Foundation Pair Kernel Gap2a Phase Bearing Transaction Residual Phase Bearing Po

The phase axis theorem

In the Recognition Science framework, a recognition event is a discrete record of a comparison between two states. The framework's library, a machine-checked collection of formal theorems, proves that each of the eight phases in a production cycle carries a posting event that lies on a specific spatial axis. The theorem named phaseBearingPostingEvent3_has_phase_axis states that for every phase, the event's axis is the one given by the Gray code schedule, a binary sequence where consecutive values differ by exactly one bit.

The proof is constructive: it builds the event, shows it belongs to the allowed posting pairs, and verifies the axis witness. The Gray code schedule itself is a bijection from the eight phases to the three spatial axes, so each phase gets a distinct axis. The theorem also guarantees that advancing through the eight phases returns to the starting point only after eight commits, not sooner, and that each commit spans exactly one tick of the recognition cycle.

In Recognition Science, the framework models physical structure as derived from these recognition events. The theorem establishes a structural fact: the phase-bearing transaction is real, complete, and tied to the spatial axes. It does not, however, select a physical action scale. The Gray state and its period are dimensionless data; adjoining the phase to the observable class still leaves two representatives with the same recognition data but different action. Consequently, neither absolute event action nor a representative-independent multiplicative source dual descends from this construction.

The complete-orbit Green aggregate is also stated exactly: for a constant per-posting magnitude q, all eight phases aggregate to 4q, because the pair-kernel Green scale is q/2. Identifying that aggregate with the static continuum source remains additional physical content, not a theorem. The no-go results, such as the absence of a selector that recovers absolute posting action, are proved; the canonical Gray schedule is a construction, not a physical selection theorem.

theorem phaseBearingPostingEvent3_has_phase_axis
    (phase : Fin 8) :
    ProductionSpatialEventWitness3
      (phaseBearingPostingEvent3 phase)
      (grayPhaseAxis3 phase) :=
  productionSpatialEvent_has_axis (grayPhaseAxis3 phase)

/-- The actual phase carrier visits all eight three-bit states exactly once. -/
theorem phaseBearingProductionPattern_bijective :
    Function.Bijective grayCycle3Path :=
  grayCycle3_bijective

/-- No positive proper prefix of the octave returns to its initial phase. -/
theorem phaseBearingAdvance3_ne_self_before_eight
    (phase : Fin 8) (commits : ℕ)
    (hpositive : 0 < commits)
    (hshort : commits < 8) :
    phaseBearingAdvance3 phase commits ≠ phase := by
  intro hreturn
  have hval := congrArg Fin.val hreturn
  simp only [phaseBearingAdvance3, Fin.val_mk] at hval
  omega

theorem phaseBearingCompleteOrbitGreenSource3_of_constant
    (postingMagnitude : Fin 8 → ℝ)
    (q : ℝ)
    (hconstant : ∀ phase, postingMagnitude phase = q) :
    phaseBearingCompleteOrbitGreenSource3 postingMagnitude =
      4 * q := by
  unfold phaseBearingCompleteOrbitGreenSource3
  simp_rw [hconstant, realGreenScaleFromPostingMagnitude]
  simp [Finset.sum_const, nsmul_eq_mul]
  ring

/-- Even on the exact eight-phase actual transaction, absolute posting action
does not descend through the current observable quotient. -/
theorem no_phaseTransaction_selector_recovers_absolutePostingAction :
    ¬ ∃ selector : PhaseTransactionAbsoluteActionSelector3,
      SelectsAllPhaseRepresentativeActions3 selector := by
  rintro ⟨selector, hselector⟩
  let left : RecognitionPhysicalValuation3.{0} 3 :=
    unitRecognitionPhysicalValuation3
      (parentPhysicalChannelCarrier3 3)
  let right : RecognitionPhysicalValuation3.{0} 3 :=
    scalePhysicalValuation3
      2 1 (by norm_num) (by norm_num) left
  let phase : Fin 8 := 0
  have hsame : SameRecognitionData3 left right := by
    dsimp [right]
    exact scalePhysicalValuation_sameRecognitionData
      2 1 (by norm_num) (by norm_num) left
  have hclass :
      physicalObservableClass3 left =
        physicalObservableClass3 right :=
    sameRecognitionData_sameObservableClass hsame
  have hleft := hselector left phase
  have hright := hselector right phase
  rw [← hclass] at hright
  have hequal :
      postingEventAction3 left.kinematics
          (phaseBearingPostingEvent3 phase).1 =
        postingEventAction3 right.kinematics
          (phaseBearingPostingEvent3 phase).1 :=
    hleft.symm.trans hright
  have hscaled :
      postingEventAction3 right.kinematics
          (phaseBearingPostingEvent3 phase).1 =
        2 *
          postingEventAction3 left.kinematics
            (phaseBearingPostingEvent3 phase).1 := by
    dsimp [right]
    simpa using
      (scalePhysicalValuation_eventAction
        2 1 (by norm_num) (by norm_num) left
        (phaseBearingPostingEvent3 phase).1)
  have hpositive :
      0 <
        postingEventAction3 left.kinematics
          (phaseBearingPostingEvent3 phase).1 :=
    mul_pos
      (left.kinematics.energy_pos _)
      (left.kinematics.duration_pos _)
  rw [hscaled] at hequal
  nlinarith

What this page does not claim

The theorem does not select a physical action scale for any posting event. The canonical Gray schedule is not a physical selection theorem, only a construction. Identifying the aggregate Green source with the static continuum source is not established by this theorem.

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/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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