Foundation Pair Kernel Physical Readout Selection S17

The readout interfaces

In Recognition Science, a recognition event, a discrete record of a comparison, must connect to physical measurements like duration, energy, and dimension. The module called foundation pair kernel physical readout selection s17 formalizes the smallest set of interfaces for making that connection. It defines four separate readouts: 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 module proves several consequences that hold independently of any particular event action. Octave-time additivity means durations combine by doubling. Complete-carrier uniqueness up to relabeling means any two exhaustive carriers are equivalent. Exact exponent five means the complete carrier has dimension five. Channel-price invariance and finite-batch energy quantization round out the list. When all four interfaces are supplied, they imply the metric and pricing package from module s16, and hence the event action and batch-action theorems from s14.

The module also proves what is not a valid readout. A stretched duration is not a physical readout, a doubled energy is not physical event energy, and a misclassified five-channel carrier is not complete. A half-composing degree channel fails self-similarity, and a doubled channel price does not apply. These counterexamples sharpen the boundaries of the interfaces.

In Recognition Science, the canonical instances are MODEL witnesses. Current recognition dynamics do not select them. The module contains no `sorry` and no new `axiom`, meaning every proof is complete within the framework's machine-checked library of formal theorems.

/-- The physical carrier is coherence-complete exactly when its classifier is
bijective for every event. Surjectivity excludes missing parent channels;
injectivity excludes extra physical channels hidden behind one parent tag. -/
def PostingCarrierCoherenceComplete3
    {N : ℕ} [NeZero N]
    (physical : PostingPhysicalChannelCarrier3 N) : Prop :=
  ∀ event, Function.Bijective (physical.classify event)

/-- Independent carrier consequence: any two complete physical carriers are
equivalent for each event, so completeness is invariant under channel
relabeling. -/
theorem completeCarriers_equivalent
    {N : ℕ} [NeZero N]
    {left right : PostingPhysicalChannelCarrier3 N}
    (hleft : PostingCarrierCoherenceComplete3 left)
    (hright : PostingCarrierCoherenceComplete3 right)
    (event : RealizedPostingEvent3 N) :
    Nonempty (left.Carrier event ≃ right.Carrier event) := by
  let leftEquiv :
      left.Carrier event ≃
        PostingConfigurationDegreeCarrier3
          (realizedPostingEventConfiguration3 event) :=
    Equiv.ofBijective (left.classify event) (hleft event)
  let rightEquiv :
      right.Carrier event ≃
        PostingConfigurationDegreeCarrier3
          (realizedPostingEventConfiguration3 event) :=
    Equiv.ofBijective (right.classify event) (hright event)
  exact ⟨leftEquiv.trans rightEquiv.symm⟩

/-- At committed spatial dimension three, every complete physical event carrier
has exactly five channels. -/
theorem completeCarrier_dimension_eq_five
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    physicalPostingCarrierDimension3 physical event = 5 := by
  rw [completeCarrier_dimension_eq_configDim hcomplete event,
    GapDerivation.configDim_at_D3]

/-- 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

What this page does not claim

The canonical readout instances are not selected by current recognition dynamics. The module does not mention exact-J field units. The physical recognition-to-linking bridge is not established by this module.

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)

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