Encyclopedia Foundation Foundation Pair Kernel Physical Valuation Initiality S19 Consumer

ARTICLE 4 claims 3 theorems 1 model

Foundation Pair Kernel Physical Valuation Initiality S19 Consumer

A machine-checked module that pins down what a physical readout can and cannot determine, proving a key non-uniqueness result.

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The S19 consumer

In Recognition Science, a physical valuation is a rule that assigns a physical meaning, such as an energy scale or a span measure, to a recognition event. The S19 consumer is a module in the framework's machine-checked library of formal theorems. It compiles and checks several results about these valuations. Its plain-language purpose is to state exactly how much physical information a recognition event carries, and to prove that some of that information is not unique.

The central theorem of the module, physicalValuationNonuniqueness_consumer, proves a striking fact. There exist two distinct physical valuations that agree on all recognition data, but disagree on the span measure and on the energy scale. Even more, there is a pair that also preserves the action of every posting pair. This means the weak physical valuation is not unique: different assignments of physical scales can be observationally equivalent at the level of recognition data, while still differing in their physical interpretation.

The module also establishes a positive result about the operational duration semantic. A theorem, canonicalOperationalReadoutConsumer_exists, shows that a model of the Tick-count duration can replace a previous residual, while the other five residuals retain their explicit hypotheses. This is a MODEL witness, not a derivation from first principles. It shows that a particular construction is coherent and exists, but it does not force that construction to be the only one.

Finally, the module recompiles the nonlinear exact-J Gauss law, the tangent Hessian, and the Green response consumer. This compilation is independent of the physical valuation normalization. The result is that these mathematical structures hold regardless of how the physical scale is chosen, reinforcing the non-uniqueness theorem: the core equations do not select a unique physical scale.

What this means for the framework is a precise boundary. The recognition data, the action, and the core equations do not pin down a single physical valuation. The framework's library proves this non-uniqueness as a theorem, and it provides a model witness for one operational semantic. The S19 consumer does not claim to have found the unique physical valuation; it proves that such uniqueness does not exist at this level.

THEOREM physicalValuationNonuniqueness_consumer · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean
physicalValuationNonuniqueness_consumer · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean:107 
/-- The S19 theorem-backed result is non-uniqueness of the weak physical
valuation, including an action-preserving inverse-scale pair. -/
theorem physicalValuationNonuniqueness_consumer :
    (∃ left right : RecognitionPhysicalValuation3.{0} 3,
      SameRecognitionData3 left right ∧
        left.span.measure 1 ≠ right.span.measure 1 ∧
        left.energyScale ≠ right.energyScale) ∧
      (∃ left right : RecognitionPhysicalValuation3.{0} 3,
        SameRecognitionData3 left right ∧
          left.span.measure 1 ≠ right.span.measure 1 ∧
          left.energyScale ≠ right.energyScale ∧
          ∀ event : PostingPair3 3,
            postingEventAction3 left.kinematics event =
              postingEventAction3 right.kinematics event) :=
  ⟨weak_signature_admits_independent_duration_energy_scales,
    weak_signature_and_action_admit_inverse_scales⟩
THEOREM physicalValuationNonuniqueness_consumer · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean
physicalValuationNonuniqueness_consumer · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean:107 
/-- The S19 theorem-backed result is non-uniqueness of the weak physical
valuation, including an action-preserving inverse-scale pair. -/
theorem physicalValuationNonuniqueness_consumer :
    (∃ left right : RecognitionPhysicalValuation3.{0} 3,
      SameRecognitionData3 left right ∧
        left.span.measure 1 ≠ right.span.measure 1 ∧
        left.energyScale ≠ right.energyScale) ∧
      (∃ left right : RecognitionPhysicalValuation3.{0} 3,
        SameRecognitionData3 left right ∧
          left.span.measure 1 ≠ right.span.measure 1 ∧
          left.energyScale ≠ right.energyScale ∧
          ∀ event : PostingPair3 3,
            postingEventAction3 left.kinematics event =
              postingEventAction3 right.kinematics event) :=
  ⟨weak_signature_admits_independent_duration_energy_scales,
    weak_signature_and_action_admit_inverse_scales⟩
THEOREM canonicalOperationalReadoutConsumer_exists · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean
canonicalOperationalReadoutConsumer_exists · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean:42 
/-- MODEL consumer showing that the NNO-backed operational duration semantic
can replace the S18 duration residual while the other five residuals retain
their explicit hypotheses. -/
theorem canonicalOperationalReadoutConsumer_exists :
    ∃ event : RealizedPostingEvent3 3,
      PostingDurationReadsOperationalTickCount3
          (canonicalCoherenceKinematics3 3) ∧
        PhysicalPostingReadoutSemantics3
          (parentPhysicalChannelCarrier3 3)
          (recognitionChannelsFromEventSurvival3
            (@canonicalPostingEventSurvival3 3 _)
            canonicalPostingEventSurvival_selfSimilar)
          (@canonicalPostingEventChannelPrice3 3 _)
          (canonicalCoherenceKinematics3 3) ∧
        PrimitivePostingRealizesOneCoherenceEvent3
          (canonicalCoherenceKinematics3 3) ∧
        postingEventAction3
          (canonicalCoherenceKinematics3 3) event.1 =
            Constants.hbar := by
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 := ⟨pair, hpair⟩
  have hcomplete :
      PostingCarrierCoherenceComplete3
        (parentPhysicalChannelCarrier3 3) :=
    parentPhysicalChannelCarrier_is_complete
  have hinjective :
      PostingCarrierParentInjective3
        (parentPhysicalChannelCarrier3 3) :=
    ((carrierComplete_iff_injective_and_surjective
      (parentPhysicalChannelCarrier3 3)).1
        hcomplete).1
  have hsurjective :
      PostingCarrierParentSurjective3
        (parentPhysicalChannelCarrier3 3) :=
    ((carrierComplete_iff_injective_and_surjective
      (parentPhysicalChannelCarrier3 3)).1
        hcomplete).2
  have hreadouts :
      PhysicalPostingReadoutSemantics3
        (parentPhysicalChannelCarrier3 3)
        (recognitionChannelsFromEventSurvival3
          (@canonicalPostingEventSurvival3 3 _)
          canonicalPostingEventSurvival_selfSimilar)
        (@canonicalPostingEventChannelPrice3 3 _)
        (canonicalCoherenceKinematics3 3) :=
    operationalTickCount_and_remainingResiduals_imply_physicalReadouts
      canonicalKinematics_operationalTickCount_model
      hinjective hsurjective
      canonicalPostingEventSurvival_selfSimilar
      canonicalPostingEventPrice_reads_JAggregate
      canonicalEnergy_has_uniformScale_one
  have hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3
        (canonicalCoherenceKinematics3 3) :=
    physicalReadouts_imply_primitiveCoherenceEvent
      hreadouts
  refine
    ⟨event,
      canonicalKinematics_operationalTickCount_model,
      hreadouts, hcoherence, ?_⟩
  exact
    primitiveCoherenceEvent_action_eq_hbar
      hcoherence event.2
MODEL s19_nonlinearGauss_tangentGreen_consumer_compiles · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean
s19_nonlinearGauss_tangentGreen_consumer_compiles · IndisputableMonolith/Foundation/PairKernelPhysicalValuationInitialityS19Consumer.lean:124 
/-- Exact-J nonlinear Gauss law, tangent Hessian, and Green response still
compile independently of the physical valuation normalization. -/
def s19_nonlinearGauss_tangentGreen_consumer_compiles :=
  PairKernelExactJNonlinearGaussS13Consumer.canonicalExactJTangentConsumer_exists

What this page does not claim

The S19 consumer does not prove that the physical valuation is unique; it proves the opposite. The module does not derive the operational duration semantic from first principles; it provides a model witness. The compilation of the Gauss law consumer does not depend on a specific physical valuation normalization.

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/PairKernelPhysicalValuationInitialityS19Consumer.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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