Masses Mass Genesis Canonical Load Factorization Support Averaged Factorized Loa

The equipartition theorem

In the Recognition Science framework, a ledger (a discrete record of events) assigns a numerical load to each occupied site of a light pattern, and the total of these loads is the particle's mass. The declaration supportAveragedFactorizedLoad_iff_uniform_and_factorization is a proved equivalence: for any light pattern with at least one occupied site, the load is support-averaged and factorized if and only if two conditions hold. The first condition, uniform site meaning load, says every occupied site carries the same load. The second, canonical primitive load factorizes, says the total integrated load equals the topology-selected MassLaw value. The theorem is proved in the machine-checked library of formal theorems, with no unproved assumptions beyond the standard logical axioms.

The equivalence matters because it splits a hard property into two checkable halves. The uniform condition is the equipartition half: it forces equal sharing across the support. The factorization condition is the value half: it pins the shared amount to the MassLaw prediction. The theorem also shows that the support-averaged property implies both halves, so the two conditions are not independent extras but exactly what the property decomposes into. This is the kind of reduction that lets the framework build upward: from a load that is evenly spread and correctly valued, the framework derives that the canonical primitive load factorizes, and from that, that the rest mass equals the predicted mass.

The theorem does not claim that every stable light pattern satisfies either condition. A separate result in the same module shows that two stable patterns with the same topology can have different integrated loads, so stability alone does not force factorization. The equivalence also does not say how the uniform or factorization conditions arise physically; it only states their logical relationship. The physical bridge from recognition to linking remains open, and the theorem says nothing about measured particle masses, only about the formal structure of loads in the framework.

theorem supportAveragedFactorizedLoad_iff_uniform_and_factorization
    (ψ : LightPattern Λ)
    (hsupp : ψ.support.Nonempty) :
    SupportAveragedFactorizedLoad ψ ↔
      UniformSiteMeaningLoad ψ ∧ CanonicalPrimitiveLoadFactorizes ψ := by
  constructor
  · intro havg
    exact
      ⟨supportAveragedFactorizedLoad_uniform ψ havg,
       supportAveragedFactorizedLoad_factorizes ψ hsupp havg⟩
  · intro h
    exact supportAveragedFactorizedLoad_of_uniform_and_factorization
      ψ hsupp h.1 h.2

/-- Equal site-load distribution over the occupied support. This is the
equipartition half of the remaining Q3/Rhat amplitude theorem. -/
def UniformSiteMeaningLoad (ψ : LightPattern Λ) : Prop :=
  ∀ x ∈ ψ.support, ∀ y ∈ ψ.support,
    siteMeaningLoad ψ x = siteMeaningLoad ψ y

theorem canonicalPrimitiveLoadFactorizes_iff_integratedMeaningLoad_eq_factorized
    (ψ : LightPattern Λ) :
    CanonicalPrimitiveLoadFactorizes ψ ↔
      integratedMeaningLoad ψ = factorizedMassLawLoad ψ := by
  constructor
  · intro h
    rw [integratedMeaningLoad_eq_support_sum ψ]
    simpa [CanonicalPrimitiveLoadFactorizes, PrimitiveLoadFactorizes,
      canonicalPrimitiveSiteLoadLaw] using h
  · intro h
    rw [integratedMeaningLoad_eq_support_sum ψ] at h
    simpa [CanonicalPrimitiveLoadFactorizes, PrimitiveLoadFactorizes,
      canonicalPrimitiveSiteLoadLaw] using h

theorem supportAveragedFactorizedLoad_uniform
    (ψ : LightPattern Λ)
    (havg : SupportAveragedFactorizedLoad ψ) :
    UniformSiteMeaningLoad ψ := by
  intro x hx y hy
  rw [havg x hx, havg y hy]

theorem supportAveragedFactorizedLoad_factorizes
    (ψ : LightPattern Λ)
    (hsupp : ψ.support.Nonempty)
    (havg : SupportAveragedFactorizedLoad ψ) :
    CanonicalPrimitiveLoadFactorizes ψ := by
  have hreal : MassLawLoadRealized ψ :=
    supportAveragedFactorizedLoad_realizes ψ hsupp
      (supportAveragedFactorizedLoad_to_masslaw_average ψ havg)
  unfold MassLawLoadRealized at hreal
  rw [integratedMeaningLoad_eq_support_sum ψ] at hreal
  simpa [CanonicalPrimitiveLoadFactorizes, PrimitiveLoadFactorizes,
    canonicalPrimitiveSiteLoadLaw, factorizedMassLawLoad_eq_predictedMass ψ]
    using hreal

What this page does not claim

The theorem does not claim that every stable light pattern satisfies the uniform or factorization conditions. It does not claim that stability alone forces factorization, since two stable patterns with the same topology can have different loads. It does not claim any physical mechanism that produces the uniform or factorization conditions.

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/CanonicalLoadFactorization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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