This paper examines the status of the Fisher selection principle H5 within the reconstruction programme. Articles 1–6, the published core of the programme, rely on five hypotheses H1–H5. While H1–H4 are geometric or regularity assumptions, H5 — the minimisation of the Fisher information functional (I(\nu|\mu_F)) — is the central working hypothesis on which the conditional results of the programme depend. The paper asks why Fisher information should be minimised. Four possible justifications are compared: perturbative stability, the Reginatto–Hall exact uncertainty principle, Frieden’s extremal Fisher information, and the Wasserstein/de Bruijn interpretation of Fisher information as the squared gradient of relative entropy. The conclusion is deliberately limited: perturbative stability provides the most autonomous internal motivation for H5, while the other arguments are structurally informative but require additional inputs or output physics. A full derivation of H5 would require a uniqueness theorem showing that Fisher information is the only admissible functional satisfying the reconstruction assumptions and a suitable stability condition. This remaining task is identified as Open Problem 2.
This paper examines the status of the Fisher selection principle H5 within the reconstruction programme. Articles 1–6, the published core of the programme, rely on five hypotheses H1–H5. While H1–H4 are geometric or regularity assumptions, H5 — the minimisation of the Fisher information functional (I(\nu|\mu_F)) — is the central working hypothesis on which the conditional results of the programme depend. The paper asks why Fisher information should be minimised. Four possible justifications are compared: perturbative stability, the Reginatto–Hall exact uncertainty principle, Frieden’s extremal Fisher information, and the Wasserstein/de Bruijn interpretation of Fisher information as the squared gradient of relative entropy. The conclusion is deliberately limited: perturbative stability provides the most autonomous internal motivation for H5, while the other arguments are structurally informative but require additional inputs or output physics. A full derivation of H5 would require a uniqueness theorem showing that Fisher information is the only admissible functional satisfying the reconstruction assumptions and a suitable stability condition. This remaining task is identified as Open Problem 2.
The Reconstruction Theorem: Minimal Kähler Completion of Fisher–Rao Dynamics This article serves as a synthesis paper for the reconstruction programme. Rather than introducing new results, it identifies the programme's deductive core and compresses the results of Articles 1–11 into a single logical structure. Purpose The goal is to determine the minimal set of assumptions required to recover Madelung–Schrödinger dynamics from information-geometric principles. The paper distinguishes clearly between established results, conditional results, and open problems. The Five Hypotheses The reconstruction programme rests on five assumptions: H1: Reparametrisation invariance. H2: Markov monotonicity. H3: Cohesion (finite Fisher volume). H4: Compact reconstruction domain. H5: Variational selection by Fisher minimisation. H1 and H2 uniquely determine the Fisher–Rao geometry. H3 and H4 provide regularity. H5 remains a working variational principle whose justification is discussed elsewhere. Main Results Theorem A (Minimal Kähler Completion). The unique minimal admissible completion of the Fisher–Rao reconstruction space is the cotangent bundle T*P(M) equipped with its canonical symplectic structure and Kähler geometry. Through the Madelung map, this completion is isometric to the projective Hilbert space CP(L²(M)) endowed with the Fubini–Study metric. Theorem B (Coupling Uniqueness). The unique coupling compatible with Fisher geometry, Wasserstein dynamics, Kähler structure, and Markov monotonicity is λ = ℏ²/(8m). Corollary. Under the standard Madelung conditions (positive density and irrotational flow), the resulting dynamics is equivalent to the Schrödinger equation. The Deductive Kernel The article condenses the programme into a single chain: Fisher–Rao geometry + Wasserstein dynamics + canonical symplectic structure + minimal Kähler completion → Madelung–Schrödinger dynamics. Within this framework, the quantum structure appears as the unique admissible completion of the reconstruction dynamics rather than as an independent postulate. Three Explicit No-Go Results The existence of an action constant is forced by the symplectic completion, but its numerical value is not derived. Physical time is not obtained directly from H1–H5 and remains associated with the causal reconstruction layer. Gauge sectors are not derived from the present framework. Conclusion Article 12 functions as the logical charter of the reconstruction programme. Its main contribution is to identify the minimal deductive pathway leading from the Fisher–Rao reconstruction hypotheses to quantum dynamics, while explicitly separating established results from open problems.
Title: The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law Description This note addresses a persistent conceptual gap in “information physics” programs that derive familiar dynamical laws by extremizing Fisher-information-based functionals: why Fisher information, rather than some other statistical functional? The proposed answer is meta-explanatory. The observer–world interface is modeled as probabilistic, and “objectivity” is formulated as Observer Equivalence (OE): invariance under reparameterization and monotonicity under coarse-graining (information-loss channels). Under these constraints, Chentsov-type characterization results single out the Fisher information metric as the unique local Riemannian geometry on the space of probability distributions (up to an overall scale). Fisher information is thus treated not as an axiom but as an invariance residue forced by OE at the probabilistic interface. To connect geometry to dynamics, the note introduces an explicit postulate—Informational Inertia—motivated as an information-geometric analogue of least-action/geodesic principles: in the absence of forcing, interface states evolve along paths of minimal statistical resistance (geodesics of the OE-fixed geometry), with physical constraints supplying the remaining “potential” structure. A single exemplar is presented: revisiting the Madelung/Reginatto (Hall–Reginatto) route to the Schrödinger equation, where the “quantum potential” arises as the geometric term associated with the Fisher penalty for reshaping the probability density. The overall scale (and thus ℏ) is explicitly acknowledged as empirical input: OE fixes the form of the geometry, not its scale. Keywords observer equivalence; Fisher information; information geometry; Chentsov theorem; coarse-graining; monotone metrics; Schrödinger equation; Madelung transformation; Reginatto; foundations of physics How to cite Ullman, G. “The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law.” Zenodo. https://doi.org/10.5281/zenodo.18495542 Notes on AI assistance This manuscript was prepared with assistance from large language models for drafting, editing, and LaTeX formatting. The author takes full responsibility for the technical content, claims, and references.
Existing information-theoretic derivations of quantum mechanics from Fisher informationlack a thermodynamic embedding: they contain no temperature, no entropy, and no free-energy balance, and therefore cannot address the quantum–classical boundary within thesame variational structure. We show that this limitation is removed by a Helmholtz-typefree-energy functional on probability distributions in which Fisher information provides theenergy and relative Shannon entropy, weighted by an effective temperature ϑ, provides theentropic cost. Chentsov’s theorem and the Shore–Johnson axioms constrain this functionalup to two coefficients, one of which is fixed by consistency with the quantum kinetic energy.At ϑ= 0 the functional reproduces the Fisher-information variational principle of Reginattoand Hall–Reginatto; extremisation of the associated action recovers the Schr¨odinger equationvia the Madelung transformation. Quantum mechanics is thereby identified as the zero-temperature fixed point of the Helmholtz structure. The finite-temperature sector ϑ > 0,which has no counterpart in previous approaches, yields three results: a gauge-invariantbalance scale Lc = ℏ/2√mϑ at which both the free-energy descent and the gradient-flowspreading law change character — the spreading exponent crossing over from the quantum-pressure value 1/4 to the diffusive value 1/2 — together with, under harmonic confinement,a genuine variationally stable minimum interpolating between the exact quantum ground-state and classical equipartition widths; a dissipation bound for localisation transitions,derived from a Wasserstein gradient flow with Lyapunov structure, from which Landauer’sprinciple follows as a corollary in the entropy-dominated regime; and a thermodynamicstability property that singles out the Born weights as the unique stable basin fractionsunder the dissipative dynamics, presented as a consistency check.
This paper extends the causal reconstruction programme from static optimal states to finite-action dynamical histories. A reconstructive history is modelled as a curve of probability measures in the mass-scaled Wasserstein space (P(M), W2,m), and admissible histories are selected by a reconstructive action combining a Wasserstein kinetic term, a Fisher information term, and a causal constraint. The resulting Euler–Lagrange equations form a coupled system: a continuity equation for the evolving density and a generalized Hamilton–Jacobi equation for the velocity potential. The central result is the Fisher–Bohm identity. For any sufficiently regular positive density ρ, the functional derivative of Fisher information satisfies δI / δρ(x) = -4 Δ√ρ / √ρ, so that the Bohm–Madelung quantum potential is exactly Q(x) = (ℏ2 / 8m) · δI / δρ(x). Thus the quantum potential is the Fisher information gradient of the probability density, scaled by ℏ2/8m. When the Fisher coupling in the reconstructive action is set to λ = ℏ2/8m, the Euler–Lagrange equations coincide with the Madelung equations and are therefore equivalent to the Schrödinger equation, under the standard assumptions of positive density and irrotational flow. This establishes a structural correspondence between reconstructive dynamics and non-relativistic quantum mechanics. The result is not presented as a derivation of quantum mechanics: the coupling λ = ℏ2/8m is matched through the Madelung Hamiltonian rather than derived from the reconstruction hypotheses. The geometric origin of this coupling remains an open problem.
The hypothesis that physical information is not fundamentally destroyed motivates a mathematical question: can localization, correlation, and effective interaction be generated from an information-based variational principle without inserting the desired interaction potential by hand? We investigate this question in a framework built on the classical Fisher information of probability densities, organized around a strict anti-circularity requirement: no Coulomb kernel, inverse-distance potential, or prescribed interaction law is introduced into the fundamental functional. Several results are established at increasing levels of generality. First, under normalization and a fixed second moment, the Fisher information obeys the sharp bound I[ρ]≥d2/R2I[ρ]≥d2/R2, with equality attained uniquely by the isotropic Gaussian; this solves the one-body constrained problem completely. Second, for any joint density ρ12ρ12 with prescribed marginals, I12[ρ12]≥I[ρ1]+I[ρ2]I12[ρ12]≥I[ρ1]+I[ρ2], with equality attained by the product density; consequently a purely local, block-additive Fisher functional cannot generate an effective interaction at any separation, Keff(R)≡0Keff(R)≡0. Third, this no-go result is shown to be far more robust than the local case alone suggests: for any hidden-variable extension Q(x1,x2,z)Q(x1,x2,z) of a two-body density, ∀ Q(x1,x2,z):I[Q] ≥ I[ρ12] ≥ I[ρ1]+I[ρ2],∀Q(x1,x2,z):I[Q] ≥ I[ρ12] ≥ I[ρ1]+I[ρ2], unconditionally, with no independence, factorization, or weak-coupling assumption placed on the auxiliary variable zz, provided the enlarged functional remains the Fisher information of the enlarged density; no claim is made about enlargements governed by an arbitrary non-Fisher coupling of zz. Within that scope, this closes off an entire class of hidden-variable evasions of the no-go theorem at once, rather than ruling out evasions one construction at a time -- a consequence obtained by combining the superadditivity of Fisher information under marginalization, established by Carlen, with a further marginalization over the auxiliary sector; the contribution of this article is the variational reading of that combination as a no-interaction obstruction, together with an exact conditional-variance identity sharpening its equality case, rather than the underlying information inequality itself. We further characterize equality in the first marginalization step: equality of the full Fisher informations forces the auxiliary sector to factorize and to carry zero Fisher information, so on an unbounded connected auxiliary domain the equality case is impossible for a proper normalized density. We then show, with an explicit lower-bound hypothesis made a formal part of the statement rather than left implicit, that exact vanishing of KeffKeff follows whenever the auxiliary extension is block-factorized across the two subsystems, the governing functional contains no explicit cross-sector coupling term prior to elimination, and the functional obeys a block-additive lower bound over the entire admissible class (not merely on factorized configurations); the Fisher bound supplies this hypothesis automatically in the Fisher case. Conversely, Keff≠0Keff=0 implies that no admissible block-factorized configuration can realize the independent optimum, though the converse implication does not hold without further hypotheses. We then exhibit a concrete construction that violates factorization: an auxiliary scalar field sourced jointly by both subsystems. The functional E[ϕ,ρ]=κ2∫∣∇ϕ∣2−g∫sρϕE[ϕ,ρ]=2κ∫∣∇ϕ∣2−g∫sρϕ produces, after eliminating ϕϕ, a nonlocal quadratic functional with the Green kernel of the Laplacian, yielding a 1/R1/R interaction in three dimensions. We show carefully that this mechanism is not explained by the nonlocality of the map ρ↦ϕρ↦ϕ: a model with two independent, equally nonlocal fields remains sector-separable, and we verify directly (rather than by importing the Fisher-specific bound into a classical field functional, where it does not automatically apply) that its infimum splits additively, giving Keff≡0Keff≡0. The operative property behind the nonzero result is instead that a single shared field is sourced by the total density, which forces a cross term ∫∇ϕA⋅∇ϕB∫∇ϕA⋅∇ϕB absent from any factorized configuration. This mechanism reaches only the third of four levels in an anti-circularity hierarchy that we introduce; whether the field itself can be derived, rather than postulated, from an information-theoretic principle remains open. The resulting framework separates four logically distinct conclusions: a proved localization theorem, a proved no-interaction theorem for local Fisher information, an unconditional lower bound ruling out entire classes of hidden-variable evasions, and a concrete non-factorized extension whose physical interpretation and information-theoretic origin remain to be established. We close by fixing this boundary precisely and organizing the remaining program around a single unresolved question — whether a non-factorized shared mediator can itself be derived from an information-theoretic principle, rather than postulated — so that the present results are used as a fixed starting constraint on that question rather than reopened by it.
This paper clarifies the role of the quantum Fisher coupling in reconstructive dynamics. Article 5 showed that the reconstructive action reproduces the Madelung equations, and hence the Schrödinger equation, when the Fisher term carries the coupling λ = ℏ2/8m. The present article argues that the fundamental object to be derived is not ℏ or m separately, but their natural combination: κ = ℏ2/m. Three results are established. First, κ is the unique coupling of dimension energy times length squared compatible with the Fisher information term in the Madelung Hamiltonian. Second, κ/2 appears as the scale of the biharmonic flow obtained by linearising the Fisher gradient flow around a flat reference state. Third, the decomposition κ = ℏ2/m requires additional geometric input: m as the inertial coefficient of Wasserstein transport, and ℏ as the prequantisation constant associated with a non-trivial symplectic loop. The paper does not derive κ from hypotheses H1–H5. Rather, it establishes that κ is the natural single object to be derived and identifies the geometric conditions under which its decomposition into ℏ and m may become accessible.