Encyclopedia Foundation Foundation Pair Kernel Physical Valuation Initiality S19 Consumer S19 Nonlinear

ARTICLE 3 claims 2 theorems 1 model

Foundation Pair Kernel Physical Valuation Initiality S19 Consumer S19 Nonlinear

A machine-checked library shows that a nonlinear Gauss law and its Green response compile independently of how physical valuations are normalized.

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The nonlinear consumer

The declaration s19_nonlinearGauss_tangentGreen_consumer_compiles is a definition in the framework's machine-checked library of formal theorems. It names a consumer: a packaged result that recompiles an earlier theorem, canonicalExactJTangentConsumer_exists, without selecting a specific physical unit for the exact-J field. In plain language, the declaration establishes that the nonlinear Gauss law, the tangent Hessian, and the Green response still compile independently of the physical valuation normalization. The library proves that these structures do not depend on which unit one chooses for the field; they hold regardless.

The declaration is a MODEL witness, not a new theorem. The canonical physical readout remains a MODEL witness. The library also proves a non-uniqueness result: there exist two physical valuations with the same recognition data but different span measures and energy scales, including an action-preserving inverse-scale pair. This means the weak physical valuation is not unique. The declaration does not claim that the nonlinear Gauss law is a theorem in the sense of being derived from the forcing chain; it claims that the consumer compiles, that is, the earlier result remains valid under the current definitions.

In Recognition Science, the framework models physical structure from a ledger of recognition events. The declaration shows that the nonlinear Gauss law and its Green response are stable: they do not depend on the choice of field unit. This is a consistency check, not a derivation of new physics. The library's theorem canonicalEnergy_has_uniformScale_one shows that the canonical energy has uniform price scale 1. The theorem canonicalOperationalReadoutConsumer_exists shows that there exists an event whose posting action equals hbar, the framework's constant. These are separate results; the declaration does not combine them into a new claim.

What the declaration does not claim: it does not claim that the nonlinear Gauss law is derived from the five conditions that force the cost function J. It does not claim that the physical valuation is unique; in fact, the library proves the opposite. It does not claim that the exact-J field unit is selected or determined. The declaration is a recompilation check, a MODEL witness that the nonlinear structures compile independently of normalization.

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
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

What this page does not claim

The nonlinear Gauss law is derived from the five conditions that force J. The physical valuation is unique; the library proves the opposite. The declaration selects or determines the exact-J field unit.

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)

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.

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