Masses Mass Genesis Load Rigidity Rest Mass Eq Predicted Mass Of Rigidity And Fa

Mass from topology

In physics, the rest mass of a particle is a number you look up. In Recognition Science, the framework's library, a machine-checked collection of formal theorems, aims to derive that number from the structure of a stable pattern. The declaration restMass_eq_predictedMass_of_rigidity_and_factorized_representative is a step in that program. It states: if a stable closed pattern has a certain rigidity property and admits a factorized representative of the same topology, then its rest mass equals its predicted mass. The theorem is proved in the framework's library; it is a formal result, not a numerical fit.

The statement is built from three ingredients. First, a stable closed pattern is a recurring, self-contained structure in the framework's model. Second, rigidity means that any two stable patterns with the same topology carry the same integrated neutral load, a quantity defined in the framework. Third, a factorized representative is a second pattern, of the same topology, whose site density has already been written in the factorized form that the mass law requires. The theorem assembles these: rigidity plus the existence of one good representative forces the mass equality for every stable pattern of that topology.

The proof is short because it reuses an earlier result. The theorem massLawLoadRealized_of_rigidity_and_factorized_representative shows that the two conditions imply the mass law load is realized. Then a previously proved closure theorem converts that load realization into the equality of rest mass and predicted mass. The declaration you asked about is the final step in that chain. It is a theorem in the library, with a formal proof that relies only on the earlier lemmas.

What the theorem does not claim is equally important. It does not assert that rigidity or a factorized representative actually exists for any given pattern. Those are separate targets, stated as definitions and open problems in the library. The theorem is conditional: if you can supply both ingredients, then the mass equality follows. It also does not assign a numerical value to any particle mass. The equality is between two framework-defined quantities; the predicted mass itself is computed elsewhere, and the comparison to measured particle masses is an empirical check, not part of this theorem.

In the broader program, this theorem reduces the task of deriving masses to two smaller tasks: prove rigidity for a topology class, and construct one factorized representative. The payoff is that no known mass enters the statement. If the two conditions are met, the mass law is forced by the structure alone. That is the sense in which the framework claims mass is derived, not input.

theorem restMass_eq_predictedMass_of_rigidity_and_factorized_representative
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ)
    (hrigid : StableTopologyLoadRigidAt ψ)
    (hrep : FactorizedSameTopologyRepresentative ψ) :
    restMass ψ = predictedMass ψ :=
  restMass_eq_predictedMass_of_load_realized ψ hψ
    (massLawLoadRealized_of_rigidity_and_factorized_representative
      ψ hψ hrigid hrep)

/-- Local rigidity target: once `ψ` is stable, every stable pattern with the
same topology has the same integrated neutral load. -/
def StableTopologyLoadRigidAt (ψ : LightPattern Λ) : Prop :=
  StableClosedLightPattern ψ →
    ∀ χ : LightPattern Λ,
      StableClosedLightPattern χ →
      SameTopology ψ χ →
      integratedMeaningLoad ψ = integratedMeaningLoad χ

theorem restMass_eq_predictedMass_of_rigidity_and_factorized_representative
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ)
    (hrigid : StableTopologyLoadRigidAt ψ)
    (hrep : FactorizedSameTopologyRepresentative ψ) :
    restMass ψ = predictedMass ψ :=
  restMass_eq_predictedMass_of_load_realized ψ hψ
    (massLawLoadRealized_of_rigidity_and_factorized_representative
      ψ hψ hrigid hrep)

What this page does not claim

The theorem does not assert that rigidity or a factorized representative exists for any pattern. The theorem does not assign a numerical value to any particle mass. The theorem does not compare its result to experimental measurements.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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