Encyclopedia Foundation Foundation Absolute Scale Event Pricing Join Operational Event Pricing Implies P
ARTICLE 5 claims 4 theorems 1 hypothesis
Foundation Absolute Scale Event Pricing Join Operational Event Pricing Implies P
A machine-checked proof shows that once events are priced by ledger work and channel energy, their duration and energy scale are forced to one specific value.
The pricing join
In physics, a coupling constant is a number that sets the strength of an interaction, and it often must be measured rather than derived. The Recognition Science framework contains a machine-checked library of formal theorems, and one of its results, operationalEventPricing_implies_productionInvariant, concerns a different kind of number: the scale at which individual events are priced. The theorem states that if two natural pricing laws hold, then the event's duration and energy are forced to take on one specific value each, with no free parameter left over.
The first law prices duration by counting the minimum number of ledger mutations, or changes to a discrete record of events, that a posting step requires. Since each posting step is proven to cost exactly one unit, the duration of a realized posting is forced to be one fundamental tick. The second law prices energy by attaching multiplicative weights to independent channel blocks, and the requirement that this composition work forces the empty-block weight to be one, which in turn fixes the unit price scale. The self-similarity condition then forces each channel to attenuate by the reciprocal of the golden ratio, so the complete carrier prices an event at phi to the negative fifth power.
The theorem does not claim that these pricing laws are derived from first principles. They are supplied physical identifications, hypotheses about how the framework's abstract structures connect to measurable quantities. What is proved is the consequence: given those two laws, the one-successor duration and the unit price scale are forced, and two decoy alternatives, a stretched duration and a doubled energy, are shown to fail the laws. The theorem also proves that any positive rescaling of both duration and energy that still satisfies the laws must have both scales equal to one.
In Recognition Science, this result closes a gap where the absolute scale of events was previously left free. It derives the native action and the coherence-event invariant from the two pricing laws. The external dimensional calibration orbit, which relates these internal scales to SI units, remains an open target.
THEOREM operationalEventPricing_implies_productionInvariant · operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- Valuation-level constructor: any weak physical valuation whose kinematics
satisfies the composed law satisfies the production-side coherence-event
invariant that the absolute-scale residuals consume as a supplied parent. -/
theorem operationalEventPricing_implies_productionInvariant
{valuation : RecognitionPhysicalValuation3.{0} 3}
{physical : PostingPhysicalChannelCarrier3 3}
{channelEnergy : PostingEventChannelEnergy3 3}
(hcomplete : PostingCarrierCoherenceComplete3 physical)
(hjoin :
OperationalEventPricingJoin3
physical channelEnergy valuation.kinematics) :
ProductionSideCoherenceEventInvariant3 valuation :=
operationalEventPricing_implies_coherenceEvent
hcomplete hjoin
/-- **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
THEOREM mutationCountPricing_implies_fundamentalTick · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- Mutation-count pricing forces the fundamental one-tick duration: the
committed dynamics prove every posting step has mutation count one. -/
theorem mutationCountPricing_implies_fundamentalTick
{N : ℕ} [NeZero N]
{kinematics : PostingEventKinematics3 N}
(hlaw : PostingDurationPricesCommittedMutationCount3 kinematics) :
RealizedPostingLastsFundamentalTick3 kinematics := by
intro event hevent
obtain ⟨A, B, hpost, _, _, hdur⟩ :=
hlaw ⟨event, hevent⟩
rw [(postingStep_implies_legalAtomicTick hpost).2] at hdur
rw [hdur]
simp [Constants.tau0, Constants.tick]
THEOREM blockEnergy_one_forced · channelBlockEnergyLaw_energy_eq_phi_zpow · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- The single-channel energy is forced to `phi⁻¹` by the reciprocal
self-similarity fixed point. -/
theorem blockEnergy_one_forced
(event : RealizedPostingEvent3 N) :
channelEnergy.blockEnergy event 1 = 1 / Constants.phi :=
Cosmology.DarkEnergyPhiDilutionDerivation.self_similar_attenuation_forced
(channelEnergy.blockEnergy_pos event 1)
(channelEnergy.blockEnergy_one_self_similar event)
/-- The energy law prices every event at `phi` to the negative carrier
dimension, with no residual scalar. -/
theorem channelBlockEnergyLaw_energy_eq_phi_zpow
{N : ℕ} [NeZero N]
{physical : PostingPhysicalChannelCarrier3 N}
{channelEnergy : PostingEventChannelEnergy3 N}
{kinematics : PostingEventKinematics3 N}
(hlaw :
PostingEnergyReadsChannelBlockEnergy3
physical channelEnergy kinematics)
(event : RealizedPostingEvent3 N) :
kinematics.energy event.1 =
Constants.phi ^
(-(physicalPostingCarrierDimension3 physical event : ℤ)) := by
rw [hlaw event]
exact channelEnergy.blockEnergy_forced event _
HYPOTHESIS PostingDurationPricesCommittedMutationCount3 · PostingEnergyReadsChannelBlockEnergy3 · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- Target-blind operational clock law. The physical duration of one realized
posting equals the real image of the ledger mutation count of one underlying
committed minimum-J posting transition realizing it, with the event's
double-entry conservation witness carried alongside. The right-hand side is
a natural-number count produced by the committed production constructor. No
named time constant, coherence energy, action quantum, or desired readout
equality occurs in this definition. -/
def PostingDurationPricesCommittedMutationCount3
{N : ℕ} [NeZero N]
(kinematics : PostingEventKinematics3 N) : Prop :=
∀ event : RealizedPostingEvent3 N,
∃ A B : LedgerPostingAdjacency.LedgerState 3,
PostingStep A B ∧
JMinimalPostingStep A B ∧
(∑ k : Fin (TorusCard3 N),
divF (elementaryPosting event.1.1 event.1.2) k) = 0 ∧
kinematics.duration event.1 =
(ledgerL1Cost A B : ℝ)
/-- Target-blind physical energy law. One realized posting's energy is its
multiplicative self-similar channel-block energy through its classified
physical carrier. No `E_coh`, `hbar`, price field, or scale field occurs. -/
def PostingEnergyReadsChannelBlockEnergy3
{N : ℕ} [NeZero N]
(physical : PostingPhysicalChannelCarrier3 N)
(channelEnergy : PostingEventChannelEnergy3 N)
(kinematics : PostingEventKinematics3 N) : Prop :=
∀ event : RealizedPostingEvent3 N,
kinematics.energy event.1 =
channelEnergy.blockEnergy event
(physicalPostingCarrierDimension3 physical event)
THEOREM operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- **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 pricing laws are not derived from the framework's axioms; they are supplied as physical identifications. The theorem does not claim that the golden ratio appears in any measured physical constant outside the framework's own definitions. The result does not provide a numerical value for any conventional physical constant such as the fine-structure constant.
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)
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.
Derived articles
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- What physical measurement would confirm the framework's predicted phi to the negative fifth power event price?
- What external dimensional calibration would connect the internal unit scale to SI units?
- What experimental observation could falsify the hypothesis that energy composes multiplicatively across channel blocks?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM operationalEventPricing_implies_productionInvariant · operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- Valuation-level constructor: any weak physical valuation whose kinematics satisfies the composed law satisfies the production-side coherence-event invariant that the absolute-scale residuals consume as a supplied parent. -/ theorem operationalEventPricing_implies_productionInvariant {valuation : RecognitionPhysicalValuation3.{0} 3} {physical : PostingPhysicalChannelCarrier3 3} {channelEnergy : PostingEventChannelEnergy3 3} (hcomplete : PostingCarrierCoherenceComplete3 physical) (hjoin : OperationalEventPricingJoin3 physical channelEnergy valuation.kinematics) : ProductionSideCoherenceEventInvariant3 valuation := operationalEventPricing_implies_coherenceEvent hcomplete hjoin/-- **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 henergyScaledThe theorem states that if two natural pricing laws hold, then the event's duration and energy are forced to take on one specific value each, with no free parameter left over. operationalEventPricing_implies_productionInvariant · operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.leanTHEOREM mutationCountPricing_implies_fundamentalTick · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- Mutation-count pricing forces the fundamental one-tick duration: the committed dynamics prove every posting step has mutation count one. -/ theorem mutationCountPricing_implies_fundamentalTick {N : ℕ} [NeZero N] {kinematics : PostingEventKinematics3 N} (hlaw : PostingDurationPricesCommittedMutationCount3 kinematics) : RealizedPostingLastsFundamentalTick3 kinematics := by intro event hevent obtain ⟨A, B, hpost, _, _, hdur⟩ := hlaw ⟨event, hevent⟩ rw [(postingStep_implies_legalAtomicTick hpost).2] at hdur rw [hdur] simp [Constants.tau0, Constants.tick]Since each posting step is proven to cost exactly one unit, the duration of a realized posting is forced to be one fundamental tick. mutationCountPricing_implies_fundamentalTick · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.leanTHEOREM blockEnergy_one_forced · channelBlockEnergyLaw_energy_eq_phi_zpow · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- The single-channel energy is forced to `phi⁻¹` by the reciprocal self-similarity fixed point. -/ theorem blockEnergy_one_forced (event : RealizedPostingEvent3 N) : channelEnergy.blockEnergy event 1 = 1 / Constants.phi := Cosmology.DarkEnergyPhiDilutionDerivation.self_similar_attenuation_forced (channelEnergy.blockEnergy_pos event 1) (channelEnergy.blockEnergy_one_self_similar event)/-- The energy law prices every event at `phi` to the negative carrier dimension, with no residual scalar. -/ theorem channelBlockEnergyLaw_energy_eq_phi_zpow {N : ℕ} [NeZero N] {physical : PostingPhysicalChannelCarrier3 N} {channelEnergy : PostingEventChannelEnergy3 N} {kinematics : PostingEventKinematics3 N} (hlaw : PostingEnergyReadsChannelBlockEnergy3 physical channelEnergy kinematics) (event : RealizedPostingEvent3 N) : kinematics.energy event.1 = Constants.phi ^ (-(physicalPostingCarrierDimension3 physical event : ℤ)) := by rw [hlaw event] exact channelEnergy.blockEnergy_forced event _The self-similarity condition then forces each channel to attenuate by the reciprocal of the golden ratio, so the complete carrier prices an event at phi to the negative fifth power. blockEnergy_one_forced · channelBlockEnergyLaw_energy_eq_phi_zpow · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.leanHYPOTHESIS PostingDurationPricesCommittedMutationCount3 · PostingEnergyReadsChannelBlockEnergy3 · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- Target-blind operational clock law. The physical duration of one realized posting equals the real image of the ledger mutation count of one underlying committed minimum-J posting transition realizing it, with the event's double-entry conservation witness carried alongside. The right-hand side is a natural-number count produced by the committed production constructor. No named time constant, coherence energy, action quantum, or desired readout equality occurs in this definition. -/ def PostingDurationPricesCommittedMutationCount3 {N : ℕ} [NeZero N] (kinematics : PostingEventKinematics3 N) : Prop := ∀ event : RealizedPostingEvent3 N, ∃ A B : LedgerPostingAdjacency.LedgerState 3, PostingStep A B ∧ JMinimalPostingStep A B ∧ (∑ k : Fin (TorusCard3 N), divF (elementaryPosting event.1.1 event.1.2) k) = 0 ∧ kinematics.duration event.1 = (ledgerL1Cost A B : ℝ)/-- Target-blind physical energy law. One realized posting's energy is its multiplicative self-similar channel-block energy through its classified physical carrier. No `E_coh`, `hbar`, price field, or scale field occurs. -/ def PostingEnergyReadsChannelBlockEnergy3 {N : ℕ} [NeZero N] (physical : PostingPhysicalChannelCarrier3 N) (channelEnergy : PostingEventChannelEnergy3 N) (kinematics : PostingEventKinematics3 N) : Prop := ∀ event : RealizedPostingEvent3 N, kinematics.energy event.1 = channelEnergy.blockEnergy event (physicalPostingCarrierDimension3 physical event)They are supplied physical identifications, hypotheses about how the framework's abstract structures connect to measurable quantities. PostingDurationPricesCommittedMutationCount3 · PostingEnergyReadsChannelBlockEnergy3 · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.leanTHEOREM operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean
/-- **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 henergyScaledThe theorem also proves that any positive rescaling of both duration and energy that still satisfies the laws must have both scales equal to one. operationalEventPricing_kills_native_rescaling · IndisputableMonolith/Foundation/AbsoluteScaleEventPricingJoin.lean