Masses Mass Genesis Anchor Amplitude Primitive Primitive Amplitude Norm Iff Prim

Two views of the same target

In the Recognition Science framework, a particle's mass is not a free parameter but a quantity that must emerge from the structure of a recognition pattern. The framework models a pattern as a discrete record of recognition events, and it assigns to each pattern a ledger, a discrete record of events, whose total load determines the mass. The declaration primitiveAmplitudeNorm_iff_primitiveFactorNorm is a theorem in the framework's machine-checked library of formal theorems. It states that two ways of writing down the same mass condition are logically equivalent: the condition that a pattern's squared norm equals the square of a certain amplitude is exactly the condition that the same norm equals a product of two sector loads.

The first condition, called the amplitude norm, says that the squared length of the pattern's neutralized window equals the square of a single amplitude, which is itself the product of two square roots. The second condition, called the factor norm, says that the same squared length equals the product of the two underlying loads directly. The theorem proves these two statements are interchangeable: if one holds, the other holds, and conversely. The proof is short because the amplitude is defined as the square root of the load, and squaring a square root returns the original load. This is not a numerical coincidence; it is a formal identity that holds for every pattern in the framework's eight-dimensional light-state geometry.

The equivalence matters because it connects two different pictures of the same physical situation. The amplitude picture treats the mass as coming from a single combined quantity, while the factor picture treats it as coming from two separate contributions: a sector base load and a phi transport factor. The theorem shows these are not competing hypotheses but two notations for the same target. It also provides a stepping stone: the framework's library uses this equivalence to show that if the amplitude condition holds, then the full chain of mass properties follows, including a positive rest mass, phi-rung quantization, and equality between predicted, inertial, and gravitational mass.

The theorem does not claim that the amplitude condition actually holds for any physical pattern. It only says that the two conditions are equivalent. Whether a real pattern satisfies them is a separate question, one that the framework's library leaves open. The theorem also does not derive the phi transport factor from deeper dynamics; it assumes that factor as part of the setup. What the theorem establishes is a bridge: once the amplitude condition is met, the factor condition follows, and with it the entire mass-genesis chain.

theorem primitiveAmplitudeNorm_iff_primitiveFactorNorm
    (ψ : LightPattern (Fin 8)) :
    AnchorPhasePrimitiveAmplitudeNorm ψ ↔ AnchorPhasePrimitiveFactorNorm ψ := by
  unfold AnchorPhasePrimitiveAmplitudeNorm AnchorPhasePrimitiveFactorNorm
  rw [primitiveClosedPatternAmplitude_sq ψ]

theorem primitiveClosedPatternAmplitude_sq (ψ : LightPattern (Fin 8)) :
    primitiveClosedPatternAmplitude ψ ^ 2 =
      primitiveAnchorSectorLoad ψ * primitivePhiTransport ψ := by
  unfold primitiveClosedPatternAmplitude
  unfold primitiveAnchorSectorAmplitude primitivePhiTransportAmplitude
  rw [mul_pow]
  rw [Real.sq_sqrt (le_of_lt (primitiveAnchorSectorLoad_pos ψ))]
  rw [Real.sq_sqrt (le_of_lt (primitivePhiTransport_pos 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 claim that any actual physical pattern satisfies the amplitude condition. The theorem does not derive the phi transport factor from the framework's dynamics. The theorem does not establish that the equivalence holds outside the framework's eight-dimensional light-state geometry.

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.

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