Encyclopedia Foundation Foundation Absolute Scale Event Pricing Join Mutation Count Pricing Implies Oper

ARTICLE 5 claims 5 theorems

Foundation Absolute Scale Event Pricing Join Mutation Count Pricing Implies Oper

A single theorem in the Recognition Science framework forces the duration of a basic event to be one tick, by tying it to the count of ledger changes that produce it.

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The duration atom

In the Recognition Science framework, a ledger is a discrete record of events. A realized posting is one primitive event that has actually happened. The theorem mutationCountPricing_implies_operationalTickCount states that if the duration of such a posting is priced by the number of ledger mutations (changes) in the single committed minimum-J posting transition that realizes it, then that duration must equal exactly one operational tick. The right-hand side is a natural-number count supplied by the committed constructor, and the committed theorem ledgerL1Cost = 1 for every posting step forces the one-successor duration. The natural-number cast into ℝ is rigid, so no positive real unit survives.

The theorem is one half of a larger join in the framework's library, a machine-checked collection of formal theorems. The other half concerns energy: the physical energy of one realized posting is a multiplicative recognition channel-block attachment. Multiplicativity forces the empty-block value to one, which is exactly the uniform event-price scale one. The fixed point forces φ⁻¹ per channel, so the complete D+2 carrier prices the event at φ⁻⁵. Neither law names Constants.tau0, Constants.E_coh, Constants.hbar, a price field, a scale field, or a desired readout equality in its definition or structure fields. Those constants appear only in derived conclusions.

The canonical mechanism satisfies both laws. The stretched-duration decoy fails the duration law against every committed posting witness, and the doubled-energy decoy fails the energy law on every carrier of every dimension because no integer power of φ equals two. The composed join forces the S14 coherence-event arrow, hence native action, and rejects both native calibration directions: any positive duration-energy rescaling of a law-satisfying kinematics that still satisfies the law has both scales one.

Honesty: the two laws are supplied physical identifications (HYPOTHESIS as physical selections), like every production-side parent before them. What is new and THEOREM is that the one-successor duration and the unit price scale are now forced from parents that carry no unit datum, and that both decoys die. The external dimensional calibration orbit of AbsoluteScaleLocalFocusingResidual is untouched and remains open.

THEOREM mutationCountPricing_implies_operationalTickCount · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
mutationCountPricing_implies_operationalTickCount · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:140 
/-- Mutation-count pricing forces S19's operational one-successor duration
readout: the physical duration is the NNO tick-count interval. -/
theorem mutationCountPricing_implies_operationalTickCount
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hlaw : PostingDurationPricesCommittedMutationCount3 kinematics) :
    PostingDurationReadsOperationalTickCount3 kinematics := by
  intro event
  obtain ⟨A, B, hpost, _, _, hdur⟩ := hlaw event
  rw [(postingStep_implies_legalAtomicTick hpost).2] at hdur
  calc
    kinematics.duration event.1 = 1 := by
      rw [hdur]
      norm_num
    _ = recognitionTickCount (postingEventEndTick3 event) -
        recognitionTickCount (postingEventStartTick3 N) := by
      rw [postingEventEndTick3_eq_succ]
      simp [recognitionTickCount, tickSucc]
THEOREM event_pricing_join · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
event_pricing_join · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:679 
theorem event_pricing_join : EventPricingJoinCert where
  canonical_mechanism_passes :=
    canonicalKinematics_satisfies_join
  coherent_valuation_passes :=
    coherentEventValuation_kinematics_satisfies_join
  duration_law_forces_operational_count :=
    fun _kinematics hlaw =>
      mutationCountPricing_implies_operationalTickCount hlaw
  energy_law_forces_unit_price_scale :=
    fun _physical _survival _price _channelEnergy _kinematics
        hself haggregate hlaw =>
      (channelBlockEnergyLaw_forces_unit_price_scale
        hself haggregate hlaw).2
  join_forces_coherence_event :=
    fun _physical _channelEnergy _kinematics hcomplete hjoin =>
      operationalEventPricing_implies_coherenceEvent
        hcomplete hjoin
  stretched_duration_rejected :=
    stretchedDuration_not_mutationCountPriced
  doubled_energy_rejected :=
    doubledEnergy_not_channelBlockEnergy
  native_rescaling_rejected :=
    fun _kinematics _physical _physical' _channelEnergy _channelEnergy'
        durationScale energyScale hdurationPos henergyPos
        hcomplete hcomplete' hjoin hjoinScaled =>
      operationalEventPricing_kills_native_rescaling
        durationScale energyScale hdurationPos henergyPos
        hcomplete hcomplete' hjoin hjoinScaled
THEOREM canonicalKinematics_satisfies_join · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
canonicalKinematics_satisfies_join · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:491 
/-- The canonical mechanism satisfies the composed law. -/
theorem canonicalKinematics_satisfies_join
    {N : ℕ} [NeZero N] :
    OperationalEventPricingJoin3
      (parentPhysicalChannelCarrier3 N)
      canonicalPostingEventChannelEnergy3
      (canonicalCoherenceKinematics3 N) :=
  ⟨canonicalKinematics_prices_mutationCount,
    canonicalKinematics_reads_channelBlockEnergy⟩
THEOREM stretchedDuration_not_mutationCountPriced · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
stretchedDuration_not_mutationCountPriced · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:172 
/-- Duration decoy rejection: the stretched-duration kinematics contradicts
the universal unit mutation cost of every committed posting witness. -/
theorem stretchedDuration_not_mutationCountPriced :
    ¬ PostingDurationPricesCommittedMutationCount3
      (stretchedDurationKinematics3 3) := by
  intro hlaw
  obtain ⟨pair, hpair⟩ := realizedPrimitivePostingPair3_exists
  obtain ⟨A, B, hpost, _, _, hdur⟩ :=
    hlaw ⟨pair, hpair⟩
  rw [(postingStep_implies_legalAtomicTick hpost).2] at hdur
  change 2 * Constants.tau0 = ((1 : ℕ) : ℝ) at hdur
  norm_num [Constants.tau0, Constants.tick] at hdur
THEOREM doubledEnergy_not_channelBlockEnergy · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
doubledEnergy_not_channelBlockEnergy · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:395 
/-- Energy decoy rejection, strong form: the doubled-energy kinematics fails
the channel-block energy law on **every** carrier of every dimension and
every attachment, because no integer power of `phi` equals two. -/
theorem doubledEnergy_not_channelBlockEnergy
    (physical : PostingPhysicalChannelCarrier3 3)
    (channelEnergy : PostingEventChannelEnergy3 3) :
    ¬ PostingEnergyReadsChannelBlockEnergy3
      physical channelEnergy (doubledEnergyKinematics3 3) := by
  intro hlaw
  obtain ⟨pair, hpair⟩ := realizedPrimitivePostingPair3_exists
  have h :=
    channelBlockEnergyLaw_energy_eq_phi_zpow hlaw ⟨pair, hpair⟩
  set n : ℕ :=
    physicalPostingCarrierDimension3 physical ⟨pair, hpair⟩ with hn
  change 2 * Constants.E_coh = Constants.phi ^ (-(n : ℤ)) at h
  rw [GapDerivation.Constants_E_coh_eq_configDim,
    GapDerivation.configDim_at_D3] at h
  have hcast : (((5 : ℕ) : ℤ)) = (5 : ℤ) := by norm_num
  rw [hcast] at h
  have h2 : Constants.phi ^ (-(n : ℤ) + 5) = 2 := by
    calc
      Constants.phi ^ (-(n : ℤ) + 5) =
          Constants.phi ^ (-(n : ℤ)) *
            Constants.phi ^ (5 : ℤ) :=
        zpow_add₀ Constants.phi_ne_zero _ _
      _ = (2 * Constants.phi ^ (-(5 : ℤ))) *
            Constants.phi ^ (5 : ℤ) := by
        rw [← h]
      _ = 2 * Constants.phi ^ (-(5 : ℤ) + 5) := by
        rw [mul_assoc, ← zpow_add₀ Constants.phi_ne_zero]
      _ = 2 := by
        rw [show (-(5 : ℤ) + 5) = 0 by norm_num, zpow_zero, mul_one]
  exact phi_zpow_ne_two (-(n : ℤ) + 5) h2

What this page does not claim

The theorem does not establish that the physical identification of duration with mutation count is empirically true. The theorem does not fix the external dimensional calibration of the framework's constants. The theorem does not derive the value of the fine-structure constant.

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

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