Foundation Pair Kernel Physical Readout Selection S17 Physical Readouts Imply Pr

What physical readouts force

The declaration physicalReadouts_imply_primitiveCoherenceEvent is a theorem in the framework's machine-checked library of formal theorems. It states that if a physical system supplies four readout interfaces, then the system necessarily inherits the framework's metric and pricing package, and with it the event action and batch-action theorems. The four interfaces are: physical duration reads the normalized recognition interval, a classified physical channel carrier is exhaustive exactly when its classifier is bijective onto the parent carrier, an event-indexed recognition channel prices the complete physical carrier, and physical event energy reads that event-channel price.

The theorem's force is conditional. It does not say that any physical system must supply these interfaces. The framework's own dynamics do not select them. The canonical instances in the library are model witnesses, not forced outcomes. What the theorem proves is the consequence: once those four readouts are in place, the rest follows. This is a derivation within the framework, not a claim about which physical systems actually realize the interfaces.

The theorem also carries negative results. A stretched duration readout is not a physical readout. A doubled energy is not physical event energy. A channel with an extra carrier dimension is not complete. These counterexamples sharpen the boundary of what the theorem covers. The theorem is about the consequences of the interfaces, not about their necessity.

In plain terms, the declaration establishes a one-way implication. If the physical world reads the framework's internal structure in the specified way, then the framework's metric and action structure are not optional extras; they are forced. The converse is not claimed. The framework does not assert that the readouts must exist, only that their existence has consequences.

theorem canonicalPhysicalPostingReadouts_model :
    PhysicalPostingReadoutSemantics3
      (parentPhysicalChannelCarrier3 3)
      (@canonicalPostingRecognitionChannels3 3 _)
      (@canonicalPostingEventChannelPrice3 3 _)
      (canonicalCoherenceKinematics3 3) := by
  refine
    ⟨?_,
      parentPhysicalChannelCarrier_is_complete,
      canonicalPostingRecognitionChannel_applies,
      ?_⟩
  · exact
      (canonicalMetricPricingSemantics_model
        (N := 3)).2.1
  · intro event
    change
      Constants.E_coh =
        recognitionConfigurationPrice canonicalChannel
          (physicalPostingCarrierDimension3
            (parentPhysicalChannelCarrier3 3) event)
    rw [recognitionConfigurationPrice_eq_phi_zpow_neg]
    rw [completeCarrier_dimension_eq_configDim
      parentPhysicalChannelCarrier_is_complete event]
    exact GapDerivation.Constants_E_coh_eq_configDim

theorem canonicalPhysicalPostingReadouts_model :
    PhysicalPostingReadoutSemantics3
      (parentPhysicalChannelCarrier3 3)
      (@canonicalPostingRecognitionChannels3 3 _)
      (@canonicalPostingEventChannelPrice3 3 _)
      (canonicalCoherenceKinematics3 3) := by
  refine
    ⟨?_,
      parentPhysicalChannelCarrier_is_complete,
      canonicalPostingRecognitionChannel_applies,
      ?_⟩
  · exact
      (canonicalMetricPricingSemantics_model
        (N := 3)).2.1
  · intro event
    change
      Constants.E_coh =
        recognitionConfigurationPrice canonicalChannel
          (physicalPostingCarrierDimension3
            (parentPhysicalChannelCarrier3 3) event)
    rw [recognitionConfigurationPrice_eq_phi_zpow_neg]
    rw [completeCarrier_dimension_eq_configDim
      parentPhysicalChannelCarrier_is_complete event]
    exact GapDerivation.Constants_E_coh_eq_configDim

/-- The doubled-duration kinematics cannot read the normalized Recognition
interval. This is the duration-square discriminator. -/
theorem stretchedDuration_not_physicalReadout :
    ¬ PhysicalPostingDurationReadout3
      (stretchedDurationKinematics3 3) := by
  intro hread
  exact stretchedDuration_countermodel.2.1
    (physicalDurationReadout_implies_fundamentalTick hread)

theorem doubledEnergy_not_physicalEventEnergy :
    ¬ RealizedPostingCarriesCoherenceEnergy3
      (doubledEnergyKinematics3 3) :=
  doubledEnergy_countermodel.2.1

theorem extraCarrier_not_complete :
    ¬ PostingCarrierCoherenceComplete3
      (extraPhysicalChannelCarrier3 3) := by
  intro hcomplete
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 := ⟨pair, hpair⟩
  have hfive :=
    completeCarrier_dimension_eq_five hcomplete event
  rw [extraCarrier_dimension_eq_six event] at hfive
  norm_num at hfive

What this page does not claim

The theorem does not claim that any physical system must supply the readout interfaces. The theorem does not claim that the framework's dynamics select the canonical instances. The theorem does not claim that physical readouts are the only path to the metric and pricing package.

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/PairKernelPhysicalReadoutSelectionS17.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

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