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\title{\textbf{The Radiation Content of General Relativity\\from a Monitored Substrate:\\Two Polarizations by Constraint Counting}}
\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 the audited-vacuum framework --- a substrate whose couplings are dynamical and whose
vacuum monitors only violations of local conservation constraints --- we derive the
radiative sector of gravitation and find it to be exactly that of General Relativity:
two transverse-traceless polarizations, massless, propagating on the matter cone,
transparent to the monitor. The derivation is forced at every step by existing data.
Four no-go computations close the naive routes (free two-phonon composites carry no
transverse-traceless spectral weight at the cone; the passive strain sector is gapped
and flat; substrate phonons are pure gauge; Gaussian noise dressing cannot soften a
metric gap). Masslessness then arises from a thermostat argument: the shear channel is
exactly invisible to the monitor, so its only cooling is threshold rearrangement, and
self-organized marginality is its unique steady state. Isotropy bounds force a bare
substrate elasticity; LARES-2 and the Hulse--Taylor pulsar independently force the
constraint set to include momentum budgets. The resulting four constraint rows ---
derived from the substrate's own conservation laws, not chosen --- leave a dark
(unmonitored, hence marginal) kernel of dimension exactly two, equal to the
$\{+,\times\}$ tensor pair at every momentum (exactly, once the edge weights are
fixed by fourth-moment isotropy --- a derived condition, not a choice), with every scalar and vector channel
watched. If a consistent massless spin-2 excitation with universal coupling is established
--- which this paper does \emph{not} establish --- Weinberg's soft-graviton theorem
and Deser's bootstrap would supply universal coupling, the Einstein--Infeld--Hoffmann
structure, the quadrupole formula, and the nonlinear completion, with one calibrated
constant ($G$). Their premises are exactly the open program below. This is a \emph{structural} closure --- the polarization content and channel
structure; positivity of the norm, causal propagation, and the amplitude-level
match to the Einstein--Hilbert action are enumerated as the remaining program,
not claimed. A full conditions ledger is included; every structural link is
falsifiable and several nearly were.
\end{abstract}

\section{The framework}
The substrate is a graph carrying a matter amplitude $\varepsilon_i$ on sites and a
coupling field $k_e = 1 + \delta k_e$ on edges; the coupling field \emph{is} the metric.
The action is
\[
S = \tfrac12\dot\varepsilon^2 - \tfrac12\textstyle\sum_e k_e (\nabla\varepsilon)_e^2
\;-\; \xi \sum_e \varepsilon_{t(e)} (\nabla\varepsilon)_e^2 \;+\; S_{\rm monitor} + S_{\rm pump},
\]
where the derivative anharmonicity $\xi$ was previously selected against a cubic
alternative by three independent gates (a Nordtvedt-type identity, Mercury's perihelion
calibration, and infrared protection of emergent Lorentz symmetry), and the monitor
couples only to \emph{violations} of the local budget constraint, so that
constraint-satisfying configurations are exact dark states. Newtonian statics, the
post-Newtonian locks, and a laboratory decoherence signature (a Lorentzian knee at
$\gamma = \alpha m c^2/\hbar$) are obtained elsewhere in the program; this paper
concerns radiation.

\section{Four no-go results}
Numerical and analytic computations on the lattice close every naive radiative route:
(i) the transverse-traceless spectral weight of free field-pairs vanishes at the
lightlike edge as $(\omega - cq)^{2.7}$ --- collinear pairs are longitudinal --- so no
free composite graviton pole exists, and none may be tuned in ($m_g < 1.3\times10^{-23}$
eV); (ii) giving the strain field inertia fails jointly: the passive stiffness spectrum
is gapped ($5.7\times10^{-3}$) and flat ($\kappa_{\rm band} = -3.7\times10^{-5}$), so no
inertia satisfies the graviton-mass and speed bounds simultaneously; (iii) substrate
displacement modes are gapless but vector-polarized and, as metric perturbations, pure
gauge; (iv) an exact lemma: an isotropically driven Gaussian block is
metric-transparent (its stationary energy is independent of the modulated stiffness),
and colored-noise dressing is convex --- always hardening. Quadratic-coupling noise of
any spectrum cannot soften a metric gap.

\section{Masslessness: the blind-channel thermostat}
The shear channel has unsigned incidence zero: $(\mathbf A\cdot{\rm shear})(r) = 0$
identically --- the monitor cannot see uniform shear, hence cannot cool it. The pump
leaks energy into every channel (through the $\xi$ vertex, slowly); the monitor is the
vacuum's only dissipation; instability above threshold rearranges at substrate rates.
With this timescale separation, the shear sector's unique steady state is the
\emph{threshold}: heated but unwatched, it ratchets soft until rearrangement sheds the
excess. Marginality --- hence masslessness --- is not tuned; it is the inevitable steady
state of the one channel the vacuum cannot cool by watching. The same invisibility
makes gravitational waves exactly transparent to the monitor. (A fifty-four--digit
fine-tuning that would otherwise afflict the bare elasticity dissolves: any bare
stiffness above the vacuum's own shear softness yields the same pinned marginal state.)

\section{Data force the constraint set}
Three observational executions fix the structure. Isotropy bounds
($\Delta c/c < 10^{-18}$) force a bare substrate elasticity: the matter-induced shear
potential alone is unstable (the vacuum energy's shear curvature is negative, an exact
concavity effect), and a slumped vacuum is anisotropic. LARES-2's one-per-mille
frame-dragging measurement and the Hulse--Taylor pulsar's $1.3\times10^{-4}$ orbital
decay independently kill the energy-budget-only monitor: a single constraint row always
leaves one scalar-polarized dark mode, and an eccentric binary sources it at
$5.5\%$ of the tensor channel --- excluded by forty years of pulsar timing. The
substrate must audit momentum as well as energy.

\section{The derived rows and the closure}
A dynamical coupling field sources violations of the matter sector's own conservation
laws; computing these from the action gives the monitored combinations directly:
\[
v_0 \;\propto\; \omega \sum_a w_a\, \delta k_a, \qquad
v_i \;\propto\; \sum_a (q\!\cdot\!\hat d_a)(\hat d_a)_i\, w_a\, \delta k_a ,
\]
with $w_a$ the vacuum stress per direction: a \emph{moving} trace is billed, a
\emph{directed flux} is billed; static configurations remain dark. On the
nine-direction complex (axes and face diagonals, all $h_{ab}$ present), these four
derived rows leave a physical dark space of dimension exactly two at every momentum:
\begin{center}
\begin{tabular}{lcccc}\toprule
$q_z$ & $P(+)$ & $P(\times)$ & $P(h_{xx}{+}h_{yy})$ & $P(h_{zz})$ \\\midrule
0.2 & 1.0000 & 1.0000 & 0.0000 & 0.0000\\
0.5 & 1.0000 & 1.0000 & 0.0000 & 0.0000\\
1.0 & 1.0000 & 1.0000 & 0.0000 & 0.0000\\\bottomrule
\end{tabular}
\end{center}
The radiative sector is $\{+,\times\}$ and nothing else: General Relativity's
radiation content is the unique dark sector of the derived constraints. With a
massless, coupled spin-2 pair propagating on the matter cone, Weinberg's theorem forces
universal coupling to $T_{\mu\nu}$ --- yielding the gravitomagnetic factor four, the
Einstein--Infeld--Hoffmann velocity terms, and the quadrupole formula --- and Deser's
bootstrap supplies the nonlinear completion. One constant, $G$, is calibrated by
Newton, as in General Relativity itself.

\section*{Discovery order, stated plainly}
The momentum constraints were adopted \emph{after} the single-constraint model's
scalar mode was excluded by pulsar timing --- the discovery order is data-first, and
we state it rather than let the reader infer it. A structure adopted under data
pressure earns the word ``derived'' only through consequences it was not fitted to:
here, the conservation-law derivation of the constraint rows, and the momentum-channel
prediction (a second decoherence channel entering the bridge as $c\,|d p_{\rm diff}/dt|$,
sharing one knee with the energy channel). If that prediction fails, the four-budget
repair was curve fitting, and this paper's central claim falls with it. We also state
what kernel counting does not establish: the propagator, positivity of the norm, the
helicity-$\pm2$ representation content, the source coupling and its amplitude, the
gravitomagnetic factor four, and the quadrupole formula are constructions this paper
does not contain; the Weinberg and Deser theorems constrain such a structure once it
exists --- they do not create it.

\section*{Related work}
Self-organized criticality as a route to emergent spacetime was proposed by Ansari and
Smolin (hep-th/0412307); composite-graviton masslessness and its entanglement with
cosmological-constant tuning appear in Carone, Erlich and Vaman (arXiv:1710.09367);
the obstructions we measure in-toy are the lattice faces of Weinberg--Witten (1980)
and of the emergent-Lorentz fine-tuning problem (Collins et al.). The closure chain is
textbook: Weinberg (1964), Deser (1970). What our searches found unclaimed is the
selection mechanism --- masslessness because the monitor cannot cool the channel it
cannot see, with marginality as the unique steady state of a heated, unwatched sector
--- and the derivation of the four-constraint structure from data pressure (isotropy,
LARES-2, PSR B1913+16) rather than by postulate. Searches are not proofs of absence.

\section{Conditions ledger and falsifiability}
The chain rests on: the thermostat premises (structural, not yet simulated in the full
driven steady state); tree-level derived rows with an isotropic vacuum (loop
corrections unexamined); the speed lock's coherence condition, met structurally because
rearrangement is stress-triggered; and the cited Weinberg--Deser theorems. Each
structural link is falsifiable and the program's public record
(\url{https://emergent-gravity.com/journal.html}) shows several nearly failing: the
scalar mode survived two suppression attempts before the momentum constraint killed it,
and one intermediate claim (a Landau-damping selection) was refuted by its own
verification run and is recorded as such. The framework's laboratory signature (zero
decoherence at rest; a Lorentzian knee at $\gamma = \alpha m_e c^2/\hbar$; the absolute amplitude
is an open derivation --- earlier milliwatt-scale targets were retracted when the
derived form factor failed its preregistered interval) distinguishes it from continuous-collapse--sourced
gravity, which carries a rest floor.
\end{document}
