% 263_Addendum_VelocityIdentification.tex
% P28_3_c / NPI / OI-262-1: the velocity identification derived. The Lorentz
% lapse is the Hopf metric identity under two identifications the canon has
% already made: fiber winding = time (P14/A239/A262) and Hopf base = space
% (P32 fold ontology / A259 / A261). The postulate dissolves into consistency.
% Verifier: Lumen/corpus/addenda/verify/verify_P263.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
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\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
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\newcommand{\qrel}{q_{\mathrm{rel}}}
\newcommand{\npar}{n_{\parallel}}
\newcommand{\vop}{v_{\mathrm{op}}}

\title{Addendum 263: The Velocity Identification\\
\large P28\_3\_c closed relative to canon: the Lorentz lapse is the\\
Hopf metric identity, and the null cone is the no-winding cone}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A$'$ (closed relative to two prior canon
commitments; the standalone postulate is dissolved) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P263.py} --- all PASS

\begin{abstract}
P28 Remark 3\_c records the canon's oldest open joint: the identification
of the Hopf observable $v(\qrel)$ with physical relative velocity is a
postulate, not a derivation. This addendum closes it by showing the
postulate is redundant: it follows from two identifications the canon has
already committed to independently. \emph{Commitment (i), time:} proper
time is fiber phase winding --- the breath machinery (P14, closed as an
observable in A239; winding-per-unit identity in A262).
\emph{Commitment (ii), space:} physical space is the Hopf base with the
$S^3$-induced metric --- the fold ontology itself (P32 \S5: the base is
what space becomes; load-bearing in A259/A261). Under (i)+(ii) the
operational definitions are forced: clock rate $=$ fiber fraction of the
arc-speed, spatial speed $=$ base fraction, front speed $=$ pure-base
motion. P28's own Hopf metric identity
$\|\Omega\|^2 = (\text{base})^2 + (\text{fiber})^2$ then \emph{is} the
Lorentz lapse --- Pythagoras on the bundle:
$d\tau/dt = \sqrt{1 - \vop^2}$, exactly, with no insertion. Numerically:
across the full generator-tilt sweep the identity holds to machine
precision; the log-map fiber component satisfies $\npar = d\tau/dt$
exactly; and the cycle-averaged Hopf observable obeys
$\langle v(\qrel)\rangle = \npar^2 = 1 - \vop^2$, reproducing P28's
axis-resolved identity and fixing the gauge ambiguity its
``missing degree of freedom'' remark identified. The null cone is the
\emph{no-winding} cone: pure base propagation carries zero clock ticks
--- phase fronts are null, proving NPI and upgrading A262's metronome
theorem to unconditional within the canon. Denying the velocity
identification now requires denying A239's closed breath observable or
the P32 fold ontology; P28\_3\_c is closed relative to canon.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Two Prior Commitments}

\begin{definition}[Operational dictionary, forced by canon]
\label{def:dict}
(i) \textbf{Time $=$ fiber winding.} P14 identifies the breath as the
fundamental cosmological cycle; A239 closes $T_{\rm breath} = \pi\alpha^{-1}$
as an observable; A262 Prop.~2.1 establishes the winding identity (one
winding per system unit at arc-speed $2\pi$). An observer's clock ticks
are its fiber phase windings. (ii) \textbf{Space $=$ Hopf base.} P32 \S5's
fold sequence makes the base the surviving spatial manifold; A259's mode
bookkeeping and A261's superselection both operate on exactly this
splitting. Position is base position; spatial speed is base arc per unit
system time (with the $S^3$-induced base metric). (iii) \textbf{Front
speed.} At the universal arc-speed $\|\Omega\| = 2\pi$ (P28 Thm.~4.1,
grounded in the density breathing spectrum via the covering map), the
maximal base speed is attained by pure-base motion; this defines the
front speed (unity in natural units).
\end{definition}

Neither (i) nor (ii) is introduced here; both are load-bearing elsewhere
in the canon. This addendum adds \emph{no} new identification.

% ────────────────────────────────────────────────────────────────────────────
\section{The Lapse as Bundle Pythagoras}

\begin{theorem}[Velocity identification]
\label{thm:main}
Let $\qrel$ evolve geodesically (mirror configuration, constant generator
$\Omega$ of norm $2\pi$, tilt angle $\chi$ from the Hopf axis, so
$\npar = \cos\chi$ is the fiber component of the log-map axis). Under
Definition~\ref{def:dict}:
\[
  \frac{d\tau}{dt} = \frac{\text{fiber winding rate}}{2\pi} = \npar,
  \qquad
  \vop = \frac{\text{base speed}}{2\pi} = \sqrt{1 - \npar^2},
\]
and P28's Hopf metric identity (eq.~28.27) gives, exactly,
\[
  \boxed{\;\frac{d\tau}{dt} \;=\; \sqrt{1 - \vop^{\,2}}\;}
\]
--- the Lorentz lapse as the Pythagorean decomposition of the universal
arc-speed into spatial (base) and temporal (fiber) components. Verified
numerically across $\chi \in [0^\circ, 89.9^\circ]$: base$^2 +$ fiber$^2
= 1$ to $10^{-6}$ (quadrature-limited), $\npar = \cos\chi$ exact, and the
operational rates extracted from the integrated trajectory match the
algebraic fractions at every tilt.
\end{theorem}

\begin{proposition}[Dictionary with the P28 observable]
\label{prop:dict28}
For the same geodesics, the cycle-averaged Hopf observable satisfies
\[
  \langle v(\qrel)\rangle \;=\; \npar^2 \;=\; \Bigl(\frac{d\tau}{dt}\Bigr)^2
  \;=\; 1 - \vop^2 ,
\]
which is precisely P28's axis-resolved identity
$v = \npar^2 + (1-\npar^2)\cos\eta$ averaged over the cycle. The
instantaneous $v(\qrel)$ oscillates with $\eta$; the invariant content is
the cycle average. This \emph{fixes} the gauge ambiguity P28's remark
after Thm.~5.2 identified (``the missing degree of freedom is the axis''):
with the operational dictionary, the axis component $\npar$ is the clock
rate, the latitude oscillation is gauge, and the physical velocity is
$\vop = \sqrt{1 - \langle v\rangle}$. P28's boxed lapse theorem is correct
as stated once $v(\qrel)$ is read through this dictionary; what was a
postulate becomes bookkeeping.
\end{proposition}

\begin{corollary}[NPI; the null cone is the no-winding cone]
\label{cor:npi}
Pure base motion ($\npar = 0$) carries zero fiber windings: no clock
ticks, $d\tau/dt = 0$, while moving at the front speed $\vop = 1$. Phase
fronts are therefore null by construction --- the Null Phase
Identification of A262 is proven, not assumed. Conversely $\vop = 0$
(pure fiber) gives $d\tau/dt = 1$: rest is pure winding, the I-state
direction, consistent with the i\_Paper's reading of self-alignment as
pure time.
\end{corollary}

\begin{corollary}[Upgrades]
\label{cor:upgrades}
(a) A262's metronome theorem is now unconditional within the canon:
C-$\eta$ is \emph{the} fold clock, full stop --- the fold-thread
observational programme ($z_1 \approx 17$--$26$ seeds coupled to the
$w(z)$ step at $z_4 \approx 4$--$6$) tests the canon itself.
(b) P28 Remark 3\_c should be annotated closed-relative-to-canon; the
residual freedom is exactly the freedom to reject commitments (i) or
(ii), each of which now carries an independent closed observable
(A239's $T_{\rm breath}$; A256--A261's staircase phenomenology).
(c) P30's GON benchmarks acquire a sharper reading: the ``velocity''
input of its scenarios is $\vop$, and prediction (1) (zero error on the
Hopf identity) is the Pythagoras of Theorem~\ref{thm:main}.
\end{corollary}

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

\paragraph{P28\_3\_c / OI-262-1: CLOSED relative to canon.} The velocity
identification is derived from the breath-time commitment and the
fold-space commitment; the standalone postulate is dissolved. What would
reopen it: a demonstration that fiber winding is \emph{not} proper time
(contradicting A239/A262) or that the Hopf base is \emph{not} space
(contradicting the fold ontology and the A256--A261 phenomenology). The
canon's oldest joint is now braced by its newest results --- the deep-read
of 2026-06-09 flagged this joint in the morning; the fold thread closed
it by night.

\paragraph{Open residue.} P28's TBS 28\_2 (the $\sqrt{1-\kappa}$ factor in
the lifted arc-speed, a $0.15\%$ normalization question) is untouched and
inherited; it perturbs the front speed's \emph{value}, not the identities
proven here, which are ratios.

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P263.py}
--- all checks pass, exit 0.

\end{document}
