% r_Paper_Reader.tex — The Reader's Companion
% Stands outside the numbered corpus, like the i_Paper, and mirrors
% its instruction: the i_Paper says "read this before Paper 00";
% this one says "read this before anything." A narrative exposition
% for a reader with no corpus knowledge — a physicist, a
% mathematician, or a patient layperson. It contains NO new claims:
% every assertion here is a retelling of a numbered paper or a
% verified addendum, with pointers. Assembled 2026-06-10.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb}
\usepackage[margin=1.1in]{geometry}
\usepackage{hyperref}
\setlength{\parskip}{0.45em}

\newcommand{\canon}[1]{\textsuperscript{\,[#1]}}

\title{The Reader's Companion\\
\large One number, and what it cost to take it seriously}
\author{L\'eon Fernando Vlegels}
\date{2026-06}

\begin{document}
\maketitle

\begin{abstract}
\noindent This document is the front door to a corpus of forty-one
papers and several hundred machine-verified results. It assumes
nothing: not the papers, not the vocabulary, not your sympathy. It
contains no new claims: everything asserted here is proved,
measured, or honestly flagged somewhere in the numbered canon, and
small bracketed pointers\canon{P36} tell you where. It is written so
that a person can follow the whole arc in one sitting: from a
three-symbol equation, through a single number read as a geometry,
to a parameter-free prediction of the most precisely measured
constant in physics, correct to about four parts in ten billion,
and to the list of measurements that will, in time, prove all of
this wrong or refuse to. The corpus's own rule applies to this
document as to everything else: nothing in it is protected from the
data.
\end{abstract}

\tableofcontents

% ════════════════════════════════════════════════════════════════════
\section{The wager, and the rules of the house}

Every theory of everything begins with a wager about what kind of
thing the universe is. This corpus's wager is unusually compact:
\emph{the universe is a geometry observing itself, and the entire
content of physics is the bookkeeping of that observation.} One
number is taken as input: the fine-structure constant, the pure
number near $1/137$ that sets the strength of light's coupling to
matter. Everything else is supposed to follow: the masses of
particles, the dark sector, the scale of galactic halos, the
texture of time. If that sounds like too much weight for one number
to carry, you are reading correctly; whether it is carried is the
question the rest of this document lays out, together with the
measurements that decide it.

Before any physics, you should know the rules this corpus plays
by, because the claims below only mean something under them.

\emph{First rule: every quantitative claim is machine-checked.}
Each result ships with a verifier, a script that recomputes the
claim from scratch and fails loudly if it drifts. At the time of
writing there are roughly six hundred such checks, all passing. You
do not have to trust the prose; you can run the arithmetic.

\emph{Second rule: every known gap is registered.} The canon
maintains a machine-readable registry of its own open problems,
misstatements, and wounds\canon{P40}. The build fails if a claimed
repair cites a document that does not exist. The number of open
items is public and can only change visibly.

\emph{Third rule: the corpus refutes itself in writing.} Three of
its own canonical claims have been formally refuted by its own
discipline (a statistical claim about prime numbers, a conjecture
about the electroweak scale, and a spectral construction whose
corrected equations do not reproduce its claimed spectrum), and the
refutations are filed with the same care as the results they
ended\canon{P40}. A
framework that can only confirm itself measures nothing; the
refutations are part of the evidence that this one measures.

\emph{Fourth rule: predictions are pre-committed.} The headline
results come with named falsifiers and a stated direction: which way
the next measurement must move, and what dies if it moves the other
way\canon{P36, P39}. There are no second knobs hidden in the prose;
where a knob exists, it is declared.

With the rules stated, here is the story.

% ════════════════════════════════════════════════════════════════════
\section{Self + Nullity = Unity}

Underneath the numbered papers there is a three-symbol equation
that the corpus's author wrote before any of them\canon{i}:
\[
\mathbf{1} + \mathbf{T} = \mathbf{0}
\qquad\text{Self} + \text{Nullity} = \text{Unity}.
\]
Read it as balanced ternary, or read it as philosophy; the corpus
means both. A \emph{self}, anything that exists as a definite
something, is always accompanied by a \emph{nullity}: that which
was not actualized, the remainder, the unchosen. Together, and only
together, they compose a unity. Nothing definite exists without a
shadow of the same size, and the shadow is not nothing; it has
structure, dynamics, consequences.

If that sounds mystical, hold on: by the end of this document the
``shadow'' will be dark matter, computed and weighed against a
hundred and seventy-five galaxies\canon{P38, P39}, and ``nullity''
will be the kernel of an explicit linear map. The corpus's deepest
habit is taking poetic structure and forcing it to do arithmetic.

The same seed was originally drawn as three hand-sketched maps,
one mathematical, one logical, one of consciousness, conjectured
to be the same map in three languages\canon{P00}. The numbered
papers are, in the author's framing, the equivalence proofs. This
document follows mostly the mathematical map, because that is where
the verifiers live; but the reader should know the other two exist,
because occasionally (when we discuss observation, or what an
``edge'' is) the consciousness map shows through the arithmetic.

% ════════════════════════════════════════════════════════════════════
\section{One number, read as a geometry}

Here is the corpus's single input. The measured fine-structure
constant has inverse
\[
\alpha^{-1}_{\rm meas} = 137.035999\ldots
\]
The corpus observes that
\[
\boxed{\;4\pi^3 + \pi^2 + \pi \;=\; 137.036304\ldots\;}
\]
and takes the near-coincidence not as numerology but as a
\emph{reading instruction}: suppose the number is this expression,
and ask what the expression describes\canon{P01--P03}.

Three terms. The corpus reads them as the three layers of a
geometry: a four-dimensional ball $B^4$ whose volume-like bulk
contributes $4\pi^3$ (about $90.5\%$ of the total), whose
three-sphere boundary $S^3$ contributes $\pi^2$ (about $7.2\%$), and
whose \emph{edge}, the thin seam where observation itself runs
out, contributes $\pi$ (about $2.3\%$)\canon{P04}. One number,
three habitats. Keep the percentages in mind; the smallest one,
$2.3\%$, returns at the climax of this story with a precision of
ten decimal places.

The same reading produces a function: the \emph{density}
\[
\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
\qquad \int_0^1 \rho(x)\,dx = 4\pi^3 + \pi^2 + \pi,
\]
a cubic whose integral is the constant itself and whose three
coefficients $(16, 3, 2)$ are the layers' fingerprints\canon{P03}.
This unassuming polynomial is the corpus's workhorse. Out of it
come, by computations you can rerun: an oscillation parameter
$\kappa = \alpha^{5/4}$ and a breathing spectrum (the geometry is
not static: it oscillates, gently, and the oscillation amplitude
is about $0.2\%$); a self-observation operator whose eigenvalues
will become particle masses (Section 9); and a chord. The ratios
$3{:}2$, $16{:}3$, $16{:}2$ are, in music's terms, a perfect fifth,
a perfect fourth over two octaves, and three exact
octaves\canon{P37}. The coefficients of the substrate sound the most
consonant chord three small integers can make, and the chord turns
out to be load-bearing: the particle masses draw on its tones
(Section 9).

One fact should be registered now, because the whole document turns
on it: \emph{the identity is not exact.} The geometric value exceeds
the measured constant by $2.2$ parts per million, roughly fifteen
thousand times the experimental error bar\canon{P36}. The deviation
is not an approximation to be excused. A theorem of the framework
requires it to be nonzero (Section 6), and a derivation fixes its
size (Section 10); the discrepancy is itself the framework's
sharpest prediction.

% ════════════════════════════════════════════════════════════════════
\section{Two observers make a world}

Why this geometry and not another? The corpus's answer is the most
philosophically radical thing in it\canon{P27}. Consider one
observer, alone. Whatever it projects, it can take back; whatever it
collapses, it can re-create. Formally: any state space whatsoever
admits operators $C$ (collapse) and $P$ (projection) with
$C \circ P = I$. The coherence condition is \emph{trivially}
satisfiable, and nothing about the world is thereby determined.

Now add a second observer. Suddenly coherence is a \emph{constraint}:
what A projects, B must be able to collapse, and the result must
reconcile. The corpus proves that demanding a shared arena for such
reconciliation forces specific geometry: the three-sphere $S^3$
is the unique compact, simply-connected, homogeneous stage on which
multiple observers can maintain coherent agreement, and it must
bound something: the private bulks become the interior
$B^4$\canon{P27, P05}. The layer fractions of Section 3 acquire
meaning as the \emph{cost of intersubjectivity}: the bulk is what is
private, the boundary is what is shared, the edge is what is
irreducibly individual, the $2.3\%$ that observation cannot
reach.

Two consequences fall out before any dynamics. \emph{Time}: with two
observers, reconciliation takes a delay, and that delay \emph{is}
time; time is not a stage but a reconciliation
protocol\canon{P27, P08}. \emph{The Big Bang}: reinterpreted as the
first disagreement, the symmetry-breaking that split one
primordial observer into many. You may take these as metaphors with
equations attached or equations with metaphors attached; the corpus
declines to distinguish.

One more thing about $C \circ P = I$, because a skeptical reader
will spot it immediately: the natural projection on $S^3$ (the Hopf
map) is two-to-one, so it has no inverse. How can collapse undo
projection? The resolution\canon{P38} is that the identity holds
\emph{on the canonical sector}, as a section law, not globally as
invertibility, and the apparent defect is the content. What the
projection cannot reconstruct (the part it annihilates, the kernel)
does not vanish. It persists, unreachable from the image.
Hold that thought; it is Section 7, and it is also the dark matter.

% ════════════════════════════════════════════════════════════════════
\section{The breath, and the number 432}

The geometry oscillates; the corpus calls it breathing. The
density's stability analysis gives a harmonic spectrum
$\omega_n = n\pi\sqrt{1-\kappa}$ with $\kappa = \alpha^{5/4} \approx
0.002$\canon{P03}: stable, gently detuned modes. The breath is the
slow cycle built on the fundamental: its period, measured in
fundamental oscillations, is
\[
T_{\rm breath} = \pi\,\alpha^{-1} = 430.51\ldots
\]
a number that any culture tuning to small-prime ratios near this period finds as $432$\canon{P14}. The correspondence is a
theorem: among the ``musically simple'' integers (those of the form
$2^a 3^b$, the just-intonation lattice), the nearest one to
$T_{\rm breath}$ is exactly $432 = 2^4 \cdot 3^3$, the bulk
coefficient times the cube of the boundary coefficient, and the
mismatch, $432/T_{\rm breath} - 1 = 0.346\%$, is not an error but a
\emph{comma}: the same kind of irreducible residue that separates
twelve perfect fifths from seven octaves and forced human music to
invent temperament\canon{P37}.

Units, briefly: the framework carries two natural clocks, the
oscillation's own time and the breath-count time, and their
conversion is derived, not chosen\canon{P35}. Once derived, every
per-breath count is exactly free of the detuning factor
$\sqrt{1-\kappa}$, and an apparent $0.1\%$ discrepancy at the
boundary between the two clocks dissolves as a unit mismatch: the
same speed quoted in two gauges. Calibrated against galactic data (Section 8), one breath
lasts $1.37$ million years, and one fundamental oscillation about
$3{,}200$ years, whose light-crossing distance, it will turn out,
is the size of every galaxy's dark core.

% ════════════════════════════════════════════════════════════════════
\section{Never closing: why exactness would be wrong}

Now the corpus's central theorem about itself, and the moment its
many small discrepancies stopped being embarrassments.

The breath advances the geometry's phase by $T_{\rm breath}$ cycles
per cycle of the slow clock: as a rotation of a circle, by the
fractional part $\rho = 0.5122452\ldots$, and $\rho$ is irrational
($\pi$ is transcendental; this fraction of it inherits the
property). An irrational rotation \emph{never closes}: no orbit
returns, no cycle coincides exactly with any rational pattern, and
\emph{every} identification of a corpus quantity with a crisp
integer or ratio must carry a nonzero residue\canon{P37}. The corpus
calls this the \textbf{Necessity of Detuning}, and it cuts both
ways: it forbids the corpus from ever claiming a perfect match,
and it indicts any perfect match the corpus might be tempted to
claim. When we reach the fine-structure constant's $2.2$ ppm
deviation in Section 10, remember: had the match been exact, this
theorem would have falsified the corpus from inside.

Here a sidebar from computer science earns its place, because it
supplies the right intuition cheaply. Hash tables, the data
structures underlying most software, scatter items by an
irrational-rotation trick, and fifty years of engineering wisdom
says the best multiplier is the golden ratio $\varphi$, the ``most
irrational'' number, the one that clings to \emph{no} rational
pattern at all. Engineering wants maximal detuning: any
near-resonance is clustering, and clustering is failure. Scored
with the engineers' own instruments, the breath's rotation is a
\emph{terrible} hash multiplier\canon{P37}: it hugs specific
rationals tightly without touching them, producing a ladder of
strong near-resonances (at $2$ breaths, at $41$, at $490$)
with response gains that jump by orders of magnitude where
$\varphi$'s creep smoothly. In one word: the breath is
\emph{tonal}. It has loud overtones, a scale of its own, preferred
cycles; a melody, where $\varphi$ is engineered noise.

Is the melody designed? No, and the answer is a
theorem\canon{P37}: among all possible rotations, perfect closure
(silence) is infinitely rare, and $\varphi$-grade atonality is rarer
still (measure zero, in the mathematician's phrase), while
\emph{some} melody is statistically guaranteed. The breath's
particular loudness is not unusual; one irrational in ten sings
louder. So the existence of the tune is necessary, its strength is
generic, and the specific melody, the particular rungs and
particular commas, is the universe's fingerprint: not chosen, not
tuned, just \emph{its}. Music is the default state of an irrational
world; only silence and noise require engineering.


% ════════════════════════════════════════════════════════════════════
\section{The dark is what projection forgets}

Recall the wound from Section 4: the projection from the
three-sphere down to ordinary observable space is two-to-one, and
what it cannot reconstruct persists outside its reach. The corpus
now states this as a definition that unifies everything it calls
dark\canon{P38}:

\begin{center}
\emph{Dark is the dynamics of the kernel of a closure map.}
\end{center}

Unpack it gently. A closure map is any operation that takes states
to their canonical, agreed-upon form: projection to the shared
boundary, rounding to the nearest chord, locking to a resonance. Its
\emph{kernel} is what it annihilates: the components that have no
image, the unchosen, the Nullity of Section 2 finally wearing
equations. Three theorems attach to every such kernel\canon{P38}. The kernel is \textbf{sealed}: the projection is
exactly symmetry-respecting, so kernel states can never convert into
visible ones. This is \emph{why dark matter is dark}: not an
assumption but a superselection theorem; it gravitates and can do
nothing else. The kernel is \textbf{paced}: it shares the visible
sector's clock; nullspaces breathe, and the dark is synchronized,
not frozen storage. And the kernel is \textbf{never empty}: by the
Necessity of Detuning, some component always fails to close;
darkness is mandatory in any universe with this geometry.

Two refinements give the picture its full shape. Dark
\emph{matter} is kernel: annihilated by the projection, present
in gravity. Dark \emph{energy} is the \emph{cokernel}: not what the
map forgets but what it fails to cover. The corpus computes
the uncovered fraction of the ambient space as
$\Lambda_0 = 1 - \pi^2/32$, the cosmological constant's geometric
face\canon{P32, P38}. The two great darks of cosmology are the two
exactness defects of a single map. And because closures compose
with their kernels \emph{adding}, a universe reached through many
layers of closure is generically dark-dominated; the observed
$\sim\!95\%$ darkness of the cosmos is, in this reading, what depth
looks like.

% ════════════════════════════════════════════════════════════════════
\section{The kiloparsec: the kernel meets the telescopes}

A sealed, gravitating kernel should be measurable the way dark
matter is measured: in the rotation of galaxies. Here the corpus
becomes ordinary falsifiable astrophysics, and this section is the
package's empirical heart\canon{P39}.

The kernel's density profile on its natural chart is fixed by the
geometry, with no fitting and exactly one length in it, the same
for every galaxy in the universe:
\[
\rho_H(r) \propto
\frac{\sin^2\!\bigl(2\arctan(r/L_h)\bigr)}{r^2},
\qquad
v_\infty^4 = a_0\,G M_b,
\quad
a_0 = \frac{\alpha^3 c^2}{\pi^2 L_h}.
\]
The second equation says: the flat rotation speed of a galaxy is set
by its \emph{ordinary} (baryonic) mass through a universal
acceleration $a_0$ built from the fine-structure constant and that
one length. Astronomers will recognize this instantly: it is the
baryonic Tully--Fisher relation, one of the tightest empirical laws
in galactic dynamics, here with a \emph{derived} coefficient.

Confronted with the standard test set (SPARC: 175 galaxies with
quality rotation curves), three things happened\canon{P39}. First,
the universality test passed: the effective acceleration shows no
trend with galaxy size (slope $0.029 \pm 0.076$), and the natural
rival (a halo scale tied to each galaxy's own radius) is
excluded at $13.5$ standard deviations. Second, the calibration:
using the gas-rich galaxies (where stellar guesswork is weakest),
$a_0$ comes out within $2.4\%$ of the value Milgrom identified
empirically forty years ago, and the one length comes out as
$L_h = 0.98$ kiloparsec. Third, a methodological caution that any one-scale model must
respect: naive per-galaxy fits show core radii \emph{apparently}
tracking each galaxy's disk size at $7.6\sigma$, which would
contradict universality. The trend is a measurement artifact:
those fits are dominated by radii where ordinary matter drowns the
signal, like measuring a candle behind a headlight. Masking to radii
where the dark component dominates collapses the false trend
monotonically while the measured core locks onto $0.89$--$1.00$ kpc
at every cut\canon{P39}. One kiloparsec,
measured three independent ways, agreeing at ten percent, with zero
per-galaxy parameters.

And the length is not arbitrary. Convert it with the breath
calibration and $L_h$ is the distance light travels in exactly one
fundamental oscillation, about $3{,}200$ light-years, and in
the chart's own units the universal acceleration \emph{is} the
fine-structure constant: $a_0 = \alpha$, exactly\canon{P35}. The
mysterious $10^{-10}\,\mathrm{m\,s^{-2}}$ that haunts galactic
dynamics and the mysterious kiloparsec are one dimensionless
statement seen from two sides.

% ════════════════════════════════════════════════════════════════════
\section{The masses: three families because the polynomial is cubic}

Particle masses, in this corpus, are eigenvalues. The geometry
observing itself defines an operator (kinetic term plus a potential
built from the density), and its discrete spectrum is the ladder of
stable observation depths: the particles\canon{P07}. The boundary
conditions are derived, not assumed (observation \emph{vanishes}
at the universe boundary; it does not run free), and the data
prefer that reading decisively\canon{P40}. With the boundaries
fixed, the spectrum has nothing to adjust.

The leading-order mass map (a power law with a derived exponent)
gets the muon right to about a percent and the tau to six:
respectable, not convincing. The residuals, however, carry
structure\canon{A287}: the corrections to the exponent, level by
level, are proportional to the \emph{coefficients of the density
polynomial}: the boundary's $3$ for the muon, the bulk's $16$
for the tau, the edge's $2$ anchoring the electron, at a
universal strength built from $\alpha^{1/4}$ and the operator's own
ground eigenvalue. With that law, both measured ratios land at one
part in ten thousand, with zero adjustable constants:
\[
\frac{m_n}{m_e} = \Bigl(\frac{\lambda_n}{\lambda_1}\Bigr)^{\beta_n},
\qquad
\beta_n = \frac{6\mu_1}{\mu_0} - c_n\,\frac{\alpha^{1/4}}{\lambda_1},
\qquad (c_e, c_\mu, c_\tau) = (\text{anchor},\, 3,\, 16).
\]
The structural corollary writes itself: the correction has exactly
as many slots as the cubic has coefficients. \emph{Three families
of matter because the density is cubic, because the
fine-structure identity has three terms.} A fourth eigenvalue
exists; it has no coefficient to draw; no fourth family forms.

Honesty requires the other half: the same law, tried on quarks,
fails completely (wrong signs, wrong magnitudes, no integer
structure), and the corpus scoped it to the charged leptons
rather than rescue it\canon{P40, A292}. The mechanism behind the
law (why each family couples to its own layer) is one of the three
named open problems at the end of this document. The law itself is
falsifiable today: improved tau-mass measurements can kill it.


% ════════════════════════════════════════════════════════════════════
\section{The constant, derived: 137.035999236}

Now the climax, where the threads braid. Recall the $2.2$ ppm
deviation from Section 3, the seed identity's ``flaw.'' Recall
the Necessity of Detuning: exactness would have been
self-falsifying. The corpus owes a derivation of the deviation's
\emph{size}, and delivers one in four steps, each independently
checkable\canon{P36}.

\emph{Step 1.} The geometry breathes: the coupling oscillates about
its geometric value with relative amplitude $\kappa = \alpha^{5/4}$.
A measurement does not catch the instantaneous value; it averages
over the cycle, and the average of an oscillating inverse sits
\emph{below} the center, by half the squared amplitude. The sign of
the deviation is thereby forced, and it is the observed sign.

\emph{Step 2.} The oscillation's energy is shared among the three
layers. In which proportion? This was the chain's one assumption
until it became a theorem: the layer-projected energies of the
oscillation modes have exact closed forms, and their shares converge
(exactly) to the layer fractions of Section 3. Equipartition
by layer measure is the ergodic limit of a computation, not a
postulate; and the data \emph{select} that limit over the
alternative (a ground-mode-dominated dressing) at thirty-one times
the experimental resolution\canon{P36}. (An aside: this is
the framework's ``exposure, not training'' principle, its claim that
the kernel samples its whole fixed architecture rather than
concentrating in a learned ground state, weighed for the first
time by a precision measurement, and preferred.)

\emph{Step 3.} One layer is unobserved. The edge, the $2.3\%$
where observation runs out, the irreducibly private seam, does
not contribute to any measured average. The measured oscillation
power is therefore the total times $(1 - \pi/\Omega)$.

\emph{Step 4.} Assemble:
\[
\boxed{\;\alpha^{-1}_{\rm meas}
= \Omega\,\sqrt{\,1 - \kappa^2\Bigl(1 - \frac{\pi}{\Omega}\Bigr)}
= 137.035999236\;}
\qquad (\Omega = 4\pi^3 + \pi^2 + \pi).
\]
No adjustable parameters: every symbol was fixed before this
calculation existed. The 2022 CODATA value is
$137.035999177 \pm 0.000000021$. The prediction lands $0.43$ parts
per billion high: within three experimental error bars of the most
precisely measured constant in physics, from $\pi$ and a geometry.

The remaining $2.8\sigma$ is not waved away. The chain has been
shown to possess \emph{no internal knob} that could absorb it
(every possible refinement pushes the wrong way), so the tension
stands as the prediction's exposed throat\canon{P36}. The
falsification is pre-committed in the canon's own text: future
CODATA editions move toward $137.035999236$, or this result dies,
and the corpus has bound itself in writing not to re-dress it.

% ════════════════════════════════════════════════════════════════════
\section{What would kill it}

A reader's trust should rest not on what a framework explains but on
what it exposes. The standing kill-list\canon{P36, P39, P40}:

\emph{The constant} (above): the CODATA direction test; a
$0.1$-ppb measurement decides outright.

\emph{The halo}: one clean, gas-rich, well-resolved galaxy whose
dark component turns over far from one kiloparsec ends universality;
a single counterexample suffices. The residual masked trend
($0.254 \pm 0.110$) must fade as dwarf-galaxy samples grow, or
persist and kill. Per-galaxy stellar masses must keep the
calibration inside its stated window.

\emph{The masses}: the lepton law must survive improving tau-mass
precision with its integer coefficients intact.

\emph{The dark's nature}: the superselection theorem permits the
hidden sector to do nothing but gravitate; any confirmed
non-gravitational dark-matter signal (direct detection,
annihilation) falsifies the identification outright.

\emph{The fold's signature}: if the early-universe fold seeded the
perturbation spectrum, a narrow stochastic gravitational-wave
background in the LISA band ($0.45$--$0.95$ mHz) is mandatory:
the theorem that sealed the dark also made this unavoidable, and it is
a discovery channel with a death clause.

These run on instruments and decades the corpus does not control,
which is what makes them worth listing.

% ════════════════════════════════════════════════════════════════════
\section{What remains, honestly}

Three named open hearts, each reduced to its sharpest form. \emph{The
integer 10}: the mass-hierarchy formula, which predicts the
twenty-two-decade ratio of the Planck mass to the electron mass at
five parts per million, contains one underived integer,
data-pinned to be exactly $10$; candidates exist (it counts, perhaps,
the metric's degrees of freedom), none derives it\canon{P40}.
\emph{The assembly}: whether the corpus's master operator is the
unique one satisfying its design requirements is open, internal
mathematics, though the kernel's operation was proven indifferent
to the answer\canon{A294}. \emph{The selection rule}: why each
particle family couples to its own geometric layer, the mechanism
behind the mass law's elegance.

And the honest ceiling, stated plainly: even complete, this package
is not yet a full theory of everything in the physicist's sense. It
does not derive the gauge group, the quark masses, neutrinos, or the
Born rule; its gravity is kinematic; its electroweak conjecture was
refuted by its own registry and awaits a successor. What it is, is a
closed, self-auditing geometric framework with parameter-free
contacts at $0.43$ ppb ($\alpha$), $5.5$ ppm (the mass hierarchy),
$10^{-4}$ (the lepton ratios), $2.4\%$ (the galactic acceleration),
and $10\%$ (the kiloparsec, thrice), every one of them attached
to a dated falsifier, every one of them recomputable from scripts
that ship with the claims.

The corpus began with three symbols: Self + Nullity = Unity.
Forty-one papers later the ledger reads: the Self is a geometry
observing itself; the Nullity is real, sealed, breathing, and
weighable in the rotation of galaxies; and the Unity is one number,
read three ways, detuned by exactly its own edge, and now
predicted back to the ten significant figures it was once merely
measured to. Whether the universe agrees is, from here, a question
for the instruments. The corpus has said what it means, signed the
arithmetic, and named what would prove it wrong; the reader is
invited to hold it to all three.

\paragraph{Where to go next.} The numbered canon: Papers i and 00
(the seed), 01--04 (the number and its layers), 27 (the second
observer), 35--39 (the modern results: units, the constant, the
breath, the dark, the branch), 40 (the corrections). The external
companion paper on the galactic results is self-contained for
readers wanting the astrophysics alone. Every paper cites its
verifiers; every verifier runs from the repository root; and the
registry of open problems is exactly as long as it says it is.

\end{document}
