Masses Mass Genesis T10 Anchor Closure C17 Axiom Audit Gap One Factor Amplitude

The factor amplitude

In the Recognition Science framework, particle masses are not free parameters but pure numbers derived from a forcing chain. The theorem ledger, a discrete record of recognition events, assigns to the gap-one worldline pattern the predicted mass 2 φ⁴², where φ is the golden ratio. That number is a fixed positive real once the sector, rung, and charge index are fixed; it carries no SI unit.

The declaration gapOne_factorAmplitude_eq_sqrt_predictedMass_div_sixteen proves that the intended factor amplitude, the positive stationary amplitude of the primitive factor, equals the positive square root of one sixteenth of that predicted mass. In symbols: amplitude = √(predictedMass / 16). This is a theorem in the machine-checked library of formal theorems, derived from the earlier identity that the factor amplitude squared equals predictedMass divided by sixteen, plus the nonnegativity of the amplitude.

What this theorem does not claim is just as important. It does not attach an SI unit to the mass or the amplitude; the value 2 φ⁴² is a pure number in the framework's native units. It does not assert that the predicted mass matches any measured particle mass; that comparison is an empirical check, not part of the theorem. It does not claim that the amplitude itself is the mass; it relates the amplitude to the mass through a fixed square-root factor.

The theorem sits inside a larger audit that shows the mass law's dependency closure reads only derived internal structure: cube-geometry integers, the coherence exponent φ⁻⁵, and the forced golden ratio. Nothing in that closure reads the amplitude, the R4 predicate, or any empirical mass input. The absolute scale is banked up to exactly one adopted model law, the load normalization, which is proved identical to the T10 residual and to R4. The freedom, the audit concludes, was never at the amplitude.

For a reader, the consequence is precise: within the framework, the factor amplitude is not an independent fitting parameter. It is a derived quantity, fixed by the same structure that fixes the mass, up to the single adopted model law that sets the load normalization. The theorem gives the exact relation; the empirical question of whether that relation matches nature remains a separate, measured question.

/-- **Positive-root form (THEOREM).** The intended gap-one factor amplitude
is the positive square root of one sixteenth of the predicted mass. -/
theorem gapOne_factorAmplitude_eq_sqrt_predictedMass_div_sixteen :
    primitivePositiveStationaryFactorAmplitude
        (worldlinePattern gapOneTwoPhaseMode) =
      Real.sqrt
        (predictedMass (worldlinePattern gapOneTwoPhaseMode) / 16) := by
  rw [← factorAmplitude_sq_eq_predictedMass_div_sixteen,
    Real.sqrt_sq
      (primitivePositiveStationaryFactorAmplitude_nonneg
        (worldlinePattern gapOneTwoPhaseMode))]

/-- **The gap-one anchor (THEOREM).** The topology-labelled gap-one
worldline pattern has the RS-native pure-number mass `2 * φ^42`. This public
receipt promotes the private arithmetic helper used by the posted-emission
wall into the paper-facing audit surface. It does not attach an SI unit. -/
theorem gapOne_predictedMass_eq_two_phi42 :
    predictedMass (worldlinePattern gapOneTwoPhaseMode) =
      2 * Constants.phi ^ (42 : ℕ) := by
  have hsec :
      sectorOf (worldlinePattern gapOneTwoPhaseMode) =
        Anchor.Sector.Electroweak := by
    simp [sectorOf, sectorFromTopology, worldlinePattern]
  have hrung : rungOf (worldlinePattern gapOneTwoPhaseMode) = 0 := by
    simp [rungOf, rungFromTopology, worldlinePattern]
  have hZ : ZOf (worldlinePattern gapOneTwoPhaseMode) = 0 := by
    simp [ZOf, ZFromTopology, sectorFromTopology, worldlinePattern]
  unfold predictedMass
  rw [hsec, hrung, hZ]
  unfold MassLaw.predict_mass
  rw [MassLaw.gap_zero_neutral]
  simp only [Anchor.yardstick, Anchor.E_coh, Anchor.B_pow_Electroweak_eq,
    Anchor.r0_Electroweak_eq]
  have hphi_ne : Constants.phi ≠ 0 := Constants.phi_ne_zero
  have h2 : (2 : ℝ) ^ (1 : ℤ) = (2 : ℝ) := by norm_num
  rw [h2]
  have hexp : (((0 : ℤ) : ℝ) - 8 + (0 : ℝ)) = (-8 : ℝ) := by norm_num
  rw [hexp]
  have hrpow :
      Constants.phi ^ (-8 : ℝ) = Constants.phi ^ (-(8 : ℤ)) := by simp
  rw [hrpow]
  have hpow :
      Constants.phi ^ (-(5 : ℤ)) * Constants.phi ^ (55 : ℤ) *
          Constants.phi ^ (-(8 : ℤ)) =
        Constants.phi ^ (42 : ℤ) := by
    rw [← zpow_add₀ hphi_ne, ← zpow_add₀ hphi_ne]
    norm_num
  conv_lhs =>
    rw [show
      (2 : ℝ) * Constants.phi ^ (-(5 : ℤ)) * Constants.phi ^ (55 : ℤ) *
          Constants.phi ^ (-(8 : ℤ)) =
        (2 : ℝ) *
          (Constants.phi ^ (-(5 : ℤ)) * Constants.phi ^ (55 : ℤ) *
            Constants.phi ^ (-(8 : ℤ))) by ring]
  rw [hpow]
  have hnat : Constants.phi ^ (42 : ℤ) =
      Constants.phi ^ (42 : ℕ) := by
    simp [← zpow_natCast]
  rw [hnat]

/-- **Positive-root form (THEOREM).** The intended gap-one factor amplitude
is the positive square root of one sixteenth of the predicted mass. -/
theorem gapOne_factorAmplitude_eq_sqrt_predictedMass_div_sixteen :
    primitivePositiveStationaryFactorAmplitude
        (worldlinePattern gapOneTwoPhaseMode) =
      Real.sqrt
        (predictedMass (worldlinePattern gapOneTwoPhaseMode) / 16) := by
  rw [← factorAmplitude_sq_eq_predictedMass_div_sixteen,
    Real.sqrt_sq
      (primitivePositiveStationaryFactorAmplitude_nonneg
        (worldlinePattern gapOneTwoPhaseMode))]

What this page does not claim

The theorem does not attach an SI unit to the predicted mass or the amplitude. The theorem does not assert that the predicted mass matches any measured particle mass. The theorem does not claim the amplitude itself is the mass; it only gives a square-root relation.

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/T10AnchorClosureC17AxiomAudit.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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