Fisher-Information No-Go Theorems for Emergent Interaction, and a Shared-Field Construction

Abstract

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.