Masses Mass Genesis Load Normalization Neutral Unit Load Realized Iff Predicted

Load normalization

In the Recognition Science framework, a light pattern is a discrete record of events, and each occupied site in the pattern carries a number called its site meaning load, a measure of how much recognition work that site contributes. The framework's ledger, the discrete record of all such events, sums these site loads into an integrated load for the whole pattern. The theorem neutralUnitLoad_realized_iff_predictedMass_eq_supportCard states a precise equivalence: if every occupied site carries a unit load, then the total integrated load equals the number of occupied sites, and this happens exactly when that count equals the pattern's predicted mass.

The theorem is proved in the framework's machine-checked library of formal theorems. Its proof is short: it rewrites the integrated load as a sum over the support, applies the unit-load assumption to replace each term with 1, and then uses the fact that summing 1 over a finite set gives its cardinality. The declaration also packages this result with two other sufficient conditions for realizing the mass law: a support-averaged load, where each site carries an equal share of the predicted mass, and a general topology-forced density that sums to the predicted value. Both are proved to realize the mass law, and for stable closed patterns, both imply the rest mass equals the predicted mass.

What the theorem does not claim is equally important. It does not assert that every pattern has unit loads at its occupied sites; that is a separate condition, called NeutralUnitLoadOnSupport, which must be supplied as a hypothesis. It does not say that the unit-load condition is the only way to realize the mass law, since the support-averaged and topology-density conditions provide alternatives. It does not claim that the predicted mass itself is derived from the support cardinality; rather, it states an equivalence under the unit-load assumption. Finally, it does not establish that any real physical pattern satisfies the unit-load condition, nor does it derive the load density from deeper principles; the docstring identifies that as the next bottom-up target.

theorem neutralUnitLoad_realized_iff_predictedMass_eq_supportCard
    (ψ : LightPattern Λ) (hunit : NeutralUnitLoadOnSupport ψ) :
    MassLawLoadRealized ψ ↔
      (ψ.support.card : ℝ) = predictedMass ψ := by
  unfold MassLawLoadRealized
  rw [integratedMeaningLoad_eq_supportCard_of_neutralUnitLoad ψ hunit]

theorem supportAveragedMassLawLoad_realizes
    (ψ : LightPattern Λ)
    (hsupp : ψ.support.Nonempty)
    (havg : SupportAveragedMassLawLoad ψ) :
    MassLawLoadRealized ψ := by
  unfold MassLawLoadRealized
  rw [integratedMeaningLoad_eq_support_sum ψ]
  have hcard_ne : (ψ.support.card : ℝ) ≠ 0 := by
    exact_mod_cast Finset.card_ne_zero.mpr hsupp
  calc
    ψ.support.sum (fun x => siteMeaningLoad ψ x)
        = ψ.support.sum
            (fun _x => predictedMass ψ / (ψ.support.card : ℝ)) := by
          apply Finset.sum_congr rfl
          intro x hx
          exact havg x hx
    _ = (ψ.support.card : ℝ) *
          (predictedMass ψ / (ψ.support.card : ℝ)) := by
          simp
    _ = predictedMass ψ := by
          field_simp [hcard_ne]

theorem stable_supportAveraged_restMass_eq_predictedMass
    (ψ : LightPattern Λ)
    (hstable : StableClosedLightPattern ψ)
    (havg : SupportAveragedMassLawLoad ψ) :
    restMass ψ = predictedMass ψ := by
  have hsupp : ψ.support.Nonempty :=
    hstable.1
  exact restMass_eq_predictedMass_of_load_realized ψ hstable
    (supportAveragedMassLawLoad_realizes ψ hsupp havg)

theorem topologyForcedLoadDensity_realizes
    (ψ : LightPattern Λ) (ρ : TopologyForcedLoadDensity ψ) :
    MassLawLoadRealized ψ := by
  unfold MassLawLoadRealized
  rw [integratedMeaningLoad_eq_support_sum ψ]
  calc
    ψ.support.sum (fun x => siteMeaningLoad ψ x)
        = ψ.support.sum ρ.density := by
          apply Finset.sum_congr rfl
          intro x hx
          exact (ρ.equals_site_load x hx).symm
    _ = predictedMass ψ := ρ.sums_to_predicted

What this page does not claim

Every pattern has unit loads at its occupied sites. The unit-load condition is the only way to realize the mass law. The predicted mass is derived from the support cardinality.

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

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