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

In the paper's words

A finite-entropy world is a finite-distinction world: uncountable precision is unphysical, while computable depth and p-adic valuation remain physically real. The state-space geometry of finite distinctions is combinatorial and ultrametric; quantum mechanics read as thermodynamics is a stochastic thermodynamics of finite alternatives, with unitary evolution and superposition conjectured as the large-distinction limit of an entropy-Hessian gradient flow (supported at finite N within the per-distinction model; 6/6 computational checks, seed 20260821). The Hilbert-space formulation over the complex numbers is that limit — a map, not the territory.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. 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.

   from 3. The Geometry of Finite Distinctions

2. 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.

   from 1. Introduction

3. 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).

   from 1. Introduction

4. 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].

   from 1. Introduction

5. L1 imported named inputs Maximum-entropy principle [14]; entropy Hessian = Fisher metric [15]; stochastic thermodynamics of discrete states [12, 16, 17]; Page–Wootters conditioning [18, 19]; ultrametricity of hierarchical order [10, 20]; semiorder structure of indistinguishability [21]; the adelic scalar family [1, 2].

   from 2. The Continuum as Infinite Information

6. Let there be N distinct alternatives, with probabilities $p = (p_{1},..., p_{N})$, entropy $S(p) = -\Sigma p_{i} \ln p_{i}$, and entropy Hessian $\nabla ^{2}S$, which is the Fisher metric of the statistical manifold [15].

   from 4. Quantum Mechanics as Stochastic Thermodynamics of Finite Alternatives

How it is done

1. 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.

   from 5. Unitarity from the Entropy Hessian

2. The acceptance criteria in the table were fixed before the final runs; two were sharpened during development (the power-law exponent from $\lt 0$ to $\lt -0.5$, and a final-deviation bound for V5) so that the vanishing claims remain falsifiable; the executed criteria are the ones in the table.

   from 9. Computational Verification and Reproducibility

3. Entropy production along a stochastic trajectory and an integral fluctuation theorem.

   from References

What would falsify it

1. An observable whose value requires an exact real at fixed finite resolution — no finite-description equivalent.

   from 8. Falsification Conditions

2. 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.

   from Abstract

3. 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.

   from Abstract

Where the claim stops

1. 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.

   from Abstract

2. 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].

   from 1. Introduction

3. 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.

   from 1. Introduction

Where to read

1. Abstract227 words · 1 min
2. 1. Introduction538 words · 2 min
3. 2. The Continuum as Infinite Information442 words · 2 min
4. 3. The Geometry of Finite Distinctions245 words · 1 min
5. 4. Quantum Mechanics as Stochastic Thermodynamics of Finite Alternatives197 words · 1 min
6. 5. Unitarity from the Entropy Hessian377 words · 2 min
7. 6. Born Weights as Max-Entropy Weights94 words · 1 min
8. 7. Relational Time from a Distinction Clock186 words · 1 min
9. 8. Falsification Conditions117 words · 1 min
10. 9. Computational Verification and Reproducibility702 words · 3 min
11. 10. What a Practitioner Can Do with This141 words · 1 min
12. 11. Conclusion95 words · 1 min