Masses Mass Genesis Load Normalization Stable Support Averaged Rest Mass Eq Pred

The mass equality condition

The declaration stable_supportAveraged_restMass_eq_predictedMass is a theorem in the framework's machine-checked library of formal theorems. It states a precise condition under which two different notions of mass agree. The first notion, restMass, is the mass of a stable, closed light pattern as measured from its internal load. The second, predictedMass, is the mass value that the framework's topology alone selects for that pattern. The theorem proves that if the pattern is stable and closed, and if every occupied site carries the same share of the total predicted mass, then the rest mass equals the predicted mass.

The condition is called "support-averaged" because it averages the predicted mass over the pattern's support, the set of sites where the pattern is active. Each occupied site carries the equal share, predictedMass divided by the number of occupied sites. The theorem shows this is a sufficient condition: it is enough to guarantee the mass equality. It is not asserted as the final physics of every pattern. The framework's own documentation calls it a "sufficient target," a concrete normalization surface that states exactly what amplitude information is missing from a simpler unit-load normalization.

The theorem builds on earlier results. A unit neutral load at every occupied site makes the integrated load equal to the support cardinality, but that alone does not recover the predicted mass. The support-averaged law supplies the missing amplitude scale. The theorem also relies on a prior result that a realized load, one whose sum equals the predicted mass, forces the rest mass to match the predicted mass for stable patterns. The new theorem combines these pieces.

The declaration does not claim that every stable pattern satisfies the support-averaged condition. It proves a conditional statement: if the condition holds, then the equality follows. The framework explicitly marks the next step as open: deriving a topology-forced load density from deeper constraints, rather than assuming it as an external calibration. The theorem is a bridge, not a final law.

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)

/-- A support-averaged load law: every occupied site carries the equal share of
the topology-selected MassLaw load. This is a sufficient target, not asserted
as the final physics of every pattern. It is useful because it states the exact
amplitude information missing from CP6 normalization. -/
def SupportAveragedMassLawLoad (ψ : LightPattern Λ) : Prop :=
  ∀ x ∈ ψ.support,
    siteMeaningLoad ψ x =
      predictedMass ψ / (ψ.support.card : ℝ)

theorem integratedMeaningLoad_eq_supportCard_of_neutralUnitLoad
    (ψ : LightPattern Λ) (hunit : NeutralUnitLoadOnSupport ψ) :
    integratedMeaningLoad ψ = (ψ.support.card : ℝ) := by
  rw [integratedMeaningLoad_eq_support_sum ψ]
  calc
    ψ.support.sum (fun x => siteMeaningLoad ψ x)
        = ψ.support.sum (fun _x => (1 : ℝ)) := by
          apply Finset.sum_congr rfl
          intro x hx
          exact hunit x hx
    _ = (ψ.support.card : ℝ) := by
          simp

What this page does not claim

The support-averaged condition is asserted as the final physics of every pattern. The theorem proves that every stable pattern satisfies the support-averaged condition. The theorem derives the load density from topology alone.

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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Derived articles

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