Masses Mass Genesis Factorized Load Density Rung Load Exponent Unfold

The rung exponent

The Recognition Science framework builds particle masses from a discrete ledger, a record of recognition events in which each event carries a forced cost. Within that framework, the mass of a stable light pattern is expressed as a product of three factors: a sector geometry term, a rung scale term, and a charge-gap term. The declaration rungLoadExponent_unfold establishes that the rung exponent in that product is simply the pattern's rung number minus eight. In symbols, for any light pattern ψ, the exponent equals (rungOf ψ) − 8.

The theorem is a definitional unfolding, proved by reflexivity: it rewrites the defined exponent in terms of the already-defined rung number. It does not, by itself, derive the rung number from anything deeper, nor does it prove that stable light dynamics forces the density to take this form. The library's own docstring states that the next theorem must derive a site load density summing to the factorized expression. The declaration is a bridge, not a destination.

What the declaration does do is remove a black box from the mass-genesis chain. The factorized load target, written as sector geometry times a phi exponent assembled from rung and charge-gap pieces, is shown to equal the predicted mass by another theorem, also proved by reflexivity. The rung exponent theorem makes the structure of that exponent explicit, so a later proof can aim at deriving the rung number itself from topology. The machine-checked library of formal theorems records this step as a theorem, not a definition, because it states an equality about the defined exponent.

The declaration does not claim that the rung number is forced by the framework's axioms, nor that the factorized density is the unique one, nor that any stable pattern's rest mass equals the predicted mass without additional hypotheses. Those claims belong to other declarations with their own proofs. What remains open is the derivation of the density itself, the object the library names as the next bottom-up target.

theorem rungLoadExponent_unfold (ψ : LightPattern Λ) :
    rungLoadExponent ψ = (rungOf ψ : ℝ) - 8 := rfl

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

What this page does not claim

The rung number is derived from deeper axioms by this declaration. Stable light dynamics forces the density to take the factorized form. The factorized density is the unique one summing to the predicted mass.

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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