Encyclopedia Masses Masses Mass Genesis T10 Organ Arrow Reduction Carrier Realized Filter Inverse Ph

ARTICLE 3 claims 3 theorems

Masses Mass Genesis T10 Organ Arrow Reduction Carrier Realized Filter Inverse Ph

A machine-checked theorem reduces a key open question in a mass-generation framework to a single, testable condition on a real number.

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The inverse-phi filter test

In the Recognition Science framework, a central open problem is to identify a specific real number, called the intended gap-one factor amplitude, from the data of a settled octave. The framework models this search as an "organ arrow": a rule that extracts some data from a settled octave, and a level function that names a real number from that data. The declaration carrierRealized_filter_inversePhiScaled_iff is a theorem about one particular test in this search.

The theorem concerns a scaled version of a base carrier pattern, scaled by the inverse of the golden ratio, φ⁻¹. It states that an organ arrow's filter realizes this inverse-phi-scaled pattern if and only if its level function returns φ⁻¹ times the intended amplitude at some octave. In plainer terms: if a candidate rule produces the inverse-phi-scaled pattern, then that rule must be naming the specific value φ⁻¹ times the target, somewhere. This is one of three similar tests, for the doubled, phi-scaled, and inverse-phi-scaled patterns, that together define a four-witness interface a candidate rule must satisfy.

The theorem is part of a larger reduction. The framework's library shows that an organ arrow meets this interface exactly when its level equals the intended amplitude at the octave of use and never equals 2f, φf, or φ⁻¹f at any octave. The question of whether any run-derived quantity can name the intended amplitude is thereby reduced to a single real-number condition. The library also shows that several specific run-derived constants, such as 1, 2, 16, cosh 1 − 1, √2, and its relatives, all miss the target. This is an audit of the constants currently in the library, not a classification of all possible constants.

What the theorem does not claim is equally important. It does not construct an organ that meets the interface. It does not establish an impossibility theorem for octave-varying organs, which could in principle name the target at the octave of use. The finite audit covers only the run-derived reals presently in the library, and the framework explicitly leaves open whether any run-derived quantity with independent ancestry can name the target. The theorem is a precise reduction and a finite audit, not a resolution of the underlying search.

THEOREM carrierRealized_filter_inversePhiScaled_iff · IndisputableMonolith/Masses/MassGenesis/T10OrganArrowReduction.lean
carrierRealized_filter_inversePhiScaled_iff · IndisputableMonolith/Masses/MassGenesis/T10OrganArrowReduction.lean:288 
/-- The pinning filter realizes the inverse-phi decoy iff the level returns
the target divided by phi at some octave. -/
theorem carrierRealized_filter_inversePhiScaled_iff
    (A : OrganArrow) (o : Q3SettledLedgerOctave) :
    CarrierRealized A.filter
        (inversePhiScaledCarrier (intendedGapOneFactorCarrier o)).pattern ↔
      ∃ o' : Q3SettledLedgerOctave,
        A.level (A.W o') = Constants.phi⁻¹ * intendedGapOneFactorAmplitude :=
  carrierRealized_filter_iff_of_pattern A _
    (inversePhiScaledCarrier (intendedGapOneFactorCarrier o)).q3_carrier
    (mul_pos (inv_pos.mpr Constants.phi_pos) intendedGapOneFactorAmplitude_pos)
    (inversePhiScaled_pattern_window o)
THEOREM organArrow_interface_iff · IndisputableMonolith/Masses/MassGenesis/T10OrganArrowReduction.lean
organArrow_interface_iff · IndisputableMonolith/Masses/MassGenesis/T10OrganArrowReduction.lean:303 
/-- Main reduction: an organ arrow meets the four-witness interface at `o`
iff its level names the target at `o` and never names a decoy scale at any
octave. -/
theorem organArrow_interface_iff (A : OrganArrow) (o : Q3SettledLedgerOctave) :
    RunPatternScaleCouplingInterface A.filter o ↔
      A.level (A.W o) = intendedGapOneFactorAmplitude ∧
        (∀ o' : Q3SettledLedgerOctave,
          A.level (A.W o') ≠ 2 * intendedGapOneFactorAmplitude) ∧
        (∀ o' : Q3SettledLedgerOctave,
          A.level (A.W o') ≠ Constants.phi * intendedGapOneFactorAmplitude) ∧
        (∀ o' : Q3SettledLedgerOctave,
          A.level (A.W o') ≠ Constants.phi⁻¹ * intendedGapOneFactorAmplitude) := by
  unfold RunPatternScaleCouplingInterface
  rw [carrierRealized_filter_doubled_iff, carrierRealized_filter_phiScaled_iff,
    carrierRealized_filter_inversePhiScaled_iff]
  constructor
  · rintro ⟨haccept, hd, hφ, hφinv⟩
    have hce : (intendedGapOneFactorCarrier o).amplitude =
        A.level (A.W (intendedGapOneFactorCarrier o).octave) := haccept
    rw [intendedGapOneFactorCarrier_amplitude,
      intendedGapOneFactorCarrier_octave] at hce
    refine ⟨hce.symm, fun o' h => hd ⟨o', h⟩, fun o' h => hφ ⟨o', h⟩,
      fun o' h => hφinv ⟨o', h⟩⟩
  · rintro ⟨hlevel, hd, hφ, hφinv⟩
    refine ⟨?_, ?_, ?_, ?_⟩
    · have hce : (intendedGapOneFactorCarrier o).amplitude =
          A.level (A.W (intendedGapOneFactorCarrier o).octave) := by
        rw [intendedGapOneFactorCarrier_amplitude,
          intendedGapOneFactorCarrier_octave]
        exact hlevel.symm
      exact hce
    · rintro ⟨o', h⟩
      exact hd o' h
    · rintro ⟨o', h⟩
      exact hφ o' h
    · rintro ⟨o', h⟩
      exact hφinv o' h
THEOREM run_derived_reals_audit · IndisputableMonolith/Masses/MassGenesis/T10OrganArrowReduction.lean
run_derived_reals_audit · IndisputableMonolith/Masses/MassGenesis/T10OrganArrowReduction.lean:492 
/-- Finite audit: the unit bridge, the per-phase settlement quantum, the
cycle total, the per-post J-cost, the holonomy √2, and the two deposit
normalizations (`postingNorm = 1/√2`, `creationDepositNorm = √2/4`) are the
run-derived constant reals currently in the library, and all miss the
target. -/
theorem run_derived_reals_audit :
    (1 : ℝ) ≠ intendedGapOneFactorAmplitude ∧
      (2 : ℝ) ≠ intendedGapOneFactorAmplitude ∧
        (16 : ℝ) ≠ intendedGapOneFactorAmplitude ∧
          (Real.cosh 1 - 1 ≠ intendedGapOneFactorAmplitude ∧
            Real.sqrt 2 ≠ intendedGapOneFactorAmplitude) ∧
          (1 / Real.sqrt 2 ≠ intendedGapOneFactorAmplitude ∧
            Real.sqrt 2 / 4 ≠ intendedGapOneFactorAmplitude) ∧
          (∑ phase : Fin 8,
            settlementLoad
              (ledgerOccupationWindow canonicalQ3SettledLedgerOctave phase)) =
            16 :=
  ⟨one_ne_intended, two_ne_intended, sixteen_ne_intended,
    ⟨coshOneSubOne_ne_intended, sqrtTwo_ne_intended⟩,
    ⟨invSqrtTwo_ne_intended, sqrtTwoDivFour_ne_intended⟩,
    canonicalCycleSettlementLoad_eq_sixteen⟩

What this page does not claim

No organ arrow is constructed that meets the interface. No impossibility theorem is claimed for octave-varying organs. The finite audit is not a classification of all possible run-derived constants.

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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/T10OrganArrowReduction.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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