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\hypertarget{the-incompleteness-of-observation}{%
\section{THE INCOMPLETENESS OF
OBSERVATION}\label{the-incompleteness-of-observation}}

\hypertarget{the-equivalence-of-quantum-mechanics-and-embedded-observation}{%
\subsubsection{The Equivalence of Quantum Mechanics and Embedded
Observation}\label{the-equivalence-of-quantum-mechanics-and-embedded-observation}}

\textbf{Author:} Alex Maybaum\\
\textbf{Date:} April 2026\\
\textbf{Status:} DRAFT PRE-PRINT\\
\textbf{Classification:} Theoretical Physics / Foundations

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{abstract}{%
\subsection{ABSTRACT}\label{abstract}}

An observer embedded in a deterministic universe cannot access the
complete state. We prove that any such observer --- coupled to a slow,
high-capacity hidden sector on a finite-dimensional configuration space
--- must describe the visible sector using P-indivisible stochastic
dynamics, mathematically equivalent to unitary quantum mechanics. The
converse also holds: any quantum system, realized as a deterministic
dilation, requires non-trivial coupling, slow-bath memory, and
sufficient hidden-sector capacity. The Schrödinger equation, Born rule,
and Bell violations emerge as structural consequences requiring no
independent quantum postulates.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{introduction}{%
\subsection{1. INTRODUCTION}\label{introduction}}

\hypertarget{the-problem-of-embedded-observation}{%
\subsubsection{1.1 The Problem of Embedded
Observation}\label{the-problem-of-embedded-observation}}

An observer embedded in a deterministic universe cannot access the
complete state: degrees of freedom beyond causal reach are permanently
hidden, and the observer's description is obtained by marginalizing over
the hidden sector. The question is what this reduction imposes on the
form of the observer's physical laws.

Prior work (QBism, relational QM, 't Hooft's cellular automaton {[}1{]})
takes observer-dependence as an interpretive starting point or derives
it from specific microphysical models. This framework differs by
identifying \emph{necessary and sufficient conditions} under which any
embedded observer in any deterministic system necessarily sees quantum
mechanics --- and proving these are the \emph{only} conditions under
which QM arises from a deterministic embedding.

\hypertarget{the-starting-point}{%
\subsubsection{1.2 The Starting Point}\label{the-starting-point}}

The framework begins with a single empirical fact that cannot be
doubted: \emph{observation occurs}. An observer records distinguishable
outcomes of interactions with a system not wholly under the observer's
control. This is the cogito of Descartes made precise --- not as
philosophy, but as a mathematical constraint.

We ask: what is the minimal mathematical structure consistent with this
fact?

\textbf{Definition.} An \emph{observation} is a triple
\((S, \varphi, V)\): a total system \(S\), a dynamics
\(\varphi: S \to S\), and an observer \(V \subsetneq S\) --- a proper
subsystem with finitely many distinguishable internal states, coupled to
the complement \(H = S \setminus V\) through \(\varphi\).

This definition formalizes three features implicit in the concept of
observation: there is a whole (\(S\)), the observer is not the whole
(\(V \subsetneq S\)), and the observer registers changes (coupling
through \(\varphi\)). From this definition, three structural properties
follow.

\textbf{Lemma 1} (Finiteness). \emph{The observer has finitely many
distinguishable internal states, so the visible configuration space
\(\mathcal{C}_V\) is finite, with a discreteness scale \(\epsilon\)
providing a finite minimal cell volume.} Any observer bounded by a
finite-area surface can couple to only finitely many modes across that
boundary; independent support comes from holographic entropy bounds
{[}2{]}.

\textbf{Lemma 2} (Causal partition). \emph{The observer is a proper
subsystem: \(V \subsetneq S\). The complement \(H = S \setminus V\) is
the hidden sector.} This partition is not a modeling choice but the
definition of embedded observation: an observer that could access all of
\(S\) would have nothing external to observe. The product decomposition

\[\Gamma = \Gamma_V \times \Gamma_H, \qquad H_{\text{tot}} = H_V + H_H + H_{\text{int}}\]

follows, with cross-partition correlations entering only through
\(H_{\text{int}}\). The product decomposition is idealized; §4.5
addresses approximation quality.

\textbf{Lemma 3} (Determinism and unique measure). \emph{\(\varphi\) is
a bijection: distinct states have distinct successors (determinism) and
distinct predecessors (reversibility). The counting measure on \(S\) is
the unique \(\varphi\)-invariant measure.} Determinism (injectivity) is
the requirement that the dynamics preserves the observer's records ---
non-injective evolution would merge distinct total states, erasing the
distinction between outcomes the observer has already recorded. On a
finite set, injectivity implies surjectivity, so \(\varphi\) is a
bijection. The invariant counting measure, in the continuum limit,
becomes the Liouville measure on \(\Gamma\):

\[\frac{\partial \rho}{\partial t} = \{H_{\text{tot}}, \rho\} \equiv \mathcal{L}\rho\]

These properties contain no quantum postulates. The claim is that
quantum mechanics \emph{emerges} under the conditions below.

\hypertarget{conditions-on-the-hidden-sector}{%
\subsubsection{1.3 Conditions on the Hidden
Sector}\label{conditions-on-the-hidden-sector}}

\textbf{(C1) Non-zero coupling.} \(H_{\text{int}} \neq 0\). The coupling
is bidirectional.

\textbf{(C2) Slow-bath timescale separation.} The visible-sector
(system) timescale \(\tau_S\) is much shorter than the hidden-sector
(bath) timescale \(\tau_B\): \(\tau_S \ll \tau_B\) --- the
\emph{inverse} of the Markovian regime. The hidden sector evolves on
timescales far exceeding those accessible to the observer.

\textbf{(C3) Sufficient capacity.} The number of hidden-sector degrees
of freedom \(N_H\) exceeds the number of visible-sector degrees of
freedom \(N_V\) by enough that visible-sector interactions do not
appreciably perturb the hidden sector's state on timescales
\(\ll \tau_B\).

\textbf{Theorem statement.} Under Lemmas 1--3 and (C1)--(C3), the
embedded observer's reduced description is mathematically equivalent to
unitarily evolving quantum mechanics (defined precisely in §3.4). The
conditions are not merely sufficient but \emph{necessary} (§3.4),
establishing full equivalence.

\hypertarget{partition-relativity}{%
\subsubsection{1.4 Partition-Relativity}\label{partition-relativity}}

\textbf{Lemma.} \emph{The emergent description is uniquely determined by
the partition. Any parameters of the emergent theory depend only on the
geometric and thermodynamic properties of the partition boundary.}

\emph{Proof.} By Lemma 3, the Hamiltonian flow \(\phi_t\) is unique. By
Lemma 2, the partition is fixed. By Lemma 3, the observer uses the
Liouville measure \(\mu\) --- the unique absolutely continuous invariant
measure on the full phase space. (An embedded observer with incomplete
knowledge of the total energy averages over energies with a smooth
prior, recovering Liouville on a phase-space band; singular invariant
measures are excluded by Lemma 3.) Letting \(\pi_V\) denote projection
onto the visible sector, the marginalized transition probabilities

\[T_{ij}(t_2, t_1) = \int_{\Gamma_H} \delta_{x_j}[\pi_V(\phi_{t_2-t_1}(x_i, h))] \, d\mu(h)\]

are therefore uniquely determined by three inputs: dynamics
(\(\phi_t\)), partition (\(\Gamma_V, \Gamma_H\)), and measure (\(\mu\)).
Since \(\phi_t\) and \(\mu\) are fixed by the definition and lemmas, the
stochastic process --- and hence any emergent quantum description (§3.1)
--- depends only on the partition. \(\square\)

\hypertarget{emergent-stochasticity-and-p-indivisibility}{%
\subsection{2. EMERGENT STOCHASTICITY AND
P-INDIVISIBILITY}\label{emergent-stochasticity-and-p-indivisibility}}

\hypertarget{emergent-stochasticity}{%
\subsubsection{2.1 Emergent
Stochasticity}\label{emergent-stochasticity}}

The total system evolves deterministically, but the visible sector alone
is stochastic. The observer knows \(x\) but not \(h\); different hidden
states \(h_k\) compatible with the same \(x\) send it to different
futures \(x'_k\). Transition probabilities are obtained by marginalizing
over the Liouville measure (§1.4). This makes the framework a hidden
variable theory; consistency with Bell's theorem is addressed in §3.3.

\hypertarget{the-slow-bath-regime}{%
\subsubsection{2.2 The Slow-Bath Regime}\label{the-slow-bath-regime}}

The Markovian limit requires \(\tau_B \ll \tau_S\) {[}3{]}. Condition
(C2) inverts this: \(\tau_S \ll \tau_B\). A slow bath must be
distinguished from a static field. By (C1), coupling is continuously
active: each visible-sector transition imprints on the hidden sector.
Because the hidden sector is slow (not static), imprints persist without
thermal overwriting. On subsequent transitions, coupling reads back
stored correlations, producing history-dependent transition
probabilities --- the non-Markovian regime {[}3{]}.

\hypertarget{p-indivisibility}{%
\subsubsection{2.3 P-Indivisibility}\label{p-indivisibility}}

By Lemma 1, the visible and hidden sectors have finite configuration
spaces \(\mathcal{C}_V\) and \(\mathcal{C}_H\) (the discrete
counterparts of \(\Gamma_V\) and \(\Gamma_H\)). On these finite sets:

\textbf{Theorem.} \emph{Let \(\mathcal{C}_V\) and \(\mathcal{C}_H\) be
finite sets with \(|\mathcal{C}_V| \geq 2\), and let
\(\varphi: \mathcal{C}_V \times \mathcal{C}_H \to \mathcal{C}_V \times \mathcal{C}_H\)
be a bijection. Define:}

\[T_{ij} = \frac{|\{h \in \mathcal{C}_H : \pi_V(\varphi(x_i, h)) = x_j\}|}{|\mathcal{C}_H|}\]

\emph{and the \(k\)-step matrix \(T^{(k)}_{ij}\) by applying
\(\varphi^k\). If \(T\) is not a permutation matrix, then the process is
P-indivisible.}

\emph{Proof.} Uses total variation distance
\(d(p, q) = \frac{1}{2}\sum_k |p_k - q_k|\). P-divisibility \(\iff\)
\(d(p(t), q(t))\) non-increasing for all initial \(p, q\) {[}4, 5{]}.

\emph{Step 1 (Recurrence).} \(\varphi\) bijective on a finite set
\(\Rightarrow\) \(\exists N: \varphi^N = \text{id}\). Thus
\(T^{(N)} = I\) and \(d(\delta_i T^{(N)}, \delta_j T^{(N)}) = 1\) for
\(i \neq j\).

\emph{Step 2 (Strict contraction).} \(T\) not a permutation
\(\Rightarrow\) \(\exists i, j, l\): \(T_{il} > 0\) and \(T_{jl} > 0\).
Then:

\[d(\delta_i T, \delta_j T) = \frac{1}{2}\sum_k |T_{ik} - T_{jk}| < 1\]

\emph{Step 3.}
\(d(\delta_i T^{(1)}, \delta_j T^{(1)}) < 1 = d(\delta_i T^{(N)}, \delta_j T^{(N)})\):
non-monotonic, hence P-indivisible. \(\square\)

The theorem requires only bijective dynamics (Lemma 3) and non-trivial
coupling (C1). Lemma 1 guarantees the finite configuration space.

\textbf{Convergence with prior work.} The theorem converges with
independent results: Pechukas {[}6{]} (reduced dynamics need not
preserve positivity), Rivas et al.~{[}4, 5{]} (divisibility failure ↔
information backflow), Pollock et al.~{[}7{]} (process-tensor
Markovianity conditions), and Strasberg and Esposito {[}8{]}
(non-Markovian reduced dynamics in the slow-bath regime of C2). Bylicka,
Johansson, and Acín {[}9{]} prove that for \emph{invertible} dynamical
maps \(\Lambda_t\) (those with \(\det\Lambda_t \neq 0\)), P-divisibility
implies CP-divisibility. The substratum dynamics \(\varphi\) is
bijective on \(\mathcal{C}_V \times \mathcal{C}_H\), but the marginal
stochastic matrix \(T(t)\) on \(\mathcal{C}_V\) is generally not
invertible --- projecting onto the visible sector loses information. The
Bylicka equivalence therefore applies only when \(T(t)\) happens to have
full rank (which includes the minimal model of §2.4 below, where
\(\det T(1) = 1/3 \neq 0\)); for the general derivation here, the
load-bearing property is P-indivisibility, which is exactly what §3.1's
stochastic-quantum correspondence requires. Milz et al.~{[}10{]} further
show that even CP-divisibility does not guarantee Markovianity in the
process-tensor sense, confirming that the non-Markovian character
identified here is robust under all current definitions.

\textbf{Continuous-time extension.} The Hamiltonian flow on
finite-dimensional phase space preserves Liouville measure on compact
energy surfaces. \(T_{ij}(t)\) is continuous with \(T(0) = I\). By (C1),
\(T(t)\) departs from the permutation class for \(t > 0\). By Poincaré
recurrence, \(\exists t_R\): the set of hidden states with
\(\pi_V(\varphi_{t_R}(x_i, h)) = x_i\) has measure \(> 1 - \delta\) for
any \(\delta > 0\). For small \(\delta\), this gives non-monotonic trace
distance, establishing P-indivisibility in continuous time.

The recurrence argument establishes P-indivisibility in principle. The
following lemma shows that under (C2), information backflow occurs on
observable timescales --- not merely at Poincaré recurrence times.

\textbf{Lemma (Accessible-timescale backflow).} \emph{Under (C1)--(C3)
with \(\tau_S \ll \tau_B\), the non-Markovian mutual information
\(I(X_{<t}; X_{>t} \mid X_t)\) is \(\mathcal{O}(1)\) for observation
windows \(t \sim k\tau_S\) with \(k\tau_S \ll \tau_B\).}

\emph{Proof.} The coupling \(H_{\text{int}}\) transfers visible-sector
information to the hidden sector at each interaction, at rate
\(\sim 1/\tau_S\). Between interactions, the hidden sector evolves under
\(H_H\) with spectral gap \(\Delta \sim 1/\tau_B\). The decay of
correlations stored in the hidden sector is governed by
\(e^{-\Delta \tau_S}\). When \(\tau_S \ll \tau_B\),
\(\Delta \tau_S \ll 1\), so the decay per visible-sector transition is
\(1 - e^{-\Delta \tau_S} \approx \Delta \tau_S \ll 1\): correlations
survive each step essentially intact. After \(k\) transitions spanning a
time \(k\tau_S \ll \tau_B\), the cumulative decay is
\(e^{-k\Delta\tau_S} \approx 1 - k\Delta\tau_S\), which remains close to
unity. The hidden sector therefore retains \(\sim k\) bits of
visible-sector history over this window, and the \((k+1)\)-th transition
reads back stored correlations through \(H_{\text{int}}\), producing
history-dependent transition probabilities --- i.e., information
backflow. More precisely, the mutual information satisfies:

\[I(X_{<t}; X_{>t} \mid X_t) \geq I_0(1 - k\Delta\tau_S) = I_0\left(1 - \frac{k\tau_S}{\tau_B}\right)\]

where \(I_0 > 0\) is the single-step information transfer from (C1). For
\(k\tau_S \ll \tau_B\), this remains \(\mathcal{O}(I_0)\) --- comparable
to the single-step coupling strength, not exponentially suppressed. The
bound saturates (C3): the maximum storable history is \(\log_2 m\) bits
(§3.4), so persistent backflow over \(K\) transitions requires
\(m \geq 2^K\), which is amply satisfied when \(N_H \gg N_V\).
\(\square\)

\textbf{Role of (C2) and (C3).} The Poincaré recurrence argument and the
accessible-timescale lemma are independent: the former establishes
P-indivisibility from (C1) and finiteness alone; the latter shows that
(C2) and (C3) promote it from a formal property to an observationally
dominant one.

\textbf{Weak-coupling and fast-bath regimes.} Known P-divisible systems
with non-zero coupling illustrate the necessity of (C2), not a failure
of the theorem. The Jaynes-Cummings model with Lorentzian spectral
density is P-divisible when dissipation dominates coupling --- the
fast-bath regime (\(\tau_B \ll \tau_S\)), violating (C2). The
Davies-Merkli Markovian convergence theorem likewise applies in the
Born-Markov (fast-bath) limit. The §3.4 necessity proof for (C2)
establishes this formally.

\hypertarget{explicit-construction-a-minimal-model}{%
\subsubsection{2.4 Explicit Construction: A Minimal
Model}\label{explicit-construction-a-minimal-model}}

Visible sector: \(x \in \{0, 1\}\). Hidden sector:
\(h \in \{1, \ldots, 6\}\). Dynamics: the permutation

\[\sigma = (0{,}1 \leftrightarrow 1{,}1)\;(0{,}2 \leftrightarrow 1{,}2)\;(0{,}3 \leftrightarrow 0{,}4)\;(0{,}5 \leftrightarrow 0{,}6)\;(1{,}3 \leftrightarrow 1{,}4)\;(1{,}5 \leftrightarrow 1{,}6)\]

satisfying (C1)--(C3). Since \(\sigma^2 = \text{id}\): at \(t = 1\), two
of six hidden states flip \(x\), giving

\[T(1,0) = \begin{pmatrix} 2/3 & 1/3 \\ 1/3 & 2/3 \end{pmatrix}\]

At \(t = 2\) (\(\sigma^2 = \text{id}\)): \(T(2,0) = I\). A Markov chain
would predict continued mixing:
\(T(1,0)^2 = \left(\begin{smallmatrix} 5/9 & 4/9 \\ 4/9 & 5/9 \end{smallmatrix}\right)\).
The actual dynamics show complete \emph{un-mixing}. Computing the
intermediate propagator:

\[\Lambda(2,1) = T(2,0) \cdot [T(1,0)]^{-1} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}\]

Negative entries --- no valid stochastic matrix exists.
\textbf{P-indivisible.} \(\square\)

\textbf{Independent check (process-tensor non-Markovianity).} The same
model also fails the Pollock et al.~{[}7{]} multi-time Markov condition
without invoking \(T(t)\)-invertibility:
\(P(x_2 = 0 \mid x_1 = 1, x_0 = 0) = 1\) versus
\(P(x_2 = 0 \mid x_1 = 1) = 1/3\). Conditioning on \(x_0\) changes the
one-step prediction from \(1/3\) to \(1\) --- a memory effect computed
directly from multi-time conditional probabilities, valid regardless of
\(\det T(t)\). The non-Markovian character is therefore confirmed under
both the divisibility-based criterion (Rivas et al.~{[}4, 5{]}; Bylicka
et al.~{[}9{]} in the invertible case) and the process-tensor criterion
(Pollock et al.~{[}7{]} in the general case).

\textbf{The mechanism.} The hidden sector acts as a memory register. The
transpositions \((0,h) \leftrightarrow (1,h)\) for \(h \in \{1, 2\}\)
flip \(x\) while preserving \(h\), so a state at \(x = 1\) with
\(h \in \{1, 2\}\) after step 1 carries the record ``I was at
\(x = 0\).'' Step 2 reads this record and flips \(x\) back ---
information backflow impossible for a memoryless process.
P-indivisibility is more extreme, not less, when memory is more
persistent.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{the-emergence-of-quantum-mechanics}{%
\subsection{3. THE EMERGENCE OF QUANTUM
MECHANICS}\label{the-emergence-of-quantum-mechanics}}

\hypertarget{the-stochastic-quantum-correspondence}{%
\subsubsection{3.1 The Stochastic-Quantum
Correspondence}\label{the-stochastic-quantum-correspondence}}

By Barandes' correspondence {[}11, 12{]}, any P-indivisible stochastic
process on a finite configuration space of size \(n\) embeds into a
unitarily evolving quantum system on \(\mathcal{H}\) of dimension
\(\leq n^3\). P-indivisible transition matrices develop negative entries
at fine time resolution; the Stinespring dilation theorem guarantees
\(\exists\) a Hilbert space and unitary \(U(t)\) with
\(T_{ij}(t) = |U_{ij}(t)|^2\) --- transition probabilities are exactly
Born-rule probabilities. The proof requires a finite-dimensional
configuration space (Lemma 1); Calvo {[}13{]} extends the correspondence
to infinite dimensions, but this is not required here. Pimenta {[}14{]}
distinguishes P-divisible and P-indivisible stochastic-quantum dynamics:
the correspondence is non-trivial only in the P-indivisible regime,
which is precisely the regime produced by §2.3. An independent
derivation using only Stinespring's dilation theorem {[}15{]} and
standard partial-trace properties {[}16{]} is given in §3.2; either
route alone suffices.

\textbf{Independent corroboration of the trace-out mechanism.} The
technical move from a deterministic substrate to non-Markovian visible
dynamics via the trace-out of inaccessible degrees of freedom ---
central to both routes --- is now established at theorem level in the
open systems literature. Brandner {[}17, 18{]} proves that for
autonomous linear evolution equations, integrating out inaccessible
degrees of freedom (whether external environments or internal
subsystems) yields well-defined non-Markovian visible dynamics in a
controlled weak-memory regime, with explicit error bounds and a
convergent perturbation scheme. This is the mathematical apparatus the
present construction relies upon, derived independently for general open
systems. Direct experimental demonstrations of the mechanism include
Mehl et al.~{[}19{]}, who showed that hidden slow degrees of freedom in
a colloidal system produce non-Markovian visible dynamics violating
naive Markovian fluctuation theorems, and Gröblacher et al.~{[}20{]},
who observed non-Markovian Brownian motion in a macroscopic
micromechanical oscillator. On the quantum side, Kim {[}21{]} shows that
monitored quantum systems are formally quantum hidden Markov models,
with a rigorous correspondence to classical hidden Markov models for
setups involving Haar-random unitaries and measurements --- the same
formal structure as the stochastic-quantum bridge. The framework's
central technical move is therefore standard rather than speculative.

\textbf{From phase space to configuration space.} The transition
probabilities \(T_{ij}\) are projections of the full phase-space flow
onto the configuration sector of \(\Gamma_V\), with momenta and
hidden-sector degrees of freedom absorbed into the Liouville
marginalization. The resulting process lives on the discrete
configuration space of Lemma 1.

Three features emerge. The Schrödinger equation arises from the
differentiability of \(U(t)\). The Born rule is the equilibrium
distribution of the indivisible process {[}11, 12{]}. The action scale
\(\hbar\) enters when defining
\(\hat{H}(t) \equiv i\hbar \, \partial_t U \cdot U^\dagger\), converting
dimensionless rates to energy units; its value requires additional
physical input from the partition.

\textbf{Phase uniqueness from continuous-time data.} The relation
\(T_{ij}(t) = |U_{ij}(t)|^2\) discards phase information at any single
time. Doukas {[}22{]} identifies this gap for discrete-time data. The
resolution: continuous-time data \(\{T_{ij}(t)\}_{t \in \mathbb{R}}\)
provides strictly more information.

\textbf{Lemma (phase-locking).} \emph{Let \(H\) be Hermitian on
\(\mathbb{C}^n\) with eigenbasis \(\{|k\rangle\}\), eigenvalues
\(\{E_k\}\), and \(V_{ik} = \langle i | k \rangle\). Assume: (G1)
non-degenerate spectrum; (G2) non-degenerate energy gaps; (G3)
\(V_{ik} \neq 0\) for all \(i, k\). Then
\(T_{ij}(t) = |\langle i | e^{-iHt} | j \rangle|^2\) for all \(i, j, t\)
uniquely determines \(H\) up to an overall energy shift and basis phase
conventions.}

\emph{Proof.} Write:

\[T_{ij}(t) = \sum_{k,l} V_{ik}\, V_{jk}^*\, V_{jl}\, V_{il}^*\; e^{-i(E_k - E_l)t}\]

By (G2), the frequencies \(\omega_{kl} = E_k - E_l\) are distinct for
distinct pairs with \(k \neq l\). Fourier-transforming yields:

\[a_{ij}^{kl} = V_{ik}\, V_{jk}^*\, V_{jl}\, V_{il}^*\]

The moduli \(|V_{ik}|\) follow from
\(a_{ii}^{kl} = |V_{ik}|^2 |V_{il}|^2\) (non-zero by (G3)): the rank-1
outer-product structure determines \(|V_{ik}|^2\) up to a global scale,
and unitarity \(\sum_k |V_{ik}|^2 = 1\) fixes the scale. The phases
carry a double-difference structure:

\[\arg(a_{ij}^{kl}) = (\varphi_{ik} - \varphi_{il}) - (\varphi_{jk} - \varphi_{jl})\]

The gauge freedom preserving all double differences is
\(\varphi_{ik} \to \varphi_{ik} + \alpha_i + \beta_k\) (physically
irrelevant basis rephasing). Fixing \(\varphi_{1k} = 0\),
\(\varphi_{i1} = 0\), each remaining phase is
\(\varphi_{ik} = \arg(a_{i1}^{k1})\), directly extracted from Fourier
data. \(H = V \, \text{diag}(E_k) \, V^\dagger\) is unique up to energy
shift and phase conventions. \(\square\)

Conditions (G1)--(G3) fail only on a measure-zero set of Hamiltonians. A
stronger argument applies whenever the partition itself breaks
symmetries of \(H_{\text{tot}}\). The genericity conditions concern the
\emph{effective} visible-sector Hamiltonian \(\hat{H}_{\text{eff}}\) ---
the operator emerging after the trace-out --- not the total Hamiltonian
\(H_{\text{tot}}\). The trace-out generically breaks the symmetries of
\(H_{\text{tot}}\) because the partition boundary does not respect them:
a partition centered on a specific observer breaks translational
invariance, rotational symmetry about distant points, and boost
invariance. Any degeneracy in \(H_{\text{tot}}\) that relies on
correlations between visible and hidden sectors --- which includes all
degeneracies associated with symmetries the partition breaks --- is
lifted in \(\hat{H}_{\text{eff}}\). The residual degeneracies of
\(\hat{H}_{\text{eff}}\) are those associated with symmetries acting
entirely within the visible sector that commute with the trace-out:
precisely the gauge symmetries of the emergent description. Gauge
degeneracies correspond to physically equivalent states and do not
affect the phase-locking argument, which operates on gauge-inequivalent
transition data. \(\hat{H}_{\text{eff}}\) therefore satisfies G1--G3 not
by typicality in an abstract ensemble, but because the partition itself
acts as a symmetry-breaking mechanism.

\hypertarget{independent-derivation-via-stinespring-dilation}{%
\subsubsection{3.2 Independent Derivation via Stinespring
Dilation}\label{independent-derivation-via-stinespring-dilation}}

An independent derivation of the emergent quantum description uses only
Stinespring's dilation theorem {[}15{]} and standard properties of the
partial trace {[}16{]}, without invoking the stochastic-quantum
correspondence of §3.1. The two routes are logically independent; either
alone suffices.

\textbf{Setup.} The finite configuration spaces
\(\mathcal{C}_V = \{x_1, \ldots, x_n\}\) and
\(\mathcal{C}_H = \{h_1, \ldots, h_m\}\) (Lemma 1) embed into Hilbert
spaces \(\mathcal{H}_V = \mathbb{C}^n\) and
\(\mathcal{H}_H = \mathbb{C}^m\) via \(|i\rangle \leftrightarrow x_i\)
and \(|k\rangle \leftrightarrow h_k\). This introduces no quantum
postulates: it is the canonical identification of probability
distributions on a finite set with diagonal density matrices.

\textbf{Lemma (permutation unitarity).} \emph{Any bijection
\(\varphi: \mathcal{C}_V \times \mathcal{C}_H \to \mathcal{C}_V \times \mathcal{C}_H\)
defines a unitary \(U_\varphi\) on
\(\mathcal{H} = \mathcal{H}_V \otimes \mathcal{H}_H\).}

\emph{Proof.} Define
\(U_\varphi |i, k\rangle = |\varphi(x_i, h_k)\rangle\). Since
\(\varphi\) is a bijection, \(U_\varphi\) permutes the orthonormal
basis, hence is unitary. \(\square\)

For continuous-time dynamics \(\varphi_t\), Stone's theorem on
\(\mathcal{H}\) yields \(U_t = e^{-i\hat{H}t}\) for Hermitian
\(\hat{H}\).

\textbf{Lemma (Reverse direction: CPTP → bijection).} \emph{Any CPTP
channel \(\Phi\) on a finite-dimensional Hilbert space
\(\mathcal{H}_V = \mathbb{C}^n\) admits, for any \(\varepsilon > 0\), a
realization as marginalization of a permutation-unitary bijection on a
larger finite configuration space: there exist finite sets
\(\mathcal{C}_V\), \(\mathcal{C}_H\) and a bijection
\(\varphi: \mathcal{C}_V \times \mathcal{C}_H \to \mathcal{C}_V \times \mathcal{C}_H\)
such that the induced channel \(\Phi_\varphi\) via partial trace (as
constructed in the forward direction) satisfies
\(\|\Phi - \Phi_\varphi\|_\diamond \leq \varepsilon\), where
\(\|\cdot\|_\diamond\) is the diamond norm.}

\emph{Proof.} Stinespring's theorem {[}15{]} gives a Hilbert-space
isometry \(V: \mathcal{H}_V \to \mathcal{H}_V \otimes \mathcal{H}_H\)
and a unitary \(U\) on \(\mathcal{H}_V \otimes \mathcal{H}_H\) extending
\(V\), with
\(\Phi(\rho) = \mathrm{Tr}_H[U(\rho \otimes \rho_0)U^\dagger]\) for some
fixed \(\rho_0 \in \mathcal{S}(\mathcal{H}_H)\). By Solovay--Kitaev
{[}28{]} (or any standard universal-gate-set density result), for any
\(\varepsilon > 0\) there exists a permutation-unitary \(U_\varphi\) on
a possibly enlarged \(\mathcal{H}_V \otimes \mathcal{H}_H'\) with
\(\|U - U_\varphi\| \leq \varepsilon\) in operator norm. The enlarged
space corresponds to a finite set
\(\mathcal{C}_V \times \mathcal{C}_H'\) via the canonical basis
identification (Setup paragraph), and \(U_\varphi\) corresponds to a
bijection \(\varphi\). Error propagation through partial trace gives
\(\|\Phi - \Phi_\varphi\|_\diamond \leq 2\varepsilon\). Replacing
\(\varepsilon\) with \(\varepsilon/2\) in the initial approximation
gives the claim. \(\square\)

\emph{Remark.} This Lemma establishes that any CPTP channel on
finite-dimensional Hilbert space can be approximated to arbitrary
precision by a bijection on a finite configuration space. The finiteness
of \(\mathcal{H}_V\) follows from the holographic bound
\(\dim \mathcal{H} \leq e^{A/4}\) applied to any bounded region. This is
the reverse direction invoked by {[}Substratum, Theorem 23, Stage 1{]}:
observed finite-dim CPTP dynamics can always be realized as the
visible-sector projection of a finite-bijection substrate. The forward
direction (bijection → CPTP) is the preceding Permutation Unitarity
Lemma; together they establish the bidirectional correspondence between
finite CPTP channels and finite bijections (up to arbitrary precision).

\textbf{The quantum channel.} The observer's ignorance of the hidden
sector (Lemma 3) corresponds to \(\rho_H = I_m/m\). The visible-sector
quantum channel is

\[\Phi(\rho_V) = \mathrm{Tr}_H\!\left[U_\varphi\,(\rho_V \otimes \rho_H)\,U_\varphi^\dagger\right]\]

This is CPTP by a standard result {[}16, Theorem 8.1{]}, with Kraus
representation
\(\Phi(\rho_V) = \sum_{k,l} K_{kl}\,\rho_V\,K_{kl}^\dagger\) where
\(K_{kl} = m^{-1/2}\langle l|U_\varphi|k\rangle_H\). The triple
\((\mathcal{H}_H, U_\varphi, \rho_H)\) is the Stinespring dilation
{[}15{]} of \(\Phi\).

\textbf{Theorem (Born rule recovery).} \emph{The classical transition
probabilities \(T_{ij}\) (§1.4) equal the Born-rule probabilities of
\(\Phi\).}

\emph{Proof.}
\(P(j|i) = \langle j|\Phi(|i\rangle\langle i|)|j\rangle = m^{-1}\sum_{k,l} |\langle j,l|U_\varphi|i,k\rangle|^2\).
Since \(U_\varphi\) is a permutation,
\(\langle j,l|U_\varphi|i,k\rangle = \delta_{(j,l),\varphi(i,k)}\). Thus
\(P(j|i) = m^{-1}\sum_k \delta_{j,\pi_V(\varphi(x_i,h_k))} = T_{ij}\).
\(\square\)

\textbf{Theorem (emergent coherence).} \emph{If \(T\) is not a
permutation matrix (condition (C1)), then \(\Phi\) generates genuine
quantum coherence: it is not entanglement-breaking.}

\emph{Proof.} If \(T\) is not a permutation, some initial state
\(|i\rangle\) maps to at least two distinct outputs, so
\(\Phi(|i\rangle\langle i|)\) has rank \(\geq 2\). By linearity,
\(\Phi\) maps some off-diagonal input \(|i\rangle\langle j|\) to a
non-zero operator, precluding the measure-and-prepare form of
entanglement-breaking channels. \(\square\)

\textbf{Theorem (CP-indivisibility).} \emph{The P-indivisibility of §2.3
implies CP-indivisibility of \(\{\Phi_t\}\): there exist
\(t_2 > t_1 > 0\) with no CPTP map \(\Lambda\) satisfying
\(\Phi_{t_2} = \Lambda \circ \Phi_{t_1}\).}

\emph{Proof.} CP-divisibility restricted to diagonal inputs reduces to
P-divisibility. The contrapositive gives the result. \(\square\)

By the Breuer-Laine-Piilo criterion {[}3, 4{]}, CP-indivisibility
implies non-monotonic trace distance (information backflow) --- the
quantum signature of non-Markovianity.

\textbf{Approximate unitarity.} On observable timescales
\(t \ll \tau_B\), the hidden-sector state is approximately frozen
(conditions (C2)--(C3)). The channel generator decomposes as
\(d\Phi_t/dt|_{t=0}(\rho_V) = -i[\hat{H}_{\text{eff}}, \rho_V] + \mathcal{D}(\rho_V)\)
with \(\mathcal{D} \sim \mathcal{O}(\tau_S/\tau_B)\). The leading-order
dynamics is the Schrödinger equation. The phase-locking lemma (§3.1)
then determines \(\hat{H}_{\text{eff}}\) uniquely from continuous-time
transition data, up to the residual diagonal-unitary rephasing freedom
that is physically trivial (basis convention).

\textbf{Comparison of routes.}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Barandes route (§3.1)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Stinespring route (§3.2)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Input & P-indivisible process on \(\mathcal{C}_V\) & Bijection on
\(\mathcal{C}_V \times \mathcal{C}_H\) + marginalization \\
Bridge & Stochastic-quantum correspondence {[}11, 12{]} & Stinespring
dilation {[}15{]} + partial trace {[}16{]} \\
Output & Unitary with \(T_{ij} = |U_{ij}|^2\) & CPTP channel with
\(T_{ij} = \langle j|\Phi(|i\rangle\langle i|)|j\rangle\) \\
Born rule & Equilibrium of indivisible process & Partial trace structure
(above) \\
Scope & Any P-indivisible process & Processes from marginalized
bijections \\
\end{longtable}

The Barandes route is more general (any P-indivisible process); the
Stinespring route requires only textbook results {[}15, 16{]} and
delivers additional structure: the tensor product
\(\mathcal{H}_V \otimes \mathcal{H}_H\), genuine quantum coherence,
CP-indivisibility, and approximate unitarity with
\(\mathcal{D} \sim \mathcal{O}(\tau_S / \tau_B) \sim 10^{-32}\). The two
routes agree on all observables --- both produce the same \(T_{ij}(t)\)
--- establishing that the emergence of QM is robust and overdetermined.
The Barandes route powers (ii) \(\implies\) (i) in the characterization
theorem; the Stinespring route supplies (iii) \(\implies\) (i)
independently.

\hypertarget{bell-inequality-violations}{%
\subsubsection{3.3 Bell Inequality
Violations}\label{bell-inequality-violations}}

The framework is a hidden variable theory evading Bell's theorem
{[}23{]} not by superdeterminism but by violating factorizability
through P-indivisible joint dynamics --- while remaining causally local.
Two subsystems interacting at preparation carry a joint transition
matrix \(T_{QR} \neq T_Q \otimes T_R\). This non-factorizability
\emph{is} stochastic entanglement {[}11, 12, 24{]}. Since the process is
indivisible, no well-defined intermediate conditional probabilities
permit Bell's factorization.

In the Jarrett decomposition, the framework violates outcome
independence while preserving parameter independence and measurement
independence --- precisely the class Fine's theorem {[}25{]} permits.
Barandes, Hasan, and Kagan {[}24{]} prove the maximum CHSH correlator is
exactly Tsirelson's bound \(2\sqrt{2}\), with independent support from
Le et al.~{[}26{]}. The bound also follows from the Stinespring route of
§3.2, which establishes full unitary QM on a tensor-product Hilbert
space; Tsirelson's original operator-norm argument {[}27{]} then yields
\(|\langle S \rangle| \leq 2\sqrt{2}\); achievability follows because
all quantum states, including maximally entangled ones, are realizable
in the emergent description.

\textbf{Fine-tuning and causal structure.} The framework does not claim
to restore Bell locality: outcome independence is genuinely violated.
What it provides is a \emph{derivation} of this nonlocality from the
causal partition, rather than treating it as axiomatic. The fine-tuning
objection (Wood and Spekkens, 2015) assumes DAG causal structure with
Markov factorization; P-indivisible processes violate the Markov
condition on any DAG, so the appropriate framework is the process tensor
{[}7{]}, within which no-signaling follows from marginalization
structure without fine-tuning.

\textbf{Parameter independence from spatial locality.} Let \(V_A\) and
\(V_B\) be space-like separated visible-sector subsystems with disjoint
coupling neighborhoods on \(G_\varphi\). The spatial Markov property of
range-1 dynamics (the area-law lemma: for spatially local bijections,
the mutual information between two regions depends only on their shared
boundary) makes \(V_A\) and \(V_B\) conditionally independent given
boundary data. Alice's measurement setting \(a\) determines which
function she evaluates on \(V_A\); it does not alter the transition
probabilities for \(V_B\), which depend only on \(V_B\)'s local coupling
neighborhood. Hence \(P(x_B \mid a, b) = P(x_B \mid b)\) --- parameter
independence holds structurally for all bounded-degree graph topologies
with range-1 coupling, with no fine-tuning of the hidden-variable
distribution. Combined with the outcome-independence violation from
stochastic entanglement, this gives the precise Jarrett decomposition.

\hypertarget{the-characterization-theorem-necessity-of-the-conditions}{%
\subsubsection{3.4 The Characterization Theorem: Necessity of the
Conditions}\label{the-characterization-theorem-necessity-of-the-conditions}}

The logical structure: Barandes' correspondence gives QM \(\iff\)
P-indivisibility. What remains is P-indivisibility \(\implies\) embedded
observation under (C1)--(C3).

\textbf{Dilation existence.} Any stochastic process on a finite
configuration space can be realized as marginalization of a
deterministic process on a larger state space --- furnishing Lemmas
1--3.

\textbf{Theorem (C1 equivalence).} \emph{The marginalized process is
P-indivisible iff \(T\) is not a permutation matrix.}

\emph{Proof.} Forward: §2.3. Reverse: if \(T\) is a permutation,
\(\Lambda(k_2, k_1) = T^{(k_2-k_1)}\) is a valid stochastic matrix for
all \(k_2 > k_1\) \(\Rightarrow\) P-divisible. \(\square\)

\textbf{Theorem (C2 necessity, conditional on ETH).} \emph{Assume the
hidden-sector Hamiltonian \(H_H\) satisfies the eigenstate
thermalization hypothesis (ETH) {[}17, 18{]}. Then in the fast-bath
regime (\(\tau_B \ll \tau_S\)), the hidden sector dephases to its
diagonal ensemble between coupling events, so the marginal dynamics on
\(\mathcal{C}_V\) is Markovian on accessible timescales
(\(T^{(k)} = T^k\)) and hence P-divisible. Contrapositively, observable
P-indivisibility in an ETH hidden sector requires
\(\tau_S \ll \tau_B\).}

\emph{Proof.} Between coupling events (separated by \(\tau_S\)), the
hidden sector evolves under its own Hamiltonian \(H_H\). The substratum
dynamics is reversible, so there is no relaxation in the strict sense
--- finite-dimensional Hamiltonian flow has purely imaginary spectrum
and Poincaré recurrence at \(t_R\) exponentially large in the system
size. What does occur, under the ETH assumption, is \emph{dephasing}:
starting from any distribution \(\mu_H(\cdot | x_i)\) conditioned on the
previous visible state, the off-diagonal matrix elements in the energy
basis decay on the timescale \(\tau_B \sim \hbar / \Delta E\), where
\(\Delta E\) is the hidden sector's typical energy spread. After
dephasing, the diagonal ensemble agrees with the microcanonical
distribution to corrections exponentially small in the hidden-sector
size, so for any few-body visible-sector observable \(O_V\) and any time
\(\tau_S \gg \tau_B\):

\[\big| \langle O_V \rangle_{\mu_H(\tau_S | x_i)} - \langle O_V \rangle_{\text{eq}} \big| \;\lesssim\; \tau_B / \tau_S\]

In the fast-bath regime (\(\tau_B \ll \tau_S\)) this is parametrically
small: from the visible sector's perspective the hidden sector ``looks''
thermalized at every coupling event regardless of its post-coupling
state, even though the underlying trajectory remains reversible and
quasi-periodic with recurrence only at \(t \sim t_R\). Each single-step
transition matrix \(T\) is therefore computed against the same effective
equilibrium distribution, so \(T^{(k)} = T^k\) --- a homogeneous Markov
chain on accessible timescales \(t \ll t_R\), hence P-divisible in the
observable sense.

Contrapositively: observable P-indivisibility requires
\(\tau_S \lesssim \tau_B\) --- the visible sector must couple back into
the hidden sector before dephasing erases the memory. For the strong,
persistent P-indivisibility of the characterization theorem, the
separation must be \(\tau_S \ll \tau_B\). The continuous-time extension
to general Hamiltonian flow replaces the collision-model picture with
the full ETH/dephasing condition; the relevant machinery for chaotic
open quantum systems in the weak-memory regime is established in {[}3,
8, 17, 18{]}. \(\square\)

\textbf{Remark (Status of ETH).} \emph{The eigenstate thermalization
hypothesis is a well-supported conjecture, not a theorem, for generic
chaotic many-body Hamiltonians. It is proved for specific models (random
matrix ensembles, certain integrable-chaotic transitions) and supported
by extensive numerical evidence across many systems {[}17, 18{]}. For
the OI framework's hidden sector --- the cosmological-horizon
complement, a generic chaotic many-body system --- ETH is the standard
assumption. The C2 necessity direction is therefore conditional on ETH;
relaxations or failures of ETH would require a separate argument. The C1
and C3 necessity theorems do not depend on ETH.}

\textbf{Remark (Consequence for the characterization theorem).}
\emph{The full biconditional (i) ⇔ (ii) ⇔ (iii) of the characterization
theorem below is similarly conditional on ETH for the (ii) ⇒ (iii)
direction specifically. The other directions --- (i) ⇔ (ii) from {[}11,
12{]}; (iii) ⇒ (ii) from §2.3; C1 and C3 necessity from their own proofs
--- are unconditional.}

\textbf{Theorem (C3 necessity).} \emph{Let \(m = |\mathcal{C}_H|\). The
non-Markovian mutual information satisfies:}

\[I(X_{<t} ; X_{>t} \mid X_t) \leq \log_2 m\]

\emph{Proof.} The total system is deterministic: \(X_{>t}\) is a
function of \((X_t, H_t)\). Conditioning on \(X_t\):
\(X_{<t} \to H_t \to X_{>t}\) is a Markov chain. By data processing:

\[I(X_{<t} ; X_{>t} \mid X_t) \leq I(X_{<t} ; H_t \mid X_t) \leq H(H_t \mid X_t) \leq \log_2 m \quad \square\]

\textbf{Corollary.} P-indivisibility across \(n\) visible configurations
requires \(m \geq n\). Sustained information backflow at rate
\(I_0 > 0\) per coupling event over \(K\) events requires
\(m \gtrsim 2^{K I_0}\) --- exponential growth in the observation
window, of which the framework's qualitative (C3) (``hidden sector large
enough'') is a conservative restatement. \(\square\)

\textbf{Definition (unitarily evolving QM).} A stochastic process
\(\mathcal{S}\) on a finite configuration space
\(\mathcal{C}_V = \{x_1, \ldots, x_n\}\) is \emph{mathematically
equivalent to unitarily evolving quantum mechanics} if there exists a
Hilbert space \(\mathcal{H}_V\) of dimension \(\leq n^3\), a Hermitian
operator \(\hat{H}\) on \(\mathcal{H}_V\), and a unitary family
\(U(t) = e^{-i\hat{H}t}\) such that \(T_{ij}(t) = |U_{ij}(t)|^2\) for
all \(i, j, t\) --- i.e., transition probabilities are exactly Born-rule
probabilities. This definition captures the Hilbert space, unitary
dynamics, and the Born rule. Additional quantum-mechanical structures
--- the tensor product decomposition for spatially separated
visible-sector subsystems, state update, and the measurement formalism
--- are addressed in the remarks following the theorem.

\textbf{Characterization theorem.} \emph{Let \(\mathcal{S}\) be a
stochastic process on a finite configuration space \(\mathcal{C}_V\)
with \(|\mathcal{C}_V| \geq 2\), and consider dynamics on accessible
timescales \(t \ll t_R\) where \(t_R\) is the Poincaré recurrence time
of any realizing substrate. Assume the hidden sector of any candidate
realization satisfies ETH {[}17, 18{]}. Then the following are
equivalent:}

\emph{(i) \(\mathcal{S}\) is mathematically equivalent to unitarily
evolving QM (in the sense of the preceding definition).}

\emph{(ii) \(\mathcal{S}\) is P-indivisible on accessible timescales ---
i.e., information backflow occurs at \(t \ll t_R\), not merely at the
formal Poincaré recurrence time.}

\emph{(iii) \(\mathcal{S}\) arises from marginalizing a deterministic
bijection on \(\mathcal{C}_V \times \mathcal{C}_H\) with (C1)
non-trivial coupling, (C2) slow-bath memory (\(\tau_S \ll \tau_B\)), and
(C3) sufficient capacity.}

\emph{Proof.} (i) \(\iff\) (ii): Barandes {[}11, 12{]}, on accessible
timescales by construction (the equivalence is formulated for the
dynamics observers can resolve, not for behavior at recurrence times).
(iii) \(\implies\) (ii): §2.3, where the accessible-timescale lemma
upgrades the bare recurrence-based P-indivisibility (which follows from
C1 alone) to information backflow on observable timescales using C2 and
C3. (ii) \(\implies\) (iii): the necessity theorems above --- C1 from
the equivalence with non-permutation \(T\), C2 from the dephasing
argument (without C2, \(T\) becomes Markovian on accessible timescales
and information backflow disappears), C3 from the data-processing bound.
\(\square\)

QM is not merely \emph{compatible with} embedded observation --- it is
\emph{equivalent to} it.

\textbf{Remark (scope of the equivalence).} The characterization theorem
establishes the Hilbert space, unitary dynamics, and Born-rule
transition probabilities. Four further structures of operational quantum
mechanics merit comment.

\emph{(i) Tensor product structure (visible--hidden).} The Stinespring
route (§3.2) constructs
\(\mathcal{H} = \mathcal{H}_V \otimes \mathcal{H}_H\) and derives the
CPTP channel \(\Phi\) by partial trace. The visible--hidden tensor
product is therefore given by the construction, not postulated.

\emph{(ii) Tensor product structure (visible-sector subsystems).}
Spatially separated subsystems within the visible sector --- e.g., two
laboratories --- correspond to subsets \(V_A, V_B \subset V\) with
disjoint coupling neighborhoods on the coupling graph. The spatial
Markov property of range-1 dynamics (the area-law lemma) guarantees
conditional independence: \(\rho_{AB} = \rho_A \otimes \rho_B\)
conditioned on boundary data. The emergent Hilbert space inherits the
factorization \(\mathcal{H}_{V_A} \otimes \mathcal{H}_{V_B}\) from the
locality of the coupling graph, with entanglement between \(A\) and
\(B\) arising from shared boundary history --- precisely the stochastic
entanglement of {[}11, 24{]}.

\emph{(iii) State update, measurement, and multi-time predictions.}
Projective measurement in the emergent description corresponds to
conditioning on a visible-sector outcome and re-marginalizing over the
hidden sector. The Lüders rule
\(\rho \to P_k \rho P_k / \text{Tr}(P_k \rho)\) is the quantum
transcription of Bayesian updating on the classical substratum (Lemma
3). No independent measurement postulate is required. Multi-time
correlation functions, weak values, and Leggett-Garg inequalities are
standard calculations within the delivered formalism (\(\mathcal{H}_V\),
\(U(t)\), Born rule); they require no additional postulates beyond what
the characterization theorem provides.

\emph{(iv) Classical non-Markovianity and quantum mechanics.} The
equivalence is not a claim that all P-indivisible processes are
``secretly quantum'' in some metaphysical sense; it is a theorem that
the mathematical structures are identical. Classical systems exhibiting
P-indivisibility --- renewal processes, semi-Markov processes on finite
state spaces --- admit a unique unitary description (phase-locking
lemma) that makes predictions beyond the transition data (energy
eigenvalues, interference, entanglement), and can be realized as
marginalization of deterministic dynamics under C1--C3 (necessity
theorems). The framework does not distinguish ``classical
non-Markovian'' from ``quantum'' --- it proves they are the same
category. The physical content of the unitary is not that of a mere
mathematical embedding: it is the \emph{specific} Hamiltonian uniquely
determined by the transition data, arising from the \emph{actual}
dilation structure of the system (§3.2).

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{discussion}{%
\subsection{4. DISCUSSION}\label{discussion}}

\hypertarget{interpretive-consequences}{%
\subsubsection{4.1 Interpretive
Consequences}\label{interpretive-consequences}}

The degrees of freedom involved in quantum experiments --- photons,
electrons, slits, detectors --- are all visible-sector objects. Their
quantum behavior is a downstream consequence of the trace-out: the
emergent quantum mechanics, once established by the theorem, governs all
visible-sector dynamics.

In the double-slit experiment, the particle traverses a single slit in
the deterministic substratum. The interference pattern arises because
opening or closing the second slit changes the boundary conditions of
the transition matrix, altering the distribution of detection events. A
which-path detector at one slit couples the trajectory to additional
visible-sector degrees of freedom, changing the transition matrix and
eliminating the interference terms. In Wigner's friend, the Friend has a
definite outcome; Wigner's superposition reflects his epistemic deficit.
The Everettian measure problem dissolves: the system evolves as a single
reality; ``branches'' are features of the compressed description. The
Born rule, often treated as an independent postulate, is derived: it is
the equilibrium distribution of the indivisible stochastic process
(§3.1), not an additional assumption.

\hypertarget{scope}{%
\subsubsection{4.2 Scope}\label{scope}}

The characterization theorem is a full equivalence: QM \(\iff\) embedded
observation under (C1)--(C3). The theorem does not identify which
physical systems satisfy the conditions; this is an empirical question.
Universality in our universe follows from the shared causal partition
defined by null geodesics.

\hypertarget{the-status-of-the-discreteness-scale}{%
\subsubsection{4.3 The Status of the Discreteness
Scale}\label{the-status-of-the-discreteness-scale}}

\(\epsilon\) does not smuggle in a quantum assumption. Lemma 1 requires
finite-dimensionality, motivated by the classical fact that finite-area
boundaries admit finitely many modes. Holographic entropy bounds {[}2{]}
provide independent, non-quantum motivation.

\hypertarget{relation-to-prior-work}{%
\subsubsection{4.4 Relation to Prior
Work}\label{relation-to-prior-work}}

\textbf{'t Hooft {[}1{]}:} Differs in mechanism (epistemic trace-out
vs.~information loss), generality (any embedded observer vs.~particular
rules), and Bell placement (outcome independence violation
vs.~superdeterminism). \textbf{QBism/relational QM:} Share
observer-relative epistemic states; this framework provides a structural
\emph{why} and quantitative predictions.

Witten's Type II algebra program achieves finite entropy without
finite-dimensional Hilbert spaces; this is compatible, since the
framework's predictions depend on finite entropy (the physical content
of Lemma 1), not on dimensional finiteness per se.

\hypertarget{assumptions-limitations-and-falsifiability}{%
\subsubsection{4.5 Assumptions, Limitations, and
Falsifiability}\label{assumptions-limitations-and-falsifiability}}

\textbf{The theorem} requires Lemma 1, the stochastic-quantum bridge
(§3.1 and §3.2; established by two independent routes), and genericity
conditions (non-degenerate spectrum, non-vanishing overlaps), which hold
for all but a measure-zero set. The P-indivisibility proof uses
finiteness via the recurrence argument (\(\varphi^N = \text{id}\)). The
accessible-timescale backflow lemma (§2.3) provides a parallel route to
P-indivisibility, establishing observable information backflow on the
timescales accessible to laboratory experiments --- physically
indistinguishable from a timeless identity for all experiments well
below the recurrence time.

\textbf{Falsifiability.} The theorem would be falsified by a failure of
both the stochastic-quantum correspondence and the Stinespring
construction for the class of processes generated here --- a possibility
excluded by the independent proofs in §3.1 and §3.2.

\textbf{A near-term mechanism-side test.} A more tractable falsification
target is provided by the recently developed operational distinction
between classical and quantum memory in non-Markovian processes {[}28,
29{]}. Some non-Markovian quantum dynamics can be simulated using only
classical stored information, while others require genuinely quantum
memory in the environment, with the dividing line set by entanglement
structure of the channel's Choi state. The framework predicts that all
\emph{fundamental} non-Markovian dynamics in nature --- those arising
from embedded observation rather than from engineered quantum baths ---
should be classical-memory simulable, because the hidden sector is by
construction a classical substrate. A physically realized fundamental
open-system process whose memory provably exceeds the classical bound
would falsify the framework's foundational ontology. This is a near-term
tractable test in a literature where the experimental and theoretical
tools are advancing rapidly.

Laboratory tests of the characterization theorem may be possible in
tabletop systems where the hidden sector capacity is tunable.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{conclusion}{%
\subsection{5. CONCLUSION}\label{conclusion}}

\textbf{The general theorem.} Under a single definition --- observation
is a proper subsystem of a deterministic whole --- and conditions
(C1)--(C3), the P-indivisibility of reduced dynamics is proved: any
bijective dynamics on a finite system with non-trivial coupling
necessarily produces P-indivisible stochastic dynamics. By the
stochastic-quantum correspondence, the observer's description is
necessarily quantum mechanics. Continuous-time transition data resolves
all gauge freedom. The Schrödinger equation, Born rule, and Bell
violations are structural consequences. The characterization theorem
establishes that the conditions are necessary: QM and embedded
observation under (C1)--(C3) are equivalent.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{acknowledgements}{%
\subsection{ACKNOWLEDGEMENTS}\label{acknowledgements}}

During the preparation of this work, the author used Claude Opus 4.6
(Anthropic) and Gemini 3.1 Pro (Google) to assist in drafting, refining
argumentation, and verifying bibliographic details. The author reviewed
and edited all content and takes full responsibility for the
publication.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\hypertarget{references}{%
\subsection{REFERENCES}\label{references}}

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