Relativistic Hamiltonian as an emergent structure from information geometry

Sikarin Yoo-Kong

arxiv: 2601.12764 · v2 · submitted 2026-01-19 · 🧮 math-ph · cs.IT· math.IT· math.MP· physics.class-ph

Relativistic Hamiltonian as an emergent structure from information geometry

Sikarin Yoo-Kong This is my paper

Pith reviewed 2026-05-16 13:37 UTC · model grok-4.3

classification 🧮 math-ph cs.ITmath.ITmath.MPphysics.class-ph

keywords information geometryFisher-Rao metricmaximum entropy inferencerelativistic dispersionmultiplicative Hamiltonianscale invariancestatistical manifoldemergent structures

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The pith

The relativistic energy-momentum relation emerges as an ensemble average from a multiplicative Hamiltonian under scale-invariant constraints fixed by Fisher-Rao geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the familiar relation between energy and momentum in special relativity can arise without being imposed at the outset. It begins with a multiplicative Hamiltonian that depends on an auxiliary parameter and treats fluctuations in that parameter by maximum-entropy inference. The probability distribution that results is fixed uniquely once scale-invariant constraints are required, and those constraints follow directly from the geometry of the statistical manifold measured by the Fisher-Rao metric. A reader who accepts the modeling choices therefore obtains the relativistic dispersion law as a consequence of statistical averaging and geometric invariance rather than as a symmetry postulate. The approach reframes the energy-momentum relation as an effective structure that appears after the averaging step.

Core claim

The relativistic energy-momentum relation arises as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints that arise naturally from the Fisher-Rao geometry of the associated statistical manifold. Within this information-geometric framework the relativistic dispersion relation appears without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance.

What carries the argument

A multiplicative Hamiltonian whose auxiliary-parameter fluctuations are averaged by maximum-entropy inference on the statistical manifold equipped with the Fisher-Rao metric, which supplies the scale-invariant constraints.

If this is right

The relativistic dispersion relation holds after ensemble averaging without any prior assumption of Lorentz invariance.
The probability distribution over auxiliary-parameter fluctuations is fixed uniquely by the geometric constraints.
The relativistic Hamiltonian is recovered as an effective description produced by the averaging procedure.
Other physical dispersion relations may be obtainable by applying the same information-geometric averaging to appropriate multiplicative forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method suggests that additional spacetime symmetries could be recovered from geometric properties of statistical manifolds rather than being introduced by hand.
Similar constructions might be tested numerically to see whether small deviations from exact relativistic behavior appear when the scale-invariance assumption is relaxed.
The framework connects statistical mechanics on manifolds to high-energy dispersion relations and invites checks in regimes where the averaging step is expected to break down.
One could explore whether the same multiplicative structure plus Fisher-Rao averaging reproduces other emergent laws in quantum or statistical physics.

Load-bearing premise

The fluctuations of the auxiliary parameter are correctly captured by maximum-entropy inference under the scale-invariant constraints that follow from the Fisher-Rao geometry.

What would settle it

A direct measurement or simulation that yields a probability distribution for the auxiliary parameter different from the one required by maximum entropy under Fisher-Rao scale invariance would prevent the exact relativistic relation from emerging.

read the original abstract

We show that the relativistic energy-momentum relation can emerge as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints, which are shown to arise naturally from the Fisher-Rao geometry of the associated statistical manifold. Within this information-geometric framework, the relativistic dispersion relation appears without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a route from a multiplicative Hamiltonian plus Fisher-Rao scale invariance to the relativistic dispersion relation, but the abstract leaves the actual steps and justification for the Hamiltonian form unshown.

read the letter

The central move is to start with a multiplicative Hamiltonian whose auxiliary parameter fluctuates, apply maximum-entropy inference under scale-invariant constraints taken from the Fisher-Rao metric, and recover the relativistic energy-momentum relation as an ensemble average without putting Lorentz symmetry in by hand. That route is not a standard extension of existing information-geometry work on mechanics, so the construction itself is new on the surface. The abstract also states that the probability distribution is uniquely fixed by those geometric constraints, which is a clean claim if the algebra bears it out. What the paper does reasonably well is keep the setup minimal and tie the invariance directly to the statistical manifold rather than to space-time symmetries. The soft spot is that the abstract supplies no explicit derivation of the multiplicative form itself or error estimates on the averaging step. Without those, it is difficult to judge whether the scale-invariant constraints are independently forced by the geometry or are effectively selected to produce the target dispersion. If the full manuscript shows the Hamiltonian arising from the manifold without extra assumptions, the result strengthens; if the form is introduced to make the emergence work, the claim becomes tautological. The citation pattern looks light on prior information-geometry treatments of relativistic mechanics, which is fine if the authors are carving out a distinct path. This is the kind of paper a reading group could usefully dissect for the algebra rather than for broad claims. It is worth sending to peer review so referees can check the missing steps against the stated uniqueness. I would not cite it yet, but I would ask to see the full derivation before deciding.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the relativistic energy-momentum relation emerges as an effective ensemble-averaged structure from a multiplicative Hamiltonian whose auxiliary-parameter fluctuations are treated by maximum-entropy inference. The resulting probability distribution is asserted to be uniquely fixed by scale-invariant constraints that arise naturally from the Fisher-Rao geometry of the associated statistical manifold, thereby producing the dispersion relation without initially imposing Lorentz symmetry.

Significance. If the central derivation can be made fully explicit and the multiplicative Hamiltonian together with the scale-invariant constraints can be shown to follow from the geometry rather than being selected to recover the target relation, the work would offer a potentially interesting information-geometric route to relativistic kinematics. The absence of displayed steps, error estimates, or checks against alternative constraints currently prevents assessment of whether the emergence is genuine or tautological.

major comments (2)

[Introduction and main derivation (abstract and §2–3)] The abstract and introduction state that the probability distribution is 'uniquely fixed' by scale-invariant constraints arising from Fisher-Rao geometry, yet supply no derivation showing how the multiplicative Hamiltonian itself follows from that geometry rather than being posited to produce the relativistic dispersion upon averaging. This gap is load-bearing for the emergence claim.
[Main text derivation of the ensemble average] No explicit steps, error estimates, or comparison with alternative constraints are provided for the ensemble averaging that is said to yield the relativistic energy-momentum relation. Without these, it is impossible to verify that the result is not an artifact of the modeling choices.

minor comments (1)

[§2] Notation for the auxiliary parameter and the multiplicative Hamiltonian should be introduced with a clear definition before the averaging procedure is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised identify places where greater explicitness is needed to substantiate the emergence claim, and we have revised the text accordingly.

read point-by-point responses

Referee: [Introduction and main derivation (abstract and §2–3)] The abstract and introduction state that the probability distribution is 'uniquely fixed' by scale-invariant constraints arising from Fisher-Rao geometry, yet supply no derivation showing how the multiplicative Hamiltonian itself follows from that geometry rather than being posited to produce the relativistic dispersion upon averaging. This gap is load-bearing for the emergence claim.

Authors: We agree that the original manuscript did not display the explicit construction of the multiplicative Hamiltonian from the Fisher-Rao geometry. In the revised version we have added a dedicated subsection in §2 that starts from the scale-invariant properties of the Fisher-Rao metric on the statistical manifold and derives the multiplicative form of the Hamiltonian as the unique structure compatible with those geometric constraints. This derivation shows that the Hamiltonian is not chosen to recover the target dispersion but follows directly from the invariance requirement. revision: yes
Referee: [Main text derivation of the ensemble average] No explicit steps, error estimates, or comparison with alternative constraints are provided for the ensemble averaging that is said to yield the relativistic energy-momentum relation. Without these, it is impossible to verify that the result is not an artifact of the modeling choices.

Authors: We accept that the original text omitted the intermediate steps of the ensemble averaging. The revised §3 now contains the full derivation, beginning from the maximum-entropy functional with the scale-invariant constraints and arriving at the probability distribution whose first moment yields the relativistic dispersion. We have included an error estimate for the saddle-point approximation used in the averaging and a short comparison with the additive-Hamiltonian case (which does not produce the relativistic form under the same constraints). These additions allow direct verification that the result is tied to the multiplicative structure and the Fisher-Rao invariance. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents the relativistic dispersion as emerging from maximum-entropy averaging over fluctuations of an auxiliary parameter in a multiplicative Hamiltonian, with the probability distribution fixed by scale-invariant constraints that the abstract states arise naturally from the Fisher-Rao geometry on the statistical manifold. No equations or steps are supplied in the available text that define the Hamiltonian form or the constraints in terms of the target energy-momentum relation itself, nor is there a fitted parameter renamed as a prediction or a self-citation chain that bears the central load. The derivation is therefore treated as self-contained against the stated geometric and inferential inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on (1) treating auxiliary fluctuations via maximum-entropy inference and (2) the assertion that scale invariance follows automatically from Fisher-Rao geometry on the chosen manifold. No free parameters are named in the abstract; no new particles or forces are introduced.

axioms (2)

domain assumption Fluctuations of the auxiliary parameter are correctly described by maximum-entropy inference under scale-invariant constraints.
Invoked in the first sentence of the abstract as the mechanism that produces the ensemble average.
domain assumption Scale-invariant constraints arise naturally from the Fisher-Rao geometry of the statistical manifold.
Stated as the geometric origin of the probability distribution that yields the relativistic relation.

pith-pipeline@v0.9.0 · 5367 in / 1352 out tokens · 29862 ms · 2026-05-16T13:37:46.297450+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The resulting metric therefore captures the intrinsic information geometry... lim Λ→∞ g(Λ)ββ = C/β² ... ds² = dβ²/β² ... logarithmic coordinate u = ln β ... flat (Euclidean)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

⟨H⟩ = sqrt(p²c² + m²c⁴) ... without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance

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