Masses Mass Genesis Load Rigidity Factorized Same Topology Representative

A shortcut to mass

A ledger, a discrete record of events, can hold patterns of light that stay closed and stable. In the Recognition Science framework, each such pattern carries a load, a number that integrates its internal structure. The question is whether that load always equals the value the framework's mass law predicts. A direct proof for every possible pattern would be a long haul. This result offers a shortcut.

The shortcut rests on two ideas. First, rigidity: within one topology class, meaning patterns that share the same connected shape, the integrated load is the same for every stable pattern. Second, a representative: at least one stable pattern in that class has already been shown to have its site density in the factorized form the mass law demands. The declaration FactorizedSameTopologyRepresentative is the formal statement of that second idea. It asserts the existence of such a representative, a stable closed pattern with the same topology whose load density has already been derived in the factorized form.

What the framework's machine-checked library of formal theorems proves is the payoff. If a stable pattern is load-rigid inside its topology class, and if a factorized representative exists for that class, then the mass law load is realized for every stable pattern in the class. The theorem massLawLoadRealized_of_rigidity_and_factorized_representative states this directly. A second theorem, restMass_eq_predictedMass_of_rigidity_and_factorized_representative, extends the result to the rest mass itself: under the same two conditions, the rest mass of any stable closed pattern equals the predicted mass. No known mass value enters either statement.

In Recognition Science, this is the load-bearing step of the mass genesis chain. It reduces a global claim about all stable patterns to a local check: prove rigidity once, then exhibit one factorized representative per topology. The global version, restMass_eq_predictedMass_of_global_rigidity_and_representatives, packages the same argument for every stable pattern on a substrate at once. The result does not by itself prove that any factorized representative exists, nor that any particular pattern is load-rigid. Those are separate targets. What it establishes is the bridge: if the two premises hold, the mass law follows for the whole class.

theorem massLawLoadRealized_of_rigidity_and_factorized_representative
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ)
    (hrigid : StableTopologyLoadRigidAt ψ)
    (hrep : FactorizedSameTopologyRepresentative ψ) :
    MassLawLoadRealized ψ := by
  rcases hrep with ⟨χ, hχdensity, hχstable, htop⟩
  have hχrealized : MassLawLoadRealized χ :=
    factorizedTopologyLoadDensity_realizes χ hχdensity
  unfold MassLawLoadRealized at hχrealized ⊢
  calc
    integratedMeaningLoad ψ = integratedMeaningLoad χ :=
      hrigid hψ χ hχstable htop
    _ = predictedMass χ := hχrealized
    _ = predictedMass ψ := (predictedMass_eq_of_sameTopology htop).symm

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)

theorem restMass_eq_predictedMass_of_global_rigidity_and_representatives
    (hrigid : StableTopologyLoadRigidOn Λ)
    (hreps : FactorizedRepresentativeForEveryStableTopology Λ)
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ) :
    restMass ψ = predictedMass ψ :=
  restMass_eq_predictedMass_of_load_realized ψ hψ
    (massLawLoadRealized_of_global_rigidity_and_representatives
      hrigid hreps ψ hψ)

What this page does not claim

No factorized representative is proven to exist for any specific topology class. No stable pattern is proven to be load-rigid. The result does not assign a numerical value to any 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/LoadRigidity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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