Encyclopedia Foundation Foundation Absolute Scale Event Pricing Join Operational Event Pricing Implies C

ARTICLE 5 claims 5 theorems

Foundation Absolute Scale Event Pricing Join Operational Event Pricing Implies C

A new theorem in the Recognition Science framework shows that two operational pricing rules, one for duration and one for energy, force a primitive posting to realize exactly one coherence event.

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The pricing join

Recognition Science models physical reality as a ledger, a discrete record of events, where each event is a posting that changes a state. The framework's central result is a cost function, J, which measures the unavoidable cost of any recognition. A primitive posting is the smallest possible change, and a coherence event is the invariant that one realized primitive posting carries a specific energy, E_coh, for a specific duration, tau0.

The theorem operationalEventPricing_implies_coherenceEvent establishes a join between two operational pricing rules. The first rule, the duration atom, states that the physical duration of a posting is priced by the ledger mutation count of the underlying committed minimum-J posting transition that realizes it. The second rule, the energy atom, states that the physical energy of a posting is a multiplicative recognition channel-block attachment, where energy through disjoint independent channel blocks composes multiplicatively, and one channel attenuates by the reciprocal self-similarity fixed point.

The theorem proves that if these two operational pricing laws hold, then a primitive posting must realize one coherence event. The proof shows that the duration law forces the one-successor duration to be one tick, because the natural-number count of mutations is rigid and cannot be rescaled. The energy law forces the unit price scale to be one, because multiplicativity forces the empty-block value to one, and the self-similarity fixed point forces the per-channel energy to be phi⁻¹, so the complete D+2 carrier prices the event at phi⁻⁵.

The theorem also kills two decoy kinematics. A stretched-duration decoy fails the duration law against every committed posting witness, and a doubled-energy decoy fails the energy law on every carrier of every dimension because no integer power of phi equals two. The join also rejects both native calibration directions: any positive duration-energy rescaling of a law-satisfying kinematics that still satisfies the law has both scales one.

In Recognition Science, this theorem is a significant step because it derives the absolute scale of duration and energy from operational pricing rules that carry no unit datum. The two laws themselves are supplied physical identifications, hypotheses as physical selections, but the theorem proves that the one-successor duration and the unit price scale are now forced from these parents. The external dimensional calibration orbit remains open.

THEOREM operationalEventPricing_implies_coherenceEvent · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
operationalEventPricing_implies_coherenceEvent · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:443 
/-- **The event-level duration and energy-pricing join, derived.** The
composed law with a coherence-complete carrier forces the exact S14
coherence-event arrow that the absolute-scale program previously supplied. -/
theorem operationalEventPricing_implies_coherenceEvent
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    {kinematics : PostingEventKinematics3 N}
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (hjoin :
      OperationalEventPricingJoin3
        physical channelEnergy kinematics) :
    PrimitivePostingRealizesOneCoherenceEvent3 kinematics :=
  ⟨channelBlockEnergyLaw_implies_coherenceEnergy
      hcomplete hjoin.2,
    mutationCountPricing_implies_fundamentalTick hjoin.1⟩
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 channelBlockEnergyLaw_forces_unit_price_scale · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
channelBlockEnergyLaw_forces_unit_price_scale · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:332 
/-- **The S18 residual closes: the uniform event-price scale is forced to
one.** Against any S18 aggregate price built from a self-similar survival on
the same carrier, a kinematics obeying the channel-block energy law reads the
price exactly, and its uniform scale representation is scale one. -/
theorem channelBlockEnergyLaw_forces_unit_price_scale
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {survival : PostingEventSurvival3 N}
    {price : PostingEventChannelPrice3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    {kinematics : PostingEventKinematics3 N}
    (hself : PostingEventSurvivalSelfSimilar3 survival)
    (haggregate :
      PostingEventPriceReadsJAggregate3 physical survival price)
    (hlaw :
      PostingEnergyReadsChannelBlockEnergy3
        physical channelEnergy kinematics) :
    PostingEnergyReadsEventChannelPrice3 price kinematics ∧
      PostingEnergyHasUniformPriceScale3 1 price kinematics := by
  have hread :
      PostingEnergyReadsEventChannelPrice3 price kinematics := by
    intro event
    rw [channelBlockEnergyLaw_energy_eq_phi_zpow hlaw event,
      eventSurvivalAndAggregate_price_eq_phi_neg_dimension
        hself haggregate event]
  refine ⟨hread, ?_⟩
  intro event
  rw [hread event, one_mul]
THEOREM phi_zpow_ne_two · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
phi_zpow_ne_two · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:86 
/-- No integer power of `phi` equals two. This is the arithmetic wall that
kills the doubled-energy decoy on every carrier dimension. -/
theorem phi_zpow_ne_two (k : ℤ) : Constants.phi ^ k ≠ 2 := by
  intro hk
  rcases le_or_gt k 1 with hle | hgt
  · have hbound : Constants.phi ^ k ≤ Constants.phi ^ (1 : ℤ) :=
      zpow_le_zpow_right₀ (le_of_lt Constants.one_lt_phi) hle
    rw [zpow_one, hk] at hbound
    linarith [Constants.phi_lt_onePointSixTwo]
  · have h2le : (2 : ℤ) ≤ k := by omega
    have hbound : Constants.phi ^ (2 : ℤ) ≤ Constants.phi ^ k :=
      zpow_le_zpow_right₀ (le_of_lt Constants.one_lt_phi) h2le
    have hphi2 : Constants.phi ^ (2 : ℤ) =
        Constants.phi * Constants.phi := by
      rw [show (2 : ℤ) = 1 + 1 by norm_num,
        zpow_add₀ Constants.phi_ne_zero, zpow_one]
    rw [hphi2, hk] at hbound
    nlinarith [Constants.phi_gt_onePointSixOne]
THEOREM operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean:560 
/-- **The composed law kills both native rescaling directions.** If a
kinematics satisfies the join over a complete carrier and its positive
duration-energy rescaling also satisfies the join over a complete carrier,
then both scales are one. This is the event-level counterpart of the S19
two-parameter orbit: the orbit cannot move a law-satisfying kinematics. -/
theorem operationalEventPricing_kills_native_rescaling
    {kinematics : PostingEventKinematics3 3}
    {physical physical' : PostingPhysicalChannelCarrier3 3}
    {channelEnergy channelEnergy' : PostingEventChannelEnergy3 3}
    (durationScale energyScale : ℝ)
    (hdurationPos : 0 < durationScale)
    (henergyPos : 0 < energyScale)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (hcomplete' : PostingCarrierCoherenceComplete3 physical')
    (hjoin :
      OperationalEventPricingJoin3
        physical channelEnergy kinematics)
    (hjoinScaled :
      OperationalEventPricingJoin3
        physical' channelEnergy'
        (scalePostingEventKinematics3
          durationScale energyScale hdurationPos henergyPos
          kinematics)) :
    durationScale = 1 ∧ energyScale = 1 := by
  obtain ⟨pair, hpair⟩ := realizedPrimitivePostingPair3_exists
  have hdurBase :=
    mutationCountPricing_implies_fundamentalTick hjoin.1 pair hpair
  have hdurScaled :=
    mutationCountPricing_implies_fundamentalTick
      hjoinScaled.1 pair hpair
  have henergyBase :=
    channelBlockEnergyLaw_implies_coherenceEnergy
      hcomplete hjoin.2 pair hpair
  have henergyScaled :=
    channelBlockEnergyLaw_implies_coherenceEnergy
      hcomplete' hjoinScaled.2 pair hpair
  change
    durationScale * kinematics.duration pair =
      Constants.tau0 at hdurScaled
  change
    energyScale * kinematics.energy pair =
      Constants.E_coh at henergyScaled
  rw [hdurBase] at hdurScaled
  rw [henergyBase] at henergyScaled
  constructor
  · refine mul_right_cancel₀ (ne_of_gt Constants.tau0_pos) ?_
    rw [one_mul]
    exact hdurScaled
  · refine mul_right_cancel₀ (ne_of_gt Constants.E_coh_pos) ?_
    rw [one_mul]
    exact henergyScaled

What this page does not claim

The theorem does not claim that the two pricing laws are derived from first principles; they are supplied physical identifications. The theorem does not claim that the external dimensional calibration orbit is closed. The theorem does not claim that the coherence event invariant is the only possible invariant.

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

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