Masses Mass Genesis Pattern Readout Equivalence Pattern Readout Equivalence Cert

The equivalence certificate

In physics, the equality of inertial mass and gravitational mass is the heart of the equivalence principle: the mass that resists acceleration is the same as the mass that feels gravity. The Recognition Science framework formalizes this idea for patterns of light. Its declaration PatternReadoutEquivalenceCert is a certificate, a packaged collection of proofs, that establishes a precise statement: if both masses are read from the same underlying cost source, then they are equal.

The framework's ledger, a discrete record of events, assigns a cost to each pattern. The certificate proves that for any stable closed pattern, the inertial readout and the gravitational readout both equal the pattern's rest mass. This is not done by definition. The certificate requires proof that each readout actually reads the same source, and that the source is the derived rest mass. The theorem pattern_single_source_identity_with_restMass states the conjunction: inertial mass equals gravitational mass, and both equal rest mass.

The certificate also connects to a topology-derived value. If the rest mass equals the predicted mass from the MassLaw, then both readouts equal that predicted mass as well. This is the theorem pattern_single_source_identity_with_predictedMass. The certificate packages these results together with the integrated load readout identity, creating a single object that can be cited as evidence for the mass genesis milestone M8.

What the certificate does not claim is just as important. It does not define the two readouts to be equal. It does not assert that any particular physical pattern satisfies the required conditions. It does not prove that the predicted mass equals the rest mass; that is a separate hypothesis, stated as a premise in the theorem. The certificate provides the logical bridge: given the single-source condition, equality follows. It is a structural result about the framework's definitions, not an empirical measurement.

theorem pattern_single_source_identity_with_restMass
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ) :
    T.readout.inertialMass ψ = T.readout.gravitationalMass ψ ∧
    T.readout.inertialMass ψ = restMass ψ ∧
    T.readout.gravitationalMass ψ = restMass ψ := by
  have hreads := pattern_single_source_reads_restMass T ψ hψ
  exact ⟨pattern_single_source_equivalence T ψ hψ, hreads.1, hreads.2⟩

theorem pattern_single_source_equivalence
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ) :
    T.readout.inertialMass ψ = T.readout.gravitationalMass ψ := by
  rw [T.inertial_from_source ψ hψ,
      T.gravitational_from_source ψ hψ]

/-- If M7 is closed for the pattern, the two readouts also equal the
topology-derived MassLaw value. -/
theorem pattern_single_source_identity_with_predictedMass
    (T : PatternSingleSourceMassTheory Λ)
    (ψ : LightPattern Λ)
    (hψ : StableClosedLightPattern ψ)
    (hM7 : restMass ψ = predictedMass ψ) :
    T.readout.inertialMass ψ = predictedMass ψ ∧
    T.readout.gravitationalMass ψ = predictedMass ψ := by
  have hreads := pattern_single_source_reads_restMass T ψ hψ
  exact ⟨hreads.1.trans hM7, hreads.2.trans hM7⟩

structure PatternReadoutEquivalenceCert where
  single_source_reads_restMass :
    ∀ {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
      (T : PatternSingleSourceMassTheory Λ)
      (ψ : LightPattern Λ),
      StableClosedLightPattern ψ →
      ReadsStablePatternLoad T.readout ψ
  single_source_identity :
    ∀ {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
      (T : PatternSingleSourceMassTheory Λ)
      (ψ : LightPattern Λ),
      StableClosedLightPattern ψ →
      T.readout.inertialMass ψ = T.readout.gravitationalMass ψ ∧
      T.readout.inertialMass ψ = restMass ψ ∧
      T.readout.gravitationalMass ψ = restMass ψ
  integrated_load_readout_identity :
    ∀ {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
      (T : StableLoadReadoutTheory Λ)
      (ψ : LightPattern Λ),
      StableClosedLightPattern ψ →
      T.readout.inertialMass ψ = T.readout.gravitationalMass ψ ∧
      T.readout.inertialMass ψ = restMass ψ ∧
      T.readout.gravitationalMass ψ = restMass ψ

What this page does not claim

The certificate does not prove that any particular physical pattern satisfies the stable closed condition. The certificate does not prove that the predicted mass equals the rest mass; that is a separate hypothesis. The certificate does not provide an empirical measurement of any mass value.

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/PatternReadoutEquivalence.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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