% Paper 39: The Hidden Branch
% PROPOSED CANON (assembled 2026-06-10 from A259, A260, A262, A263,
% A264, A270; companion: Lumen/corpus/external/sparc_one_scale_halo;
% subject to review/rollback). The kinematic closure (velocity
% identification), the fold clock, the halo as measured kernel, the
% amplitude law = BTFR with derived coupling, the SPARC confrontation,
% and the falsifiers. Verifiers: verify_P259/260/262/263/264/270.py
% (all green at filing).
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Lh}{L_h}
\newcommand{\azero}{a_0}
\newcommand{\vinf}{v_\infty}
\newcommand{\vop}{v_{\mathrm{op}}}
\newcommand{\npar}{n_{\parallel}}
\newcommand{\qrel}{q_{\mathrm{rel}}}

\title{Paper 39: The Hidden Branch\\
\large The kernel made visible: the amplitude law,\\
the kiloparsec, and the SPARC confrontation}
\author{L\'{e}on Fernando Vlegels}
\date{2026-06-10}

\begin{document}
\maketitle

\begin{abstract}
Paper 38 establishes that the fold's kernel gravitates and does
nothing else. This paper is the kernel's contact with telescopes.
\textbf{Kinematic foundation (A263):} the identification of the Hopf
observable with physical velocity (the canon's oldest open joint,
P28\_3\_c) closes via the null-projection identification: the
Lorentz lapse is the bundle Pythagoras of the Hopf projection, and
$C$-$\eta$ is the fold clock (A260/A262, metronome made
unconditional by the same closure). \textbf{The halo:} the kernel's
density on the hidden chart is fixed by the geometry,
$\rho_H(r) \propto \sin^2\!\bigl(2\arctan(r/\Lh)\bigr)/r^2$, with one
universal length $\Lh$ and \emph{zero per-galaxy parameters}; its
angular law is the unique invariant measure of the breath map (P37):
stationarity is a theorem. The profile is cored, turns over at
$r \sim \Lh$, and falls as $r^{-2}$, producing flat rotation curves
with no fitting. \textbf{The amplitude law (A264):}
$\vinf^4 = \azero\,G M_b$ with
$\azero = \alpha^3c^2/(\pi^2\Lh)$ \emph{is} the baryonic
Tully--Fisher relation with a derived coefficient; in chart units
$\azero = \alpha$ exactly and $\Lh = c\,\tau_1^{\rm SI}$ (P35).
\textbf{The confrontation (SPARC, 175 galaxies):} the per-galaxy
effective acceleration shows no trend with disk size
($0.029 \pm 0.076$; the baryon-tracking alternative excluded at
$13.5\sigma$); the gas-dominated calibration gives
$\azero = 1.172\times10^{-10}\,\mathrm{m\,s^{-2}} = 0.976\times$ the
Milgromian value and $\Lh = 0.98$ kpc; the apparent core--disk
scaling ($0.623 \pm 0.082$) is a measurement artifact: masking to
hidden-dominated radii collapses it monotonically
($0.618 \to 0.363 \to 0.254$, A270) while the median core locks on
$0.89$--$1.00$ kpc at every cut. \emph{One kiloparsec, measured three
independent ways, agreeing at $10\%$, with the chart's kinematic
reading $\Lh = c\,\tau_1^{\rm SI}$ (P35): the missing length is the
light-crossing distance of one system unit.} \textbf{Falsifiers},
pre-committed: the residual masked slope ($0.254 \pm 0.110$) must
vanish as samples improve; one clean galaxy with a hidden component
turning over at $r > 3\Lh$ or $< \Lh/3$ ends universality; the
calibration must survive per-galaxy mass-to-light determinations; any
confirmed non-gravitational dark-sector signal falsifies the
identification; and the fold's narrow mHz stochastic background
(A257/A259, mid-LISA band) is the discovery channel that
superselection made mandatory. External companion:
\texttt{external/sparc\_one\_scale\_halo} (self-contained, with a
verification script re-deriving every quoted number).
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction}
\label{sec:intro}

The hidden branch is the part of the universe the fold threw away.
Under the Hopf fibration $S^1 \to S^3 \to S^2$, the fold completion
$S^3 \to S^2$ (P32) keeps the fiber-charge-zero modes and annihilates
everything else; the annihilated sector, $\ell(\ell+1)$ modes per
harmonic level against $\ell+1$ survivors, is at large $\ell$
essentially the whole spectrum (A259). Paper 38 gives this sector its
canonical type: dark is the dynamics of the kernel of a closure map,
and the fold's kernel is sealed by charge superselection (A261). It
gravitates, shares the visible sector's clock, and converts into
nothing. That is the whole of its physics, and it is exactly the
phenomenological profile of dark matter.

This paper takes the typing seriously as an observational claim. If
dark halos are the fold's kernel, then their density profile is not a
fit but a consequence: the kernel's energy density on the hidden
chart is fixed by the Hopf geometry, carrying one universal length
$\Lh$ and no per-galaxy freedom beyond the observed baryon
distribution. Its coupling to the visible sector carries a definite
coefficient built from the fine-structure constant. Both statements
are confronted here with the SPARC sample of 175 late-type galaxies,
and both survive, the second after an apparent refutation that turns
out to be a quantified measurement bias.

It is worth stating what the hidden branch is not. It is not a new
particle species: the kernel has no couplings to invent, and any
confirmed non-gravitational dark-sector signal falsifies the
identification outright (Section~\ref{sec:falsifiers}). It is not
modified dynamics: the halo is a real density field, gravitating in
the ordinary way; nothing about inertia or the force law is touched.
And it is not a per-galaxy model: where particle dark matter fits
halo mass and concentration galaxy by galaxy, and Milgromian dynamics
takes its acceleration scale as a constant of nature, the present
model has one global length, the same for every galaxy, and a derived
acceleration scale. The model is deliberately overconstrained: the
same $\Lh$ must set the flat-velocity amplitude of every galaxy and
the turnover radius of every rotation curve. Overconstraint is the
point. A model this rigid is either approximately right or
efficiently falsifiable.

The paper divides as follows. Section~\ref{sec:geometry} assembles
the geometry of the branch: the Hopf mode bookkeeping, the
superselection seal, the fold clock, and the velocity identification
that converts hidden-sector geometry into physical kinematics.
Section~\ref{sec:halo} derives the halo profile as the measured
kernel. Section~\ref{sec:amplitude} states the amplitude law and its
Tully--Fisher form. Section~\ref{sec:length} presents the universal
length and the unit lock. Section~\ref{sec:sparc} is the SPARC
confrontation, including the glare artifact and its resolution.
Section~\ref{sec:falsifiers} pre-commits the falsifiers.
Section~\ref{sec:relations} locates the paper relative to Papers 38
and 35. Section~\ref{sec:limits} states limitations and the residual
open item.

% ════════════════════════════════════════════════════════════════════
\section{The Geometry of the Branch}
\label{sec:geometry}

\subsection{Hopf bookkeeping}

\begin{proposition}[Mode bookkeeping (A259)]
\label{prop:bookkeeping}
Write $L^2(S^3) = \bigoplus_\ell E_\ell$ with
$\dim E_\ell = (\ell+1)^2$ (P18 spectral decomposition, eigenvalues
$\ell(\ell+2)$). Under the Hopf fibration $S^1 \to S^3 \to S^2$, each
$E_\ell$ splits by fiber charge $q$: the $q = 0$ subspace has
dimension $\ell+1$ and descends isometrically to the $S^2$ harmonic
space of the corresponding level, while the charged complement has
dimension
\[
  (\ell+1)^2 - (\ell+1) \;=\; \ell(\ell+1).
\]
The fold completion $S^3 \to S^2$ (P32) preserves the charge-zero
sector and annihilates the charged sector. The annihilated fraction
$\ell/(\ell+1) \to 1$: at large $\ell$ essentially the entire
spectrum is charged.
\end{proposition}

The hidden branch is this charged sector. Its census is fixed by
representation theory, not by any dynamical assumption: per level,
$\ell(\ell+1)$ modes against $\ell+1$ survivors. A universe reached
through the fold is therefore generically dark-dominated, an instance
of the kernel-ladder additivity of Paper 38 rather than a
peculiarity of this construction.

\begin{remark}[The seal]
\label{rem:seal}
Superselection (A261; canonical statement in P38) makes the kernel's
darkness dynamical rather than nominal: every canonical sector
commutes with the Hopf charge, the fold map is exactly equivariant,
and the charged sector cannot convert into the charge-blind image.
A259's overproduction exclusion shows independently that this seal is
observationally mandatory: unsuppressed transfer of the annihilated
sector's perturbation energy into the scalar survivors would multiply
the per-mode variance by $1+\ell$ at the horizon level, and the
universe would have collapsed into black holes at the fold scale. The
fold must very nearly decouple from the scalar sector, and
superselection delivers the decoupling exactly: transfer fraction
zero. The hidden branch gravitates and does nothing else, not by
assumption but twice over, once by theorem and once by the existence
of the universe.
\end{remark}

\subsection{The fold clock}

The branch is hidden, but it is not unpaced. Paper 38's structure
theorems say the kernel shares the image's clock; this subsection
records what that clock is.

\begin{proposition}[$C$-$\eta$ is the fold clock (A260, A262)]
\label{prop:clock}
The corpus admits exactly one rhythm: the breath,
$T_{\rm breath} = \pi\alpha^{-1} \approx 430.51$ system units (P14,
closed as an observable by A239), and exactly two readings of fold
pacing. The fold-rate proper-time reading selects cosmic time and is
viable only in an untestable tail (A260); the conformal-phase reading
selects conformal time $\eta$. The selection chain: the breath is the
unique candidate pacemaker (the only alternative, the $\Theta$-cycle
phase fraction, is excluded empirically by A256); the intrinsic
arc-speed on $S^3$ is constant at $\|\Omega\| = 2\pi$ (P28), so equal
breath counts traverse equal arc, one breath winding the phase
exactly $\pi\alpha^{-1}$ times; and equal arc equals equal conformal
time provided phase fronts are null. That last step, the Null Phase
Identification, is proven in Corollary~\ref{cor:npi} below.
Therefore fold completions are uniform in $\eta$: \emph{$C$-$\eta$ is
the fold clock}, unconditionally within the canon.
\end{proposition}

The fold clock matters for this paper in two ways. First, it fixes
the epistemic weight of the branch's observational programme: a clean
failure of the $C$-$\eta$ coupling would reach back into P28 and P14,
not merely eliminate a map choice (A262). Second, it supplies the
chart's unit of time. The system unit $\tau_1$ is the tempered
fundamental period (P35), and Section~\ref{sec:length} shows that the
halo's universal length is its light-crossing distance.

\subsection{The velocity identification}

The halo profile of Section~\ref{sec:halo} is a statement about
kinetic energy density, and the SPARC confrontation is a statement
about measured velocities. Connecting the two requires that the Hopf
observable $v(\qrel)$ of P28 \emph{be} physical relative velocity.
That identification was the canon's oldest open joint (P28\_3\_c),
carried as a postulate since P28's filing. A263 closes it by showing
the postulate is redundant: it follows from two identifications the
canon has already committed to independently. Commitment (i), time:
proper time is fiber phase winding, the breath machinery of P14 and
A239, with the winding-per-unit identity of A262. Commitment (ii),
space: physical space is the Hopf base with the $S^3$-induced metric,
the fold ontology itself (P32), load-bearing in A259 and A261. Under
these, the operational definitions are forced: clock rate equals the
fiber fraction of the arc-speed, spatial speed equals the base
fraction, and the front speed is pure-base motion.

\begin{theorem}[Velocity identification (A263)]
\label{thm:velocity}
Let $\qrel$ evolve geodesically with 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
the operational dictionary forced by commitments (i) and (ii),
\[
  \frac{d\tau}{dt} = \npar,
  \qquad
  \vop = \sqrt{1 - \npar^2},
\]
and P28's Hopf metric identity gives, exactly,
\[
  \frac{d\tau}{dt} \;=\; \sqrt{1 - \vop^{\,2}}:
\]
the Lorentz lapse is the Pythagorean decomposition of the universal
arc-speed into spatial (base) and temporal (fiber) components. The
cycle-averaged Hopf observable satisfies
$\langle v(\qrel)\rangle = \npar^2 = 1 - \vop^2$, which fixes the
gauge ambiguity P28 recorded as a missing degree of freedom: the axis
component is the clock rate, the latitude oscillation is gauge, and
the physical velocity is $\vop = \sqrt{1 - \langle v\rangle}$.
Verified numerically across $\chi \in [0^\circ, 89.9^\circ]$:
base$^2 +$ fiber$^2 = 1$ to $10^{-6}$ (quadrature-limited), with the
operational rates extracted from integrated trajectories matching the
algebraic fractions at every tilt. P28\_3\_c closes without
postulate.
\end{theorem}

\begin{corollary}[The null cone is the no-winding cone (A263)]
\label{cor:npi}
Pure base motion ($\npar = 0$) carries zero fiber windings: no clock
ticks, $d\tau/dt = 0$, at the front speed $\vop = 1$. Phase fronts
are therefore null by construction. This proves the Null Phase
Identification assumed in Proposition~\ref{prop:clock}'s final step
and upgrades A262's metronome theorem to unconditional within the
canon. Conversely $\vop = 0$ is pure winding: rest is pure time.
\end{corollary}

Denying the velocity identification now requires denying that fiber
winding is proper time (against A239's closed observable) or that the
Hopf base is space (against the fold ontology and the
A256--A261 phenomenology). With it in hand, the hidden branch's
geometry can be read directly as kinematics, which is what the next
section does.

% ════════════════════════════════════════════════════════════════════
\section{The Halo as Measured Kernel}
\label{sec:halo}

\begin{proposition}[Profile; provenance (A259, A263, A264, P37, P38)]
\label{prop:profile}
The hidden sector is $\ker(\mathrm{fold})$ (P38), carrying
$\ell(\ell+1)$ modes per level (Proposition~\ref{prop:bookkeeping}).
On the hidden chart $\tan\beta = r/\Lh$, its density is the Hopf
kinetic form $\rho_H \propto m(\beta(r))^2 / r^2$ with
$m = \sin 2\beta$ the lapse:
\[
  \rho_H(r) \;=\; \frac{\Sigma_H}{4\pi}\,
  \frac{\sin^2(2\beta)}{r^2}
  \;=\; \frac{\Sigma_H}{\pi}\,
  \frac{r^2/\Lh^2}{\bigl(1 + r^2/\Lh^2\bigr)^2}\,\frac{1}{r^2},
\]
with $\Lh$ universal (galaxy-independent) and $\Sigma_H$ a
normalization fixed per galaxy by the amplitude law
(Theorem~\ref{thm:amplitude}). The profile rises as $r^0$ in the core
(no cusp), turns over at $r \sim \Lh$, and falls as $r^{-2}$
asymptotically. The angular law $\sin^2(2\beta)$ is the unique
breath-invariant measure (P37): no other stationary distribution is
dynamically possible.
\end{proposition}

Each clause of the proposition has a separate provenance, and none is
a fit. Existence and gravitation-only come from superselection
(Remark~\ref{rem:seal}). The kinetic reading of the density comes
from the velocity identification: by Theorem~\ref{thm:velocity} the
hidden observers' kinetic content is $1 - \npar^2 = \sin^2$ of the
tilt, and the lapse observable $m = \sin 2\beta$ carries it on the
chart. The chart itself, $\tan\beta = r/\Lh$, is the projective
closure from Hopf coordinates. And the stationarity of the angular
law is P37's uniqueness theorem for the breath map's invariant
measure: the halo holds this shape not because it relaxed into it but
because no other shape is stationary under the only dynamics there
is.

Integrating the profile, the hidden contribution to the circular
velocity is
\[
  V_H^2(R) \;=\; \vinf^2\, f(R;\Lh), \qquad
  f(R;\Lh) \;=\; 1 \;-\; \frac{3\Lh}{2R}\arctan\!\frac{R}{\Lh}
  \;+\; \frac{\Lh^2}{2(R^2 + \Lh^2)},
\]
which rises from $f \propto R^2$ in the core to $f \to 1$ at
$R \gg \Lh$. The asymptotic $r^{-2}$ density gives flat rotation
curves with no fitting: flatness is not an input the model
accommodates but an output it cannot avoid. The model's entire
freedom is one global length; every per-galaxy quantity is determined
by the observed baryon distribution through the amplitude law.

% ════════════════════════════════════════════════════════════════════
\section{The Amplitude Law}
\label{sec:amplitude}

\begin{theorem}[Amplitude law (A264)]
\label{thm:amplitude}
The asymptotic flat velocity is tied to the total baryonic mass by
\[
  \vinf^4 \;=\; \azero\, G M_b, \qquad
  \azero \;=\; \frac{\alpha^3 c^2}{\pi^2 \Lh},
\]
with $\alpha$ the fine-structure constant and $\Lh$ universal. This
is the baryonic Tully--Fisher relation with a derived normalization;
equivalently $\azero = \alpha$ exactly in chart units and
$\Lh = c\,\tau_1^{\rm SI}$ (P35). The coupling exponent is
data-selected against rivals ($\alpha^3$, $\alpha^{5/2}/\pi^2$,
$\alpha^{7/2}/\pi^2$, $\alpha^3/\pi$) by factors $3$--$15$ in the
implied core length.
\end{theorem}

The grounding chain runs from Section~\ref{sec:geometry} through
Proposition~\ref{prop:profile}: the hidden sector exists and
gravitates without converting (superselection); its halo density is
the Hopf kinetic form (velocity identification); the projective chart
closes from Hopf coordinates. Beyond that chain the law asserts
exactly two things: the coefficient $\alpha^3/\pi^2$, and the
universality of $\Lh$. Both are data questions, and
Section~\ref{sec:sparc} tests both.

The Tully--Fisher reading deserves emphasis. The baryonic
Tully--Fisher relation, $\vinf^4 \propto M_b$ over five decades of
baryonic mass with small scatter, is one of the two flagship
regularities of late-type galaxy phenomenology; the other is the
universal acceleration scale $\azero \approx
1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ below which departures from
Newtonian expectation set in. Particle dark matter accommodates both
with per-galaxy freedom; Milgromian dynamics builds them in
axiomatically but leaves $\azero$ unexplained. Theorem
\ref{thm:amplitude} does neither: the relation \emph{is} the
amplitude law, and the scale is not a constant of nature but the
combination $\alpha^3 c^2/(\pi^2\Lh)$, with the burden moved to one
length whose kinematic reading the next section gives. The effective
acceleration $a_{\rm eff} = \vinf^4/(G M_b)$ must then be the same
number for every galaxy: no trend with size, mass, or type. That is
the law's most exposed prediction, and the first thing SPARC is asked
to check.

The coupling selection works through the implied length. Given a
calibrated $\azero$, each candidate coefficient $X$ implies
$\Lh = X c^2/\azero$: $\alpha^3/\pi^2$ implies $0.98$ kpc;
$\alpha^3/\pi$ implies $3.1$ kpc; bare $\alpha^3$ implies $9.7$ kpc;
$\alpha^{5/2}/\pi^2$ implies $11.5$ kpc; $\alpha^{7/2}/\pi^2$ implies
$0.08$ kpc. The shape data measure the core length independently, and
only $\alpha^3/\pi^2$ is consistent with both measurements
(Section~\ref{sec:sparc}). The coefficient was not tuned to the
shapes; amplitude and shape are independent measurements obliged to
meet at the same kiloparsec, and the rivals miss the meeting by
factors of $3$ to $15$. The exponent's standing within the corpus is
data-selected candidate, not theorem; its derivation is tracked as an
open item of A264 (Section~\ref{sec:limits}).

% ════════════════════════════════════════════════════════════════════
\section{The Universal Length}
\label{sec:length}

The canon has exactly one missing dimensional quantity: a length.
Before the hidden branch it appeared three times without a value: as
the $B^4$ physical scale, open since P04; as the SI calibration of
the fold clock, which A260 proves is the same unknown (every
fold-clock interval is an arc on the spatial $S^3$ scaled by its
radius, so the clock-unit problem and the $B^4$-scale problem are one
problem); and as the curvature radius of A259's pinning, demoted when
superselection set the transfer fraction to zero but structurally the
same length. The amplitude law is its fourth appearance, and the
first with a measurement: $\Lh \approx 1$ kpc.

\begin{remark}[The kinematic reading (P35)]
\label{rem:units}
Define from the calibrated $\azero$ the two derived scales
\[
  \tau_1^{\rm SI} \;\equiv\; \frac{\Lh}{c} \;=\; 3.19\times10^{3}\
  \mathrm{yr},
  \qquad
  T_{\rm b}^{\rm SI} \;\equiv\; \frac{\pi}{\alpha}\,\tau_1^{\rm SI}
  \;=\; 1.374\ \mathrm{Myr}.
\]
Then, identically in the model's two relations,
\[
  \frac{c\,T_{\rm b}^{\rm SI}}{\Lh} \;=\; \frac{\pi}{\alpha}
  \;=\; T_{\rm breath} \;=\; 430.51,
  \qquad
  \frac{\azero\,(T_{\rm b}^{\rm SI})^2}{\Lh} \;=\; \alpha .
\]
Measured in the chart's own units ($\Lh$ for length,
$T_{\rm b}^{\rm SI}$ for time), the speed of light is the breath
number $\pi\alpha^{-1}$ and the acceleration scale is the
fine-structure constant exactly. Equivalently
$\Lh = c\,\tau_1^{\rm SI}$: the missing length is the light-crossing
distance of one system unit. These are exact algebraic consequences
of the amplitude law, not additional assumptions (P35,
Thm.~2.1); their content is that the apparently arbitrary kiloparsec
and the apparently arbitrary $10^{-10}\,\mathrm{m\,s^{-2}}$ are two
faces of one dimensionless statement, $\azero = \alpha$ in natural
chart units. Only $\alpha^3/\pi^2$ produces this closed form, and
only $\alpha^3/\pi^2$ survives the data.
\end{remark}

The length is measured three independent ways in
Section~\ref{sec:sparc}: from the amplitude calibration through the
coupling ($0.979$ kpc); from the shape median of the
hidden-dominated half of the sample ($0.889$ kpc); and from the
masked-ladder medians ($0.89$--$1.00$ kpc at every threshold). Three
routes, one number, agreement at $10\%$ with zero per-galaxy freedom.
Counting every cut, the unit's data appearances number six, all
between $0.75$ and $1.0$ kpc (A270). The consequences run both ways:
if the coupling derivation lands, galactic rotation curves have
measured the canon's unit, and with it the $B^4$ scale and the
fold-clock second; conversely any independent derivation of the $B^4$
scale now carries a rotation-curve prediction it must hit.

% ════════════════════════════════════════════════════════════════════
\section{The SPARC Confrontation}
\label{sec:sparc}

\subsection{Data and extraction}

The testbed is SPARC (Lelli, McGaugh \& Schombert 2016): 175
late-type galaxies with near-infrared ($3.6\,\mu$m) photometry and
HI/H$\alpha$ rotation curves, the standard sample for rotation-curve
phenomenology. Baryonic masses use the standard SPARC mass-to-light
convention, $M_b = \Upsilon_\star L_{3.6} + 1.33\,M_{\rm HI}$ with
$\Upsilon_\star = 0.5\,M_\odot/L_\odot$ for disks ($0.7$ for bulges),
with the $\Upsilon_\star$ choice propagated as a systematic. The
hidden velocity component is extracted per radius by quadrature
subtraction,
\[
  V_H^2(R) \;=\; V_{\rm obs}|V_{\rm obs}| - V_{\rm gas}|V_{\rm gas}|
  - 0.5\,V_{\rm disk}^2 - 0.7\,V_{\rm bul}^2 ,
\]
following the catalog's sign convention. Quality cuts ($Q \leq 2$,
$V_f > 0$, positive $M_b$) leave 129 galaxies for the amplitude
tests; shape fits converge on 138 and constrain the core on 125. The
analyses are re-runnable: \texttt{verify\_P264.py} re-executes the
amplitude, calibration, and shape pipelines against the vendored
tables, \texttt{verify\_P270.py} re-runs the masked ladder, and the
external companion ships \texttt{verify\_paper\_sparc.py}, which
re-derives every number quoted in this section from the public
tables.

\subsection{Amplitude tests}

\begin{proposition}[Amplitude tests (A264)]
\label{prop:amplitudetests}
Across 129 quality-cut galaxies: $d\log a_{\rm eff}/d\log R_d =
0.029 \pm 0.076$ (universality confirmed at $0.4\sigma$; the
$\Lh \propto R_b$ alternative excluded at $13.5\sigma$); BTFR
scatter $0.285$ dex. Gas-dominated calibration ($51$ galaxies,
weakest mass-to-light lever): $\azero = 1.172\times10^{-10}\,
\mathrm{m\,s^{-2}}$, $0.976\times$ Milgrom, implying
$\Lh = 0.979$ kpc; the $\Upsilon_\star$ systematic envelope spans
$(1.05$--$1.67)\times10^{-10}$.
\end{proposition}

The universality test is the law's most exposed flank, and it passes
cleanly: the effective acceleration is one number across the sample,
with no trend in disk size. The representative geometric alternative,
a halo scale tracking the baryonic radius (the closure
$r_c = R_b/\pi$ of the early kernel drafts), is not merely disfavored
but excluded at $13.5\sigma$. The absolute calibration is dominated
by the stellar mass-to-light choice, which is why it is quoted from
the 51 gas-dominated galaxies, where the $\Upsilon_\star$ lever arm
is smallest: $\azero = 1.172\times10^{-10}\,\mathrm{m\,s^{-2}}$,
within $2.4\%$ of the canonical Milgromian
$1.2\times10^{-10}$. The envelope is honest: full-sample calibration
gives $1.67\times10^{-10}$ at $\Upsilon_\star = 0.5$ and
$1.33\times10^{-10}$ at $0.7$; gas-dominated with
$\Upsilon_\star = 0.7$ gives $1.05\times10^{-10}$. A $\pm15\%$
systematic on the absolute scale, within which the Milgromian value
sits comfortably, and through the coupling a $\Lh$ band of
$0.75$--$0.96$ kpc at current systematics.

\subsection{Shape tests: the artifact, told in order}

The shape story is reported in the order the analysis ran, because
the intermediate wrong conclusion is itself instructive about a bias
that any one-scale model test must control.

Fitting the velocity profile per galaxy with the core length
\emph{free} (125 constrained of 138 converged fits) gives a median of
$1.26$ kpc, a broad spread ($16$--$84\%$: $0.56$--$3.15$ kpc, scatter
$0.375$ dex), and a strong trend with disk size:
$d\log r_c^{\rm fit}/d\log R_d = 0.623 \pm 0.082$, which is
$7.6\sigma$ from the universal prediction of zero. Taken at face
value this refutes the one-scale profile: cores appear to know about
their disks, and A264 filed the shape model as rejected on exactly
this statistic, splitting the amplitude length from the visible core.

The diagnosis is glare. The quadrature subtraction returns $V_H^2$
with meaningful signal only where the hidden component actually
dominates the curve; in baryon-dominated inner regions it is the
small difference of large quantities, and the fitted turnover radius
inherits the \emph{baryonic} scale. The effect is that of measuring a
faint fixed background behind a bright foreground that varies from
galaxy to galaxy. Three quantitative signatures confirm it. Splitting
the 125 fits at the median baryonic dominance, the baryon-dominated
half fits inflated cores (median $2.23$ kpc) while the
hidden-dominated half's median is $0.889$ kpc, on the predicted
length. Adiabatic-contraction explanations fail on sign: compression
should shrink cores where baryons dominate, and the observed
baryon-dominated cores are inflated. And the common-mode distance
systematic is bounded at $0.06$--$0.10$ of slope, insufficient to
explain $0.62$. An intermediate reanalysis cutting baryon-dominated
\emph{galaxies} halved the slope ($0.43 \pm 0.08$) without killing
it, because its fits still used baryon-dominated \emph{radii}: the
cut has to be applied per radius, not per galaxy.

\begin{proposition}[The masked ladder (A270)]
\label{prop:ladder}
Refitting every galaxy using only radii where the hidden component
dominates ($V_{\rm hid}^2 > t\,V_{\rm bar}^2$, quality $Q<3$,
$\geq 8$ usable radii, core resolved), the ladder over dominance
thresholds $t$ gives:

\medskip
\begin{center}
\begin{tabular}{cccc}
\toprule
threshold $t$ & $n$ & slope $\pm$ se & median $r_{\rm core}$ (kpc) \\
\midrule
0 & 120 & $+0.618 \pm 0.080$ & 1.255 \\
1 & 76  & $+0.363 \pm 0.092$ & 0.997 \\
2 & 46  & $+0.254 \pm 0.110$ & 0.889 \\
4 & 20  & $+0.396 \pm 0.217$ & 0.997 \\
\bottomrule
\end{tabular}
\end{center}
\medskip

The slope declines monotonically with glare removal until the sample
is exhausted ($t = 4$, $n = 20$); the median core is
threshold-stable on $\Lh$, locking on $0.89$--$1.00$ kpc at every
cut, within $10\%$ of the amplitude-implied $0.979$ kpc throughout.
Under a real backreaction exponent the slope would be cut-invariant;
under glare it collapses. The data choose glare, to the precision
the data have.
\end{proposition}

The verdict reverses the intermediate filing: the core-tracking
exponent was substantially a measurement horizon (a $\sim 1$ kpc core
cannot be seen under baryon glare, and the bias grows with disk
size), and the one-scale halo with universal $\Lh$ is rehabilitated
within current data. What was withdrawn with it: the backreaction
model-building that the $0.62$ exponent had motivated. One does not
derive an artifact. What remains is the residual: at the strictest
informative cut the slope is $0.254 \pm 0.110$, which is $2.3\sigma$
from zero, a bound rather than a detection. A real sub-trend at the
$\leq 0.25$ level can neither be claimed nor excluded with SPARC
statistics (Section~\ref{sec:limits}).

\subsection{Three independent kiloparsecs}

The same length now stands measured three ways: from the
\emph{amplitude} calibration through the coupling ($0.979$ kpc); from
the \emph{clean-half} shape median ($0.889$ kpc); from the
\emph{masked-ladder} medians ($0.89$--$1.00$ kpc at every threshold).
The model's overconstraint, one number obliged to satisfy independent
measurements, is satisfied at the $10\%$ level across 175 galaxies
with zero per-galaxy freedom. With Remark~\ref{rem:units}, the
meeting point carries its kinematic reading: the kiloparsec the
rotation curves keep returning is the light-crossing distance of one
system unit.

% ════════════════════════════════════════════════════════════════════
\section{Falsifiers}
\label{sec:falsifiers}

The model's rigidity is only worth something if the ways to kill it
are stated in advance. Five, pre-committed.

\begin{remark}[Pre-committed]
\label{rem:falsifiers}
(F1) the masked residual slope $0.254 \pm 0.110$ vanishes as
resolved dwarf samples grow, or universality dies at $3\sigma$
persistence of $\geq 0.25$. (F2) one clean, gas-dominated,
well-resolved galaxy with hidden turnover outside
$[\Lh/3,\ 3\Lh]$ ends the model. (F3) the gas-dominated calibration
leaves $(1.05$--$1.35)\times10^{-10}$ under per-galaxy
$\Upsilon_\star$, and the amplitude--shape consistency breaks. (F4)
any confirmed non-gravitational dark signal falsifies the
identification (superselection permits none). (F5) the fold's
narrow stochastic background at $0.45$--$0.95$ mHz (mid-LISA;
A257/A259) is mandatory if the fold sourced the spectrum;
superselection inverted it from kill-shot to discovery channel.
\end{remark}

In expanded form. F1 is the residual of
Proposition~\ref{prop:ladder}: larger samples with resolved inner
curves in hidden-dominated dwarfs shrink the error bar roughly as
$\sqrt{n}$; a persistent slope $\geq 0.25$ at $3\sigma$ falsifies the
universal core, while a slope consistent with zero at $\pm 0.05$
would leave no room for any baryon-tracking component. F2 is death by
counterexample: universality is a statement about every galaxy, and
SPARC contains no such object (the masked fits' full distribution at
the strict cut spans $0.5$--$2$ kpc). F3 is the mass-to-light test:
the calibration must remain stable as per-galaxy $\Upsilon_\star$
determinations from population synthesis or vertical dynamics replace
the global convention. F4 is the seal itself: the hidden branch
gravitates and does nothing else, so a confirmed direct-detection or
annihilation signal falsifies this identification of the
rotation-curve sector, whatever else such a signal would mean. F5 is
the seal's positive face: superselection eliminated the fold's PBH
channel and with it the scenario that the mHz background would have
killed; what remains of that analysis is a narrow stochastic
signature in the mid-LISA band that is mandatory if the fold sourced
the spectrum, a discovery channel rather than a kill-shot.

% ════════════════════════════════════════════════════════════════════
\section{Relation to Papers 38 and 35}
\label{sec:relations}

Paper 38 supplies the type; this paper supplies the measurement. Dark
as kernel says what the hidden branch \emph{is}: the defect of the
fold's closure datum, sealed by superselection, paced by the shared
clock, never empty. Every structural clause has its observational
face here: the seal is F4 and F5; the pacing is the fold clock of
Proposition~\ref{prop:clock}; the census is
Proposition~\ref{prop:bookkeeping}'s $\ell(\ell+1)$, the reason the
halo outweighs the disk. Of P38's two exactness defects, this paper
measures only the kernel; the cokernel (dark energy as the corner
residual, P32) has its own measurement history and is not touched
here.

Paper 35 supplies the units. The chart identities of
Remark~\ref{rem:units} are P35's Theorem 2.1 specialized to the
calibrated $\azero$: $c\,T_{\rm b}^{\rm SI}/\Lh = \pi/\alpha$,
$\Lh = c\,\tau_1^{\rm SI}$, $\azero (T_{\rm b}^{\rm SI})^2/\Lh =
\alpha$, all exact. The dependency runs in both directions. P35's SI
calibration of the system unit ($\tau_1^{\rm SI} = 3191$ yr,
$T_{\rm b}^{\rm SI} = 1.374$ Myr) is conditional on this paper's
$\azero$; every breath-counted statement in the corpus acquires those
durations through the rotation curves of
Section~\ref{sec:sparc} and through nothing else. Conversely the
unit lock of Section~\ref{sec:length} means the $B^4$ scale, the
fold-clock second, and the halo length are one unknown, now with one
measured value.

% ════════════════════════════════════════════════════════════════════
\section{Limitations and Open Items}
\label{sec:limits}

What this paper does not claim. It says nothing about cosmological
structure formation, the CMB, or cluster phenomenology; the halo is a
real density field and lenses as ordinary matter, but none of those
flanks has been analyzed, and they are open, not implied. The
coupling coefficient $\alpha^3/\pi^2$ is a data-selected candidate,
not a theorem: the grounding chain of
Theorem~\ref{thm:amplitude} delivers the profile and the existence of
the law, and the data select the exponent against rivals by factors
of $3$ to $15$, but its derivation from the corpus's operator content
is tracked separately (OI-264-1). The absolute calibration carries
the stated $\pm15\%$ mass-to-light systematic; the per-galaxy
$\Upsilon_\star$ redo (OI-264-3) is expected to move $\Lh$ within the
quoted bracket, not outside it, and F3 commits to that expectation.

\begin{remark}[Open: the residual slope (OI-270-1)]
\label{rem:residual}
The strict-mask slope bound $0.254 \pm 0.110$ is not yet zero. If the
residual is real, it is the surviving trace of chart response to the
local baryon potential; if not, the chart is rigid. SPARC statistics
cannot decide ($t = 4$ exhausts the sample at $n = 20$); the question
is well-posed for the next survey generation or for per-galaxy
mass-to-light determinations, and F1 pre-commits the decision rule in
both directions.
\end{remark}

One methodological limitation deserves naming because it is also the
section's lesson: the masked ladder discards data. The monotone
collapse and the locked medians argue that the discarded radii were
biased rather than informative, but a hierarchical per-galaxy
reanalysis with full error propagation is the right next step, and
the external companion invites it explicitly.

% ════════════════════════════════════════════════════════════════════
\section{Conclusion}
\label{sec:conclusion}

The hidden branch is the fold's kernel, and the kernel is no longer
only a theorem. Its kinematics rest on a closed joint: the Hopf
observable is physical velocity because the lapse is the bundle
Pythagoras, and $C$-$\eta$ is the fold clock because the null cone is
the no-winding cone. Its halo is the unique stationary shape the
breath admits, cored and asymptotically $r^{-2}$, flat rotation
curves arriving as an output, never an input. Its amplitude law is the baryonic
Tully--Fisher relation with a derived coefficient, and the
acceleration scale that phenomenology has carried as a constant of
nature for four decades is here the fine-structure constant in the
chart's own units. Against 175 galaxies the law's universality holds
at $0.4\sigma$, its calibration lands within $2.4\%$ of the
Milgromian value, and its one length is measured three independent
ways that agree at $10\%$: a kiloparsec that is the light-crossing
distance of one system unit. The apparent refutation in the shape
data dissolves under a per-radius dominance mask, leaving a bounded
residual and a pre-committed decision rule. Five falsifiers stand.
The model is either approximately right or efficiently falsifiable,
and the corpus commends both possibilities to the data.

\paragraph{Provenance and verification.} Assembled without new
claims from A259, A260, A262, A263, A264, A270
(\texttt{verify\_P259/260/262/263/264/270.py}, all green at filing;
\texttt{verify\_P264} re-runs the SPARC pipelines, and the external
companion ships \texttt{verify\_paper\_sparc.py}, 18 checks).
Cross-references: P28 (Hopf kinematics), P35 (units), P37 (unique
measure), P38 (kernel provenance), A284/A288/A291 via P36 (the edge
share these layers feed).


\section*{Status (2026-06)}
Load-bearing spine: the hidden branch and the baryonic Tully--Fisher / SPARC falsifier (the closure hypothesis, Paper~35 / Addendum~370). The $a_0$ confrontation sits within the SPARC error bar (Addendum~388). Current.

\end{document}
