hamiltonian_equivalence
plain-language theorem explainer
The theorem states that whenever the small-deviation hypothesis holds for a recognition reality field and metric at a point, the recognition Hamiltonian density lies within ε³ of some standard Hamiltonian for all ε below 0.1. Researchers recovering classical field theory from recognition dynamics would cite it to justify the linear-limit form ½(Π² + (∇Ψ)² + m²Ψ²). The proof is a direct one-line application of the hypothesis definition.
Claim. Let ψ be a recognition reality field and g a metric tensor. If the small-deviation equivalence hypothesis holds at spacetime point x, then for every ε < 0.1 there exists a standard Hamiltonian H_std such that |HamiltonianDensity(ψ, g, x) − H_std(x)| < ε³.
background
The module derives the Hamiltonian for the Recognition Reality Field (RRF), an abbrev for maps from Fin 4 coordinates to reals. HamiltonianDensity is defined as (1/2) times the sum of squared time and spatial partial derivatives of the field, recovering the standard kinetic-plus-potential expression in the linear limit. The hypothesis H_HamiltonianEquivalence is the empirical claim that this density approximates conventional field-theory Hamiltonians inside a cubic error bound for ε < 0.1.
proof idea
The proof is a one-line wrapper that directly applies the hypothesis h of type H_HamiltonianEquivalence.
why it matters
This result supplies the small-deviation bridge that lets the recognition Hamiltonian reduce to the standard energy density used in quantum field theory. It supports the module objective of obtaining energy conservation from time-translation symmetry once the linear limit is justified. No downstream theorems are recorded, leaving open its integration with the full conservation statements.
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