Masses Mass Genesis Anchor Amplitude Primitive Primitive Amplitude Cp6 Iff Primi

The amplitude equivalence

In the Recognition Science framework, a particle's rest mass is not a free parameter but a derived quantity: a pattern of eight complex numbers, called a light pattern, carries a load that the framework's dynamics preserve. The theorem primitiveAmplitudeCP6_iff_primitiveAmplitudeNorm states that two descriptions of this load are exactly equivalent. The first description, the CP6 form, says the pattern can be written as a single amplitude multiplied by a unit chord, a geometric statement about direction. The second, the norm form, says the squared length of the pattern equals the square of that same amplitude, an algebraic statement about magnitude.

The theorem proves the two statements imply each other for every light pattern. If the geometric description holds, the algebraic one follows; if the algebraic one holds, the geometric one follows. This is a genuine equivalence, not a one-way implication. The proof is short: it uses the fact that multiplying a vector by a real number scales its squared length by the square of that number, and that a unit chord has squared length one. The machine-checked library of formal theorems contains this result as a single declaration, and it is tagged THEOREM because it is proved with no unverified assumptions.

The equivalence matters because it connects two different routes to the same target. The CP6 form is the geometric route, closer to the framework's picture of a neutral chord in a meaning manifold. The norm form is the algebraic route, closer to the computations that produce a predicted mass. Having the equivalence means a proof that establishes one form automatically establishes the other. The library also shows that either form, together with a closed-pattern evidence condition, forces the full mass chain: the pattern's integrated load stays constant under evolution, the rest mass is positive, it is quantized on the phi rung, and it equals the predicted mass and both the inertial and gravitational readouts.

What the theorem does not claim is just as important. It does not prove that any actual particle satisfies either form; it only proves the two forms are equivalent. It does not derive the chord from the underlying Q3/Rhat dynamics; the docstring says explicitly that this still does not derive the chord from those dynamics. The equivalence is a landing surface, not a derivation. It also does not, by itself, produce a numerical mass for any particle; that requires the additional closed-pattern evidence and the full chain theorem. The theorem is a precise logical bridge, and the framework's own documentation is careful to say what remains open.

theorem primitiveAmplitudeCP6_iff_primitiveAmplitudeNorm
    (ψ : LightPattern (Fin 8)) :
    AnchorPhasePrimitiveAmplitudeCP6 ψ ↔
      AnchorPhasePrimitiveAmplitudeNorm ψ := by
  constructor
  · exact primitiveAmplitudeNorm_of_primitiveAmplitudeCP6 ψ
  · exact primitiveAmplitudeCP6_of_primitiveAmplitudeNorm ψ

theorem primitiveAmplitudeCP6_iff_primitiveAmplitudeNorm
    (ψ : LightPattern (Fin 8)) :
    AnchorPhasePrimitiveAmplitudeCP6 ψ ↔
      AnchorPhasePrimitiveAmplitudeNorm ψ := by
  constructor
  · exact primitiveAmplitudeNorm_of_primitiveAmplitudeCP6 ψ
  · exact primitiveAmplitudeCP6_of_primitiveAmplitudeNorm ψ

theorem full_chain_for_primitiveAmplitudeCP6
    (T : StableLoadReadoutTheory (Fin 8))
    {ψ : LightPattern (Fin 8)}
    (E : Q3ClosedPatternEvidence ψ)
    (hcp6 : AnchorPhasePrimitiveAmplitudeCP6 ψ) :
    (∀ k : ℕ,
      integratedMeaningLoad (evolvePattern k ψ) = integratedMeaningLoad ψ) ∧
    0 < restMass ψ ∧
    PhiRungQuantized ψ ∧
    restMass ψ = predictedMass ψ ∧
    T.readout.inertialMass ψ = restMass ψ ∧
    T.readout.gravitationalMass ψ = restMass ψ :=
  full_chain_for_primitiveFactorNorm T E
    (primitiveFactorNorm_of_primitiveAmplitudeCP6 ψ hcp6)

What this page does not claim

The theorem does not prove that any actual particle satisfies either the CP6 or the norm form. The theorem does not derive the chord from Q3/Rhat dynamics. The theorem does not by itself produce a numerical mass for any particle.

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/AnchorAmplitudePrimitive.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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