Encyclopedia Foundation Foundation Pair Kernel Gap2a Phase Bearing Transaction Residual

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Gap2a Phase Bearing Transaction Residual

A machine-checked construction shows an eight-phase transaction that cannot finish early, while proving that no physical scale can be selected from it.

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The eight-commit transaction

In Recognition Science, a ledger (a discrete record of events) advances through transactions. This construction builds the smallest explicit transaction requested after a prior production-orbit residual. It has exactly eight phases, and each phase carries a posting event and one committed Tick operation. The transaction state advances compositionally around a theorem-backed three-bit Gray carrier, a binary sequence where consecutive states differ by one bit. It returns after eight primitive commits and cannot return sooner.

That construction does not select a physical action scale. The Gray state and its exact period are dimensionless data. Adjoining the phase to the current observable class still leaves two representatives with the same Recognition data and different action. Consequently neither absolute event action nor a representative-independent multiplicative source dual descends. The no-go results are THEOREM, proved in the machine-checked library of formal theorems. No sorry; no new axiom.

The complete-orbit Green aggregate is also stated exactly. For a constant per-posting magnitude q, all eight phases aggregate to 4 * q, because the pair-kernel Green scale is q / 2. Identifying that aggregate with the static continuum source remains additional physical content.

The canonical Gray schedule below is a construction, not a physical selection theorem. The no-go results are THEOREM. No sorry; no new axiom.

The theorems establish that the transaction is actual and exact: it has eight phases, each with a posting event on a Gray phase axis, and the phase advance is additive and returns after eight commits. Theorems also prove that no selector can recover absolute posting action, no selector can be an action dual for all representatives, and no current boundary scale-breaking law exists. These are the no-go results: the construction is real, but it does not choose a physical scale.

THEOREM phaseBearingAdvance3_add · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean
phaseBearingAdvance3_add · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean:125 
/-- Transaction composition is addition of primitive commit counts. -/
theorem phaseBearingAdvance3_add
    (phase : Fin 8) (first second : ℕ) :
    phaseBearingAdvance3
        (phaseBearingAdvance3 phase first) second =
      phaseBearingAdvance3 phase (first + second) := by
  apply Fin.ext
  simp only [phaseBearingAdvance3, Fin.val_mk]
  omega
THEOREM phaseBearingAdvance3_eight · phaseBearingAdvance3_ne_self_before_eight · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean
phaseBearingAdvance3_eight · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean:142 
/-- Every transaction state returns after the complete eight-commit orbit. -/
theorem phaseBearingAdvance3_eight
    (phase : Fin 8) :
    phaseBearingAdvance3 phase 8 = phase := by
  apply Fin.ext
  simp [phaseBearingAdvance3,
    Nat.mod_eq_of_lt phase.isLt]
phaseBearingAdvance3_ne_self_before_eight · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean:150 
/-- 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 no_phaseTransaction_selector_recovers_absolutePostingAction · no_phaseTransaction_sourceSelector_is_actionDual_for_all_representatives · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean
no_phaseTransaction_selector_recovers_absolutePostingAction · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean:222 
/-- 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
no_phaseTransaction_sourceSelector_is_actionDual_for_all_representatives · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean:286 
/-- Phase decoration also cannot make the positive action-dual source
representative-independent. -/
theorem no_phaseTransaction_sourceSelector_is_actionDual_for_all_representatives :
    ¬ ∃ selector : PhaseTransactionActionDualSourceSelector3,
      IsActionDualForAllPhaseRepresentatives3 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 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)
  rw [hscaled] at hright
  rw [eulerJActionUnit_eq_one] at hleft hright
  nlinarith [hleft.1, hleft.2, hright.2]
THEOREM phaseBearingCompleteOrbitGreenSource3_of_constant · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean
phaseBearingCompleteOrbitGreenSource3_of_constant · IndisputableMonolith/Foundation/PairKernelGap2aPhaseBearingTransactionResidual.lean:334 
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

What this page does not claim

The module does not claim that the eight-phase schedule is a physical selection theorem. The module does not claim that absolute event action or a source dual can be derived from the transaction. The module does not claim that the Green aggregate equals a continuum source; that identification remains additional physical content.

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)

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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