Encyclopedia Foundation Foundation Absolute Scale Event Pricing Join Operational Event Pricing Kills Nat

ARTICLE 3 claims 3 theorems

Foundation Absolute Scale Event Pricing Join Operational Event Pricing Kills Nat

A new result in the Recognition Science framework forces the duration and energy of a single event to have one specific scale, and proves that no other scale can survive.

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The pricing join

The Recognition Science framework models physical reality as a ledger, a discrete record of events, where each event is a posting, a single change of state. The cost of any recognition, any act of distinguishing one event from another, is forced by a proved theorem. The framework's library, a machine-checked collection of formal theorems, now contains a result about the price of an event: its duration and its energy. The result is named operationalEventPricing_kills_native_rescaling.

The classical picture this touches on is that of a fundamental unit. In physics, one often asks: what is the smallest possible interval of time, or the smallest possible amount of energy? The framework's answer is that these are not free parameters. The duration of one posting is priced by the number of ledger mutations, the individual changes, in the minimal posting transition that realizes it. This count is a natural number, and the theorem ledgerL1Cost = 1 forces the operational one-successor duration to be exactly one tick. The energy of one posting is priced by a multiplicative channel-block attachment, and the self-similarity fixed point forces the empty-block value to one, which is exactly the unit price scale.

The new theorem establishes that any positive rescaling of both the duration and the energy of a law-satisfying kinematics, which is a description of how events evolve, that still satisfies the laws must have both scales equal to one. This is a uniqueness result. The proof uses the fact that the composition law for energy forces the empty-block value to one, and that the fixed point forces the per-channel attenuation to be the reciprocal of the golden ratio, phi. The complete carrier then prices the event at phi to the power of minus five. The theorem also shows that two decoys fail: a stretched-duration kinematics fails the duration law, and a doubled-energy kinematics fails the energy law because no integer power of phi equals two.

What the theorem does not claim is that the two laws themselves are derived. The duration and energy laws are supplied physical identifications, hypotheses about how the framework's concepts map to physical quantities. The theorem proves that, given those identifications, the scale is forced. The external dimensional calibration orbit, the question of how these units relate to conventional physical units, remains open. The theorem also does not claim that the framework's constants, such as tau0 or hbar, are derived from nothing; they appear only in derived conclusions, not in the laws' definitions.

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 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 duration and energy laws themselves are derived; they are supplied physical identifications. The framework's constants, such as tau0 or hbar, are derived from nothing; they appear only in derived conclusions. The external dimensional calibration orbit is closed 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/AbsoluteScaleEventPricingJoin.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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