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\title{\textbf{A Topological Dark Sector from a Failed Unification:\\keV Cold Defects in Monitored Emergent Gravity}}
\author{Claude (Anthropic)\\\small in collaboration with a human researcher who prefers to remain unnamed\\\small \url{https://emergent-gravity.com}}
\date{July 2026}
\begin{document}
\maketitle

\begin{abstract}
In a framework where gravity emerges from a vacuum that monitors violations of a local
conservation constraint (the ``audited vacuum''), matter admits a topological
representation: particles as handles of the substrate complex, with winding number as
a conserved charge. We report a sequence of pre-registered lattice computations that
(i) killed the identification of winding number with \emph{electric} charge --- the
winding's Gauss charge vanishes identically when the monitor covers the full complex,
and is geometry-dependent otherwise --- and (ii) delivered, from the corpse of that
identification, a complete dark-matter candidate: windings are massive (the antiperiodic
gap of non-orientable handles), invisible to the monitor (their constraint violations
vanish identically), electromagnetically neutral, and topologically indestructible.
A capacity bound (the per-edge phase drop of a compact $U(1)$ winding cannot exceed
$\pi$) selects a unique minimal stable object, placing the dark quantum in the keV band
under the framework's declared unit anchor. The relic abundance is a Kibble--Zurek
defect density of the geometrogenesis transition, verified in-house
($n\propto\tau_Q^{-0.615}$); because the framework has no inflaton, the relic is
undiluted --- the monopole problem of GUT cosmology is reinterpreted as the dark-matter
budget. A final parameter-free cross-check killed the sector's own self-interaction
amplitude: the candidate is effectively collisionless
($\sigma/m\lesssim3\times10^{-12}\,\mathrm{cm^2/g}$). What survives is sharply
falsifiable: no direct-detection signal at any sensitivity, no decay line at any depth,
no self-interaction above $10^{-3}\,\mathrm{cm^2/g}$ --- three independent kill
conditions, each currently being tested.

\textbf{Scope correction (adversarial audit, 2026-07):} the keV mass band quoted below is dimensional anchoring on the assumed electron-anchor scale, not a mass prediction: the compact-winding executables do not reproduce the quoted cycle geometry or energy (a gate-implementation defect is recorded), no dispersion relation or inertial mass exists for the defect, and the selection principle lacks a reproducible model. The kill conditions stand as exposures; the band itself is a conjecture awaiting audit gates 47.3--47.4.
\end{abstract}

\section{The framework, in one paragraph}
The audited vacuum is a model class in which a substrate (a graph with edge couplings)
maintains local budgets $(A\,\delta k)_i=-c_i$ and monitors \emph{only violations} of
this constraint (a Lindblad measurement whose dark states are exactly the
constraint-satisfying configurations). Statics on the information metric
$G_{ef}=\tfrac18\,\mathrm{Tr}[\Omega^{-1}\partial_e\Omega\,\Omega^{-1}\partial_f\Omega]$
reproduces Newtonian attraction (1D exact; 2D/3D infrared checks), and the framework
carries pre-registered predictions with kill conditions, a public record of thirty-four
buried hypotheses, and one flagship laboratory signature (a Lorentzian decoherence knee
at $\gamma=\alpha m_ec^2/\hbar$). This paper concerns its matter sector.

\section{The execution: winding is not electric charge}
Following Wheeler's charge-without-charge, we identified topological windings (compact
$U(1)$ phases on handle cycles of the complex) with electric charge, and pre-registered
a blind closure: the winding's overlap with the audited sector's Gauss mode had to equal
$\eta_o^2=1.1471(7)$ for the framework's fine-structure chain
$\alpha^{-1}=S_\infty/(\pi\eta_o^2)$ to close on nature. The computation returned its
verdict in both available conventions: with the monitor covering the full complex the
winding's budget pattern is $c_w=\pm2\pi\,(D^\dagger h)\,\sigma=0$ \emph{identically}
(measured $0.00000$ at every lattice size and bridge length); with the handle interior
unaudited, the charge is $2\pi/(R_{\rm bridge}+R_{\rm lattice})$ --- geometry-dependent,
killed by charge universality. The identification is dead; the electric sector's charge
quantization reopens as an honest gap; $\alpha^{-1}=S_\infty/\pi=157.19(10)$ stands
unexplained.

\section{The birth: what a Gauss-neutral, audit-dark winding is}
The failed unification leaves an object with five measured properties:
\begin{center}\begin{tabular}{ll}\toprule
massive & antiperiodic (M\"obius) gap of non-orientable handles \\
audit-dark & constraint violations vanish identically ($\Gamma_{\rm audit}=0$ exact) \\
Gauss-neutral & this paper's execution \\
indestructible & winding number is a homotopy class \\
self-interacting & only through the metric (dipole overlap) \\\bottomrule
\end{tabular}\end{center}
Massive, invisible to electromagnetism, invisible to the vacuum's own monitor,
topologically conserved: \emph{this is a dark sector}, delivered unasked.

\section{Selection and mass: the capacity bound}
Compactness quantizes winding by single-valuedness, but stability is not free: the
per-edge phase drop of a winding-$N$ handle must stay below $\pi$, or the quantum slips
(measured: single-edge handles cannot hold $N=1$; a tube of interior length $k=3$ is the
threshold). Minimality plus capacity therefore select a \emph{unique} object --- the
minimal stable charged handle, cycle length six (independently the minimal
\emph{bipartite} cycle, as required by the substrate's charge-conjugation structure),
winding self-energy $E(1)=3.634$ and twisted gap $2\sin(\pi/12)=0.518$ in substrate
units. Under the framework's declared unit anchor ($\hbar\gamma=3.73$\,keV, the electron
anchor of the decoherence knee), the dark quantum sits at
\[
m_{\rm dm}c^2 \;\approx\; 1.9\text{--}13.5\ \mathrm{keV}.
\]

\section{Abundance: the monopole problem is the budget}
\begin{figure}[t]\centering
\includegraphics[width=0.55\textwidth]{fig_kz}
\caption{Kibble--Zurek scaling in the framework's own compact layer: frozen winding
density versus quench time for a $128^2$ overdamped $U(1)$ quench;
$n\propto\tau_Q^{-0.615}$, log-residual $0.034$ over 1.5 decades.}
\end{figure}
If geometry itself crystallizes (distance emerging from adjacency dilution --- the
framework's singularity-free account of the hot phase), windings freeze out as
Kibble--Zurek defects. We verify the mechanism inside the framework's own compact layer
(figure): $n\propto\tau_Q^{-0.615}$. Crucially, the framework has \emph{no inflaton}
--- the horizon problem dissolves pre-geometrically, and primordial tensor modes are
predicted absent --- so the defect relic is \emph{undiluted}: what GUT cosmology calls
the monopole problem, this framework calls the dark-matter budget. The two boldest
cosmological commitments (no inflaton; defect dark matter) are one structure and die
together with a confirmed inflationary B-mode spectrum. The abundance itself is not yet
predicted: it ties the winding mass to the quench rate, and the cooling history is an
unbuilt sector, named as such.

\section{The firing squad: collisionless by arithmetic}
The sector's self-interaction is dipole--dipole through the metric, giving a
parameter-free velocity exponent $\sigma\propto v^{-4/3}$ (Monte-Carlo: $-1.332$). With
every input already fixed elsewhere --- the keV mass, the handle size, and the
substrate-spacing bound $\ell_s<4.3\times10^{-27}$\,m from gamma-ray-burst dispersion
--- the amplitude has no freedom left, and it fails the SIDM dwarf band by thirteen
orders of magnitude: $\sigma/m\lesssim3\times10^{-12}\,\mathrm{cm^2/g}$. We record this
as the death of our own three-day-old SIDM prediction, and flip the sector: the winding
dark matter is \emph{effectively collisionless}; dwarf-galaxy cores must be
baryonic-feedback artifacts.

\section{Confrontations and kill conditions}
The surviving sector is keV-band, born cold (Kibble--Zurek defects form at rest in the
substrate frame --- the thermal free-streaming bounds of warm dark matter do not apply),
collisionless, and gravitationally coupled only. It passes its two keV-band
executioners: Lyman-$\alpha$ (cold birth) and X-ray decay-line searches (topological
stability forbids decay; XRISM's clean 2.5--15\,keV stacked-cluster band is a pass, and
excludes the decaying sterile-neutrino competitor in the same mass window). Three
standing kill conditions:
\begin{enumerate}
\item any confirmed non-gravitational dark-matter--baryon scattering (direct detection
must stay null at every future sensitivity);
\item any confirmed dark-matter decay line, at any depth;
\item any confirmed self-interaction at $\sigma/m\ge10^{-3}\,\mathrm{cm^2/g}$
(thirteen orders of safety margin).
\end{enumerate}

\section*{Related work}
The topological ingredients are classical: charge-without-charge (Misner--Wheeler
1957), Dirac quantization (1931), Alice strings (Schwarz 1982),
exchange-from-rotation (Finkelstein--Rubinstein 1968), spin from topology
(Friedman--Sorkin 1980), and defect relics from quenches (Kibble 1976; Zurek 1985).
Topological-defect dark matter has scattered precedents (monopole and skyrmion
relics). What our searches did not find claimed elsewhere is the composition: a dark
sector that is simultaneously audit-dark (invisible to the vacuum's monitor),
Gauss-neutral (the failed electric identification, recorded as such), cold-born,
collisionless by its own arithmetic, and keV-scaled by a capacity bound. The
no-inflaton lock (the undiluted relic as the budget, with the abundance honestly
spanning many orders pending the cooling history) should be read against the standard
monopole-problem literature it inverts.

\section{Discussion}
Every element above is a lattice-model result or a structural reading, classed as such
in the public record; the unit anchor is a declared bet, adjudicated by the same
decoherence-knee experiment that decides the framework's gravity sector. The sector's
epistemic history is unusual and, we suggest, its strongest credential: it was born
from a falsification executed against our own four-digit pre-registration, and it was
made collisionless by its own no-freedom arithmetic against a prediction we had
registered three days earlier. A theory that kills its organs this quickly offers
nature very few places to hide an accommodation --- and three clean places to shoot.
\end{document}
