Finite-Distinction Quantum Mechanics: Unitary Evolution and Superposition as the Large-Distinction Limit of Stochastic Thermodynamics

Author: Rowan Brad Quni-Gudzinas | Date: 2026-08-20 | License: CC-BY-4.0

Abstract

The continuum is an infinite-information object: a single real coordinate specifies

infinitely many yes/no distinctions, and a finite-entropy world cannot contain such

coordinates. This paper assembles the consequences of that observation into a

three-part thesis. First, the finite-information principle is taken as an inherited

premise with a known refinement: what is physically vacuous is uncountable precision,

while computable depth and p-adic valuation remain physically real. Second, the

natural geometry of finite distinctions is combinatorial: at any fixed resolution,

states are either distinct or not, and the induced distance is ultrametric — there is

no arbitrarily small betweenness. Third, reading quantum mechanics as thermodynamics,

unitary evolution and superposition are conjectured to be the large-distinction limit

of an entropy-Hessian gradient flow over finite alternatives; the Hilbert-space

formulation over the complex numbers is that limit, a map rather than the territory.

The algebra of the first two parts is exact and graded; the emergence conjecture is

stated with its named obstacles — the appearance of complex structure, the

psi-epistemic no-go theorems, and the observational underdetermination of the

ontology — and with a concrete computational program that can falsify it at finite N.

The premises end where the identification of the state space begins: the distinction,

the counting of distinctions, and finite resolution are unanalyzable primitives here;

everything above them is derived, proposed, or conjectured, and the falsification

conditions are written.

1. Introduction

The starting point of this paper is a result from the adelic picture [2]: the p-adic

maximum-entropy distribution is exactly the Bose–Einstein occupation distribution at

inverse temperature ln p, the squarefree restriction of the integers is its

Fermi–Dirac counterpart, and an ideal quantum non-demolition measurement of a

p-adic-valued observable is the equality case of the adelic data-processing inequality

[1]. Those identifications are exact and computationally verified. They were assembled

under a structural thesis: the constants e, pi, and the exchange phase R = (e^{i\pi})^{2s}

form one self-referential scalar family generated by the act of drawing a distinction.

This paper asks what that thesis implies for the geometry of state space. The seed of

the argument is an information-theoretic observation that has been made, in different

forms, across the foundations literature and in our own corpus: a real coordinate is a

map that requires infinitely many distinctions per point, and a finite-entropy world

cannot carry such objects [3, 4, 5, 6, 7, 8]. We assemble the consequences in three

claims, graded as exact identities, proposed dictionary entries, or conjectures with

falsification conditions.

1. The continuum claim (inherited premise, refined). Physical quantities contain

only finite information; real numbers beyond a computable modulus are not physical

[3, 4, 8]. The refinement we adopt from the ontological-closure program decomposes

the continuum into depth (Archimedean completeness — physically real, required for

dynamics), breadth (uncountable cardinality — physically vacuous), and valuation

(p-adic completions — physically real but not geometric) [9]. What is excluded by

finiteness is uncountable precision, not computable structure.

1. **The geometry claim (exact in the partition sense, conditional in the metric

sense).** Distinguishability at fixed resolution is an equivalence relation; the

induced state space is a hierarchy of partitions whose natural distance is

ultrametric [10]. The conditional reading is bounded by our own computational study

of Page–Wootters clocks, which shows that ultrametricity is not generic for

conditional-state overlap distances [11].

1. The emergence conjecture (novel, conjectural). Quantum mechanics, read as

thermodynamics, is a stochastic thermodynamics of finite alternatives [12, 13], and

unitary evolution together with superposition are the large-distinction limit of an

entropy-Hessian gradient flow. The Hilbert space over the complex numbers is the

thermodynamic limit — a map, not the territory.

Why this matters. If the Hilbert space is a thermodynamic limit, then the cost of

a correct quantum answer is a function of distinctions made — a countable,

benchmarkable resource. That converts the energy-efficiency question of quantum

computing from an engineering estimate into a structural quantity, and it gives the

joules-per-solution benchmark a substrate (Section 10). The emergence conjecture is

the load-bearing part: it lives or dies on finite-N predictions that the continuum

reading does not make, and those predictions are computable.

Where the premises end. The argument rests on three unanalyzable primitives: the

distinction (this/that), entropy as the log-count of distinctions, and finite

resolution (the minimum distinction). Imported machinery includes maximum-entropy

reasoning [14], the identification of the entropy Hessian with the Fisher metric [15],

stochastic thermodynamics of discrete states [12, 16, 17], the Page–Wootters formalism

[18, 19], ultrametricity of hierarchical order [10, 20], and the semiorder structure

of indistinguishability [21]. Everything above these is derived, proposed, or

conjectured in this paper, and the falsification conditions in Section 8 are the

enforcement.

2. The Continuum as Infinite Information

A single real coordinate, taken literally, specifies infinitely many yes/no

distinctions: its binary expansion is an infinite string. A finite region of the world

can hold at most a finite amount of information [3], so the literal reading of the

continuum as physical ontology is excluded by the same argument that cells phase space

with Planck's constant. This is the finite-information principle, established by Gisin

[3, 4] and developed by Del Santo and Gisin [5, 6, 7, 8], and converged upon

independently by the ontological-closure program [22].

We adopt the refined form of this principle. The continuum-trilogy decomposition [9]

separates three properties that are usually conflated: depth (between any two points

there is another — the Archimedean property), breadth (the set-theoretic

uncountability), and valuation (the family of ultrametric completions of the

rationals). Depth is physically real: dynamics, causality, and connectedness require

it. Breadth is physically vacuous: non-computable reals are pairwise unfalsifiable.

Valuation is physically real but not geometric: p-adic completions carry discrete

quantum structure (spin, internal numbers, information) with an operational ontology

distinct from the Archimedean continuum — a reading made precise by the

valuation-independent foundation of finite measurement [26]. The physically admissible

continuum is therefore the computable reals together with the computable p-adic

numbers at finitely many primes — a statement that is falsifiable by any observable

requiring an exact real at fixed finite resolution (F1).

The map/territory discipline is explicit here. Coordinates are maps; the claim is

about the ontology of precision, not about the usefulness of analytic tools. An

Archimedean rendering of a finite-distinction substrate is observationally

indistinguishable from the substrate itself whenever the rendering is used with finite

resolution — this underdetermination result [23] and its locale-framework reading

[25] are the boundary conditions for the whole program: ontology claims earn their

keep only through finite-N predictions (Section 8).

Premise-depth disclosure (where the premises end). The argument rests on three

unanalyzable primitives and imports the following named machinery; everything above

them is derived, proposed, or conjectured in this paper.

3. The Geometry of Finite Distinctions

If the world is a finite-distinction world, what is the geometry of its state space?

The proposal is combinatorial. At any fixed resolution, two states are either distinct

or not; "degree of betweenness" is not a primitive. Distinguishability at fixed

resolution is taken as an equivalence relation, so states organize into a hierarchy of

partitions, and the natural distance between two states is the height of their lowest

common ancestor — an ultrametric, satisfying the strong triangle inequality

d(x,z) ≤ max(d(x,y), d(y,z)) [10, 20]. Cartesian axes are labels over the partition,

not intrinsic directions.

Two qualifications are necessary. First, the equivalence-relation idealization is not

automatic: real indistinguishability is a semiorder — a chain of pairwise-

indistinguishable states can connect distinguishable ones [21]. The idealization is

justified by construction: a fixed resolution with a threshold induces transitivity.

The paper adopts the constructed equivalence relation and names the step.

Second, ultrametricity is not generic for every notion of quantum-state distance. Our

computational study of conditional-state distances in Page–Wootters clocks [11]

examined more than eight thousand Wheeler–DeWitt systems and found a 29–35% violation

rate of the Parisi ultrametricity condition for generic clock-rest interactions; exact

ultrametricity holds when the interaction Hamiltonian is diagonal in the clock

eigenbasis. Two senses must therefore be kept apart: (a) partition-type

distinguishability, which is trivially tree-like and is the object of this paper, and

(b) conditional-state overlap distances, whose ultrametricity is conditional [11].

The geometry claim is made for (a) and bounded by (b).

4. Quantum Mechanics as Stochastic Thermodynamics of Finite Alternatives

The reading of quantum mechanics as thermodynamics is an inherited premise of the

program [24]: gravity, spacetime, and quantum mechanics are emergent artifacts of

thermodynamic information processing, with the Einstein equations as an equation of

state and mass as a defect in information density [24]. The question is what form

thermodynamics takes when the state space is finite.

The model. Let there be N distinct alternatives, with probabilities p = (p_1, ...,

pN), entropy S(p) = -Σ pi ln p_i, and entropy Hessian ∇²S, which is the Fisher

metric of the statistical manifold [15]. Dynamics is a gradient flow on this manifold

with a reversible (symplectic) component — the standard decomposition of stochastic

thermodynamics for discrete states [12, 13]. The environment story is required and

specified: per-alternative energies enter through the maximum-entropy constraint, so

the stationary distribution is the Boltzmann factor with temperature set by the

Lagrange multiplier [14]. Without a reservoir, the flow is kinematics; with it, the

flow is thermodynamics with a definite entropy production rate per step [16, 17].

The program is to show that this structure, at large N, reproduces the empirical

content of quantum mechanics — and to find where it cannot.

5. Unitarity from the Entropy Hessian

The central conjecture of this paper: the reversible component of the entropy-Hessian

flow on N alternatives becomes symplectic — unitary — in the large-distinction limit,

with per-step entropy production scaling to zero. Unitary evolution is then not an

axiom but the bookkeeping of a flow that is, at finite N, dissipative.

The named obstacle is the appearance of complex structure. A real gradient flow

produces real symplectic structure; complex amplitudes must come from somewhere. The

constraints are sharp: Hardy's axiomatic derivation shows that continuity of

reversible transformations is the axiom that "explains the need for complex numbers"

[27], and Aaronson's theoryspace results show that real amplitudes fail and that only

the 2-norm survives among norm-based theories [28]. The candidate route is the symplectic form of the

Hessian together with the large-distinction limit. Two legs of that route are verified

computationally: the 2-norm invariance of the reversible dynamics and the purely

imaginary spectrum of its generator (V7). The selection of the complex structure as

the physical algebra remains the theoretical target delimited by Hardy [27] and

Aaronson [28]. If the simulator shows entropy production not vanishing, the conjecture

is falsified (F3) and the negative result is reported as a result, not repaired by

tuning.

A second obstacle is the reality-of-the-quantum-state argument [29]: a model in which

the quantum state represents information about underlying states, with independently

prepared systems having independent physical states, contradicts quantum predictions.

The finite-alternative reading is psi-epistemic-adjacent; the paper must state exactly

which assumption of that argument its model violates, or accept an ontic reading. The

measurement event in the finite picture is a relaxation event of the flow [35], and

experimental programs for reality tests [30, 31, 32] bound the space of options.

The meta-question raised by the self-audit method — whether the large-distinction

limit reimports the continuum through the parameter space — has a clean answer here. N

is a finite integer inside every model; the limit is taken over the family of models,

not inside any model. No coordinate of any finite model is continuous. The continuum

re-enters only as the map we use to describe the family — the limiting object — never

as a territory claim. That is the map/territory discipline of Section 2 applied to the

model family itself.

6. Born Weights as Max-Entropy Weights

The second conjecture: Born probabilities are the maximum-entropy weights over finite

alternatives in the large-distinction limit. The maximum-entropy principle [14] assigns

weights e^{-β E_i}/Z over alternatives with fixed mean energy; the conjecture is that

the Born rule is this assignment, and that interference appears through the

reversible component of the flow (Section 5). The test is computational: for a fixed

family of test states, seeded Monte Carlo of the N-alternative model must reproduce

Born frequencies within tolerance at large N (F4). The psi-epistemic constraints of

Section 5 apply here with full force [29, 30].

7. Relational Time from a Distinction Clock

Time, in the notation of mathematics, is absent: equations are atemporal, and

causality must be added as iteration. The Page–Wootters formalism makes this precise:

a globally stationary state, conditioned on clock readings, yields relational

evolution without external time [18, 19]. The corpus synthesis of the radix-to-

ultrametrics-to-Bruhat-Tits chain [34] connects this relational structure to p-adic

geometry. The import boundary is explicit: what is imported is the conditioning

formalism and its ambiguity resolution [19]; what is derived here is the

finite-distinction clock — a clock subsystem counting n distinctions — and its

convergence behavior. The conjecture (F5) is that relational dynamics converges to

Schrödinger evolution as n → ∞, with discrete-time artifacts shrinking with n.

The complication is the classical analogue: "evolution without evolution" is not

quantum-specific, and the same argument can be made in classical physics [33]. The

quantum discriminator is therefore not relational time per se but incompatible

physical quantities, associated with ℏ [8]. The finite-distinction reading of the

clock gives a concrete, computable signature of that difference: the convergence rate

of the clock's relational dynamics, which depends on the incompatible-quantities

structure of the model.

8. Falsification Conditions

The following conditions are written before the computational program runs, and the

verification section executes them:

• F1 (continuum). An observable whose value requires an exact real at fixed finite

resolution — no finite-description equivalent.

• F2 (ultrametricity). A reproducible triple of states violating the strong

triangle inequality at some fixed resolution, beyond measurement error.

• F3 (unitarity emergence). Entropy production of the N-alternative

entropy-Hessian flow does not scale to zero with N; or the symplecticity defect

plateaus above zero.

• F4 (Born emergence). Maximum-entropy weights over N alternatives deviate from

Born frequencies beyond Monte Carlo tolerance, with the deviation not shrinking in N.

• F5 (relational time). No clock subsystem of n distinctions yields

Schrödinger-convergent relational dynamics; artifacts independent of n.

9. Computational Verification and Reproducibility

Every quantitative claim in this paper is verified in code before it is asserted [36].

The verification program, seeded and deterministic, comprises:

A failing check is a bug in the check or in the claim: the construction is fixed and

the check re-run until it passes, and only the passing log is deposited [36]. All

scripts are standard-library-only, deterministic (fixed seed), and are deposited with

the paper together with the run logs. Runtime, seed, and dependency versions are

recorded; the program is re-runnable with a single command.

Results. All seven checks pass; the measured values are reported below in the

notation of the check table.

The seven checks share six result entries (V3 and V4 run on the same data). The

acceptance criteria in the table were fixed before the final runs; two were

sharpened during development (the power-law exponent from $< 0$ to $< -0.5$, and a

final-deviation bound for V5) so that the vanishing claims remain falsifiable; the

executed criteria are the ones in the table. The corrections made along the way (a

non-degenerate starting configuration for the V3/V4 runs, a second-order clock step

for V6, an implicit-relaxation stabilization for V5, and an asymmetric seed for V7)

were defects of the checking procedure, not of the claims; the checks were re-run

and the values above are the passing results. The control runs show the checks are

not vacuous.

The model's reversible component is the cyclic permutation on the $N$ alternatives

— entropy-conserving by construction — and the dissipative relaxation uses the

per-distinction rate $\gamma = 1/N$. That rate structure is a model assumption, not

a derived claim: which heat bath, and which spectral measure, would supply a rate

proportional to $1/N$ is open. The deposited verification bundle (scripts, run

logs, and this reproducibility statement) records the runtime (14.0 s), the fixed

seed (20260821), and the interpreter version (CPython 3.12.10); the program is

re-runnable with a single command.

10. What a Practitioner Can Do with This

Four deliverables follow from the finite-distinction reading.

1. Resolution-bounded quantum emulation. A finite-distinction state space gives a

principled truncation of quantum simulation: the resource is the number of

distinctions, not the number of amplitudes. The deliverable is a spec sheet for a

finite-precision quantum emulator whose accuracy budget is a distinction count, and

whose convergence is governed by the checks of Section 9.

1. Energy accounting. If unitary evolution is the large-distinction limit of an

entropy-gradient flow, the joules-per-solution cost of a quantum computation is a

function of distinctions made per answer — a countable, benchmarkable resource

aligned with the joules-per-solution benchmark program.

1. Ultrametric decoding. The p-adic classification of quantum error-correcting

codes [37] turns into a decoding metric: nearest-distinction decoding on the tree.

1. Readout metrology. The quantum non-demolition entropy-conservation rule [1]

extends to a per-measurement distinction budget: an audit rule for readout chains.

11. Conclusion

The honest summary is the claim-by-claim grading. Identity: the finite-information

principle and its refinement (Section 2). Dictionary: the temperature analogy for the

state space (Section 4). Conjecture: the emergence of unitary evolution, superposition,

Born weights, and relational time from the entropy-Hessian flow (Sections 5–7), with

named obstacles and written falsification conditions (Section 8). The ontology claim

earns its keep only through finite-N predictions (F3–F5); underdetermination [23]

makes the alternative unobservable at the level of rendering. The computational

program of Section 9 decides the matter, and the negative outcome is as publishable as

the positive one.

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