposition}

\begin{proposition}[Reading 2: conformal phase $\Rightarrow$ C-$\eta$]
\label{prop:r2}
In P28, all dual-observer dynamics is phase propagation on $S^3$: the
intrinsic arc-speed is constant ($\|\Omega\| = 2\pi$, P28 \S numerical
experiments) and observables are functions of traversed arc. On an FRW
background with spatial slice $S^3$ of radius $\Rc$, equal arc intervals
$\Delta\chi$ correspond to equal conformal-time intervals
$\Delta\eta = \Rc\,\Delta\chi$. A fold completion --- a global phase-locked
event of the $S^3$ boundary --- paced uniformly in traversed arc is
therefore paced uniformly in $\eta$: the C-$\eta$ map.
\end{proposition}

\begin{theorem}[Conditional selection]
\label{thm:select}
The corpus admits exactly the two readings above; $\ln a$ (e-fold counting)
appears nowhere in the canon as a dynamical clock. Since Reading 1 is
confined to the untestable tail (Prop.~\ref{prop:r1}), the corpus-derivable
fold clock in the testable window is conformal:
\emph{if the fold clock is derivable from existing structure, C-$\eta$ is
selected.}
\end{theorem}

\begin{corollary}[Status of C-$\ln a$]
\label{cor:clna}
The C-$\ln a$ survivor of A256 requires clock structure the canon does not
currently contain (NEW THEORY flag). Its observational signatures --- the
$10^{24}$ g PBH band (A256), the mandatory 0.45--0.95 mHz SIGW spike
(A257), and the $\alpha^6$/$\Omega_k$ pinning (A259) --- remain valid as
stated, but a detection would now carry a stronger implication: it would
demonstrate new clock structure beyond the present corpus, not merely
confirm it. Conversely the canonical near-term tests of the
corpus-derivable branch are C-$\eta$'s: JWST seed demographics at
$z_1 \approx 17$--$26$ coupled to a $w(z)$ step at $\zfour \approx 4$--$6$.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{The Unit Reduction}

\begin{theorem}[One unit problem, not two]
\label{thm:reduce}
Absolute calibration of the fold clock requires converting system units to
SI. The corpus's clock constants ($T_{\rm breath} = \pi\alpha^{-1}$, A239;
the $\Theta$-cycle phase ledger, P28) are dimensionless or in system units;
the only corpus-native object that can carry an SI length is the radius
$\Rc$ of the spatial $S^3$ --- which is precisely the $B^4$ physical scale
whose absence P04 records as open (Schwarzschild connection). Under
Reading 2 the conversion is explicit: $\Delta\eta = \Rc\,\Delta\chi$, so
every fold-clock interval is an arc on $S^3$ scaled by $\Rc$. The fold-clock
unit problem and the $B^4$-scale problem are therefore the same problem.
\end{theorem}

\begin{remark}[Conditional closure via A259]
\label{rem:anchor}
A259's curvature pinning gives the conditional anchor: if the transfer
fraction is $s = \alpha^6$, then $\Rc = 7.5\times10^{4}$ Mpc. The
present-day conformal age is then a measured arc,
\[
  \chi_0 \;=\; \frac{\eta_0}{\Rc} \;\approx\; \frac{1.42\times10^{4}\ \rm
  Mpc}{7.5\times10^{4}\ \rm Mpc} \;\approx\; 0.19\ \text{rad},
\]
and with $\zfour = 5$ (mid-window) the four boundary folds each subtend
$\chi(\zfour)/4 \approx 0.022$ rad of $S^3$ arc. The proximity of this
per-fold arc to the edge fraction $\pi/\alpha^{-1} = 0.0229$ is recorded as
an unexplored observation, not a claim (PSLQ-honesty convention); it is
exact only at one point of the $\zfour$ window. A failed $\Omega_k$ test of
A259 (Rem.~3.2 there) voids this anchor without affecting
Theorems~\ref{thm:select} and \ref{thm:reduce}.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Verdict and Open Items}

\paragraph{OI-256-1 verdict: reduced and conditionally selected
(Case B$+$).} The clock is C-$\eta$ if corpus-derivable
(Thm.~\ref{thm:select}); its absolute units are the $B^4$ scale
(Thm.~\ref{thm:reduce}); the scale is conditionally $7.5\times10^4$ Mpc via
A259's $\alpha^6$ candidate (Rem.~\ref{rem:anchor}). Full closure requires
either the OI-259-1 transfer rule (which would simultaneously fix $s$,
$\Omega_k$, $\Rc$, and the clock unit --- four locks, one key) or an
independent derivation of $\Rc$.

\paragraph{Open items.}
\begin{enumerate}
  \item \textbf{OI-260-1.} Derive the fold-completion pacing from the
        kernel (why uniform arc per fold --- candidate: the Wheel's
        Plateau/Oscillate alternation on the $\Delta_{S^1}$ sector as the
        fold metronome). This would upgrade Thm.~\ref{thm:select} from
        conditional to unconditional.
  \item \textbf{OI-260-2.} The per-fold arc vs.\ edge-fraction proximity
        (Rem.~\ref{rem:anchor}): coincidence audit once $\zfour$ is
        measured.
\end{enumerate}

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P260.py}
--- all checks pass, exit 0. Task: \texttt{Codex/Write/TASK\_031} (executed
by Cowork directly, 2026-06-09).

\end{document}
