Wave-function collapse and the Born rule in generalized trace dynamics: requirements, a fluctuation--dissipation condition, and a near-equilibrium prediction

v2: for changes compared to v1 please see Revision Notes in accompanying zip bundle.

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Generalized trace dynamics (GTD) proposes that quantum theory is the thermodynamic
equilibrium of a deterministic matrix dynamics, and that the collapse of the wave function is
what controlled departures from that equilibrium produce. In the same framework the Standard
Model and gravitation are sought as emergent structures, so that one programme aims at
unification and at the measurement problem together. We ask, carefully, what such a programme
must deliver to explain \emph{both} state-vector reduction \emph{and} the Born rule---and we
argue that the right target is collapse-with-Born statistics, not the specific phenomenological
law of continuous spontaneous localization (CSL), which may or may not be the correct law. We
first set out the four ingredients any dynamical account of collapse requires, recall the sense in
which CSL's stochastic structure is forced by its premises, and locate the premise a foundational
theory most naturally questions. We then identify what GTD already supplies---an intrinsically
fermionic, anti-self-adjoint reduction channel---and what it does not: a derived noise, a
spatial collapse operator, and the Born weights. On the simplest GTD-relevant model we establish
several facts. A static distribution over a state-independent drift variable
cannot reproduce the Born rule; the randomness must be a state-coupled process. For continuous
reduction that completes, the Born rule is the statement that the branch-weight process is a
driftless It\^o martingale. Because the collapse noise in GTD is emergent from a
\emph{deterministic} substrate (hence smooth), its simplest white-noise limit is
\emph{Stratonovich}, which carries a spurious drift; we show this both spoils the martingale at
order one (independently of the noise correlation time) and prevents the collapse from completing.
This is not a claim that non-white noise generically spoils Born---it does not---but that a
\emph{bare} smooth-noise branch-weight equation, lacking a response counterterm, fails. The Born
rule is recovered only if a dissipative drift---a candidate being the response part of the
\emph{projected} Adler--Millard charge fluctuation---cancels this Stratonovich drift; we present
this as a precise, conjectural acceptance test, not a derived identity. We further show that this
required cancellation is equivalent to linearity of the noise-averaged dynamics, and hence to
no-signalling, so the Born condition acquires three equivalent faces; in generalized trace dynamics,
however, no-signalling is itself an emergent, off-equilibrium property rather than a fundamental
input, so this sharpens the open problem rather than resolving it. Ordinary
fluctuation--dissipation relations are exact at thermal equilibrium; \emph{if} the projected
Adler--Millard equilibrium relation induces the specific nonlinear counterterm required, then
the Born rule is exact at equipartition and violations are controlled by the mismatch from that
balance---a quantity distinct from the overall strength of collapse, and small near equilibrium.
Finally we work out, on the model, what a non-zero mismatch $\epsilon$ would look like. The
completed-outcome probabilities are given in closed form by a regularized incomplete beta function,
$P_+=I_{p_0}(1-\epsilon,1-\epsilon)=p_0+\epsilon\,g(p_0)+O(\epsilon^2)$ with
$g(p)=(1-p)\ln(1-p)-p\ln p$: every outcome probability is pushed toward one half. Applied to the two
wings of an entangled pair---through a sequential local map, and through a two-qubit stochastic
equation with simultaneous collapse---this gives Bell--CHSH correlations $S=2\sqrt2-1.17\,\epsilon$,
a \emph{sub}-Tsirelson deviation, with $S=2$ exactly at half cancellation; a supra-quantum value would
require an over-cancelling response, for which no mechanism is known. The same mismatch produces
statistical signalling (setting dependence of remote marginals, of order $0.1\,\epsilon$ for
non-maximally entangled states, with maximally entangled states protected by a symmetry) and a
dependence of joint statistics on the collapse ordering. These are the counterparts, in GTD, of the
three signatures of Valentini's quantum non-equilibrium, here with the sign fixed toward classicality.
We discuss what it would take to detect such a violation, and show that existing tests of
higher-order (Sorkin) interference already bound $\epsilon\lesssim10^{-3}$ under a stated reading.
This corrects an earlier claim of this programme that trace dynamics predicts a violation of the
Tsirelson bound. We separate throughout what is established from what is conjectural, and identify
the single calculation---the interacting, projected Adler--Millard fluctuation propagator---on which
the programme now turns.