Masses Mass Genesis Factorized Load Density Stable Factorized Topology Load Dens

The factorized mass target

In the Recognition Science framework, particle masses are not free parameters. The framework's library of machine-checked formal theorems derives a predicted mass for each stable light pattern, a discrete record of events. The theorem named stable_factorizedTopologyLoadDensity_restMass_eq_predictedMass states a conditional identity: if a pattern is stable and if a certain load density exists, then the pattern's rest mass equals its predicted mass.

The load density is the central object. It assigns a nonnegative number to each site in the pattern's support, the set of positions where the pattern is active. The density must match the site's meaning load and its total must equal the factorized mass law load. That factorized load is the product of three pieces: a sector geometry factor from the pattern's topology, a rung exponent from its closed-pattern scale, and a charge-gap exponent from its topology-derived charge. The theorem proves the identity by rewriting the factorized load as the predicted mass, a definitional equality.

The theorem does not prove that stable light dynamics forces the density to exist. That derivation remains open. The density is an assumption, not a consequence. The framework has factored the mass target into concrete pieces, but the next theorem must derive a site load density summing to this exact factorized expression from stable dynamics.

What the theorem does establish is a clean reduction. It shows that the mass prediction reduces to a single sufficient condition: exhibit a density with the stated properties. The factorized form gives the target a concrete shape, splitting the mass law value into sector, rung, and charge-gap contributions. This is the remaining black box in the mass genesis chain, and the theorem names exactly what must be supplied.

theorem stable_factorizedTopologyLoadDensity_restMass_eq_predictedMass
    (ψ : LightPattern Λ)
    (hstable : StableClosedLightPattern ψ)
    (ρ : FactorizedTopologyLoadDensity ψ) :
    restMass ψ = predictedMass ψ :=
  stable_topologyForcedLoadDensity_restMass_eq_predictedMass ψ hstable
    (topologyForcedLoadDensity_of_factorized ψ ρ)

/-- The factorized MassLaw load target, written as sector geometry times one
phi exponent assembled from rung and charge-gap pieces. -/
noncomputable def factorizedMassLawLoad (ψ : LightPattern Λ) : ℝ :=
  sectorLoadFactor ψ *
    Constants.phi ^ (rungLoadExponent ψ + chargeGapLoadExponent ψ)

theorem factorizedMassLawLoad_eq_predictedMass
    (ψ : LightPattern Λ) :
    factorizedMassLawLoad ψ = predictedMass ψ := by
  rfl

What this page does not claim

The theorem does not prove that stable light dynamics forces the load density to exist. The theorem does not derive the predicted mass formula from first principles; it assumes the density. The theorem does not state that the equal-share density is the physically realized one.

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/FactorizedLoadDensity.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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