Masses Mass Genesis Canonical Load Factorization

The load factorization theorem

In the Recognition Science framework, a particle is a stable pattern of recognition events, and the framework's ledger records how much meaning load each occupied site carries. The central question for mass genesis is whether a particle's actual, integrated load matches the value its topology alone would predict. The canonical load factorization theorem, proved in the framework's machine-checked library of formal theorems, answers this with an exact equivalence: the load factorizes precisely when the integrated meaning load equals the topology-selected MassLaw value.

The theorem is not a claim that every stable pattern factorizes. The library proves a sharp obstruction: two stable patterns with the same topology can carry different integrated loads, so stability alone cannot force factorization. What the theorem establishes is the reduction: factorization is equivalent to the load realizing the predicted value. This is the exact bridge between the abstract topology and the measurable mass.

The proof works by showing that factorization is equivalent to an equipartition condition. When the load factorizes, every occupied site carries the same meaning load, equal to the total factorized load divided by the number of occupied sites. The library proves this equivalence in both directions, and from it derives the key physical consequence: for any stable pattern whose load factorizes, the rest mass equals the predicted mass.

The module also records two sufficient routes to factorization. One requires a neutral unit load on the support with the support cardinality matching the factorized load. The other requires the topology-scaled CP6 load condition, where each occupied site's neutralized window is a fixed amplitude times a neutral unit chord. Both routes are proved to imply factorization, and the library shows they are equivalent to the equipartition condition.

What this establishes in plain language is a precise criterion for when a particle's mass is what its topology says it should be. The framework does not assert this happens for all stable patterns; it proves the equivalence and the obstruction. The factorization theorem is the load-bearing link that turns a topological prediction into an actual mass, and the library certifies exactly when that link holds.

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 not_allStableCanonicalPrimitiveLoadFactorizes_of_sameTopology_different_load
    {ψ χ : LightPattern Λ}
    (hψ : StableClosedLightPattern ψ)
    (hχ : StableClosedLightPattern χ)
    (htop : SameTopology ψ χ)
    (hload : integratedMeaningLoad ψ ≠ integratedMeaningLoad χ) :
    ¬ (∀ η : LightPattern Λ,
        StableClosedLightPattern η →
          CanonicalPrimitiveLoadFactorizes η) := by
  intro Hall
  have hψR : MassLawLoadRealized ψ :=
    (canonicalPrimitiveLoadFactorizes_iff_massLawLoadRealized ψ).1
      (Hall ψ hψ)
  have hχR : MassLawLoadRealized χ :=
    (canonicalPrimitiveLoadFactorizes_iff_massLawLoadRealized χ).1
      (Hall χ hχ)
  exact sameTopology_different_load_obstructs_both_realized
    hψ hχ htop hload ⟨hψR, hχR⟩

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

theorem restMass_eq_predictedMass_of_stable_and_canonicalPrimitiveLoadFactorizes
    (ψ : LightPattern Λ)
    (hstable : StableClosedLightPattern ψ)
    (hfactor : CanonicalPrimitiveLoadFactorizes ψ) :
    restMass ψ = predictedMass ψ :=
  restMass_eq_predictedMass_of_load_realized ψ hstable
    ((canonicalPrimitiveLoadFactorizes_iff_massLawLoadRealized ψ).1 hfactor)

What this page does not claim

The theorem does not prove that all stable patterns factorize their load. The module does not derive the factorized MassLaw load value from first principles. The equivalence does not identify which topology-selected value is realized by actual particles.

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)

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