Masses Mass Genesis Support Symmetry Uniform Site Meaning Load Of Support Rhat T

The equipartition theorem

In the Recognition Science framework, a light pattern is a discrete record of events, each event assigned to a site and carrying a numerical load. The declaration uniformSiteMeaningLoad_of_supportRHatTransitive is a theorem: if a pattern's occupied sites admit a symmetry that (1) maps occupied sites to occupied sites, (2) preserves each site's load, and (3) can move any occupied site to any other occupied site in finitely many steps, then every occupied site carries the same load. The proof is short: pick two sites, use the symmetry to move one to the other, and since the symmetry preserves load, the two loads are equal.

This is an equipartition result. It says that a transitive symmetry, one that connects the whole occupied support, forces uniformity. The framework's library, a machine-checked collection of formal theorems, records the proof in Lean. The theorem is a step in a larger reduction: it combines with a factorization condition to show that a stable pattern's rest mass equals its predicted mass, and that a topology-scaled load exists. The theorem itself, however, does not construct the symmetry. It assumes the symmetry as a premise.

The declaration does not claim that such a symmetry exists for any particular physical pattern. The framework's own documentation states that instantiating this symmetry from the actual Q3 carrier, rather than postulating it for a generic finite substrate, remains hard future work. The theorem is conditional: given a transitive load-preserving action, uniformity follows. It does not prove that the action exists.

/-- The equipartition theorem: transitive load-preserving Rhat support action
forces uniform neutral load across the occupied support. -/
theorem uniformSiteMeaningLoad_of_supportRHatTransitive
    {ψ : LightPattern Λ}
    (A : SupportRHatAction ψ)
    (htrans : SupportRHatTransitive A) :
    UniformSiteMeaningLoad ψ := by
  intro x hx y hy
  rcases htrans x hx y hy with ⟨n, hn⟩
  have hload := A.load_iterate n hx
  rw [hn] at hload
  exact hload.symm

theorem restMass_eq_predictedMass_of_supportRHatSymmetry_and_factorization
    {ψ : LightPattern Λ}
    (hstable : StableClosedLightPattern ψ)
    (_A : SupportRHatAction ψ)
    (_htrans : SupportRHatTransitive _A)
    (hfactor : CanonicalPrimitiveLoadFactorizes ψ) :
    restMass ψ = predictedMass ψ :=
  restMass_eq_predictedMass_of_stable_and_canonicalPrimitiveLoadFactorizes
    ψ hstable hfactor

What this page does not claim

The theorem does not prove that any physical pattern admits the required symmetry. The theorem does not define the load function siteMeaningLoad. The theorem does not establish the factorization condition.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/MassGenesis/SupportSymmetry.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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