Foundation Pair Kernel Physical Valuation Initiality S19 Consumer Canonical Ener

Uniform price scale

A ledger is a discrete record of events, and in Recognition Science each event carries a price. The declaration canonicalEnergy_has_uniformScale_one is a machine-checked theorem stating that a particular, canonical way of assigning energy to posting events uses a single uniform price scale. In plain terms, it establishes that the framework's standard energy readout does not change its unit of measure from one event to the next; the scale is fixed at one.

The theorem is part of a larger consumer module that compiles two results: the scale-independence of a two-model comparison and a target-blind operational duration semantic. The canonical physical readout remains a MODEL witness, meaning it is a definitional choice rather than a derived necessity. The theorem itself is tagged as a THEOREM in the framework's library, meaning it is proved in the machine-checked collection of formal theorems.

What the declaration does not claim is equally important. The framework's own library proves a non-uniqueness result: there exist two distinct physical valuations that share the same recognition data but differ in their span measure and energy scale. Even stronger, there exists a pair with the same recognition data and the same action on every event, yet different energy scales. This means the uniform scale property holds for the canonical choice, but it does not single out that choice as the only possible one.

In Recognition Science, the theorem-backed result is non-uniqueness of the weak physical valuation, including an action-preserving inverse-scale pair. The canonical energy readout is one consistent choice, not the unique one. The uniform scale theorem establishes internal consistency of that choice, while the non-uniqueness theorem establishes that the framework does not overclaim uniqueness where none exists.

theorem canonicalEnergy_has_uniformScale_one :
    PostingEnergyHasUniformPriceScale3
      1
      (@canonicalPostingEventChannelPrice3 3 _)
      (canonicalCoherenceKinematics3 3) := by
  intro event
  change
    Constants.E_coh =
      1 * canonicalPostingEventChannelPrice3 event
  rw [one_mul,
    completeCarrier_channelPrice_eq_E_coh
      parentPhysicalChannelCarrier_is_complete
      canonicalPostingRecognitionChannel_applies
      event]

/-- 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⟩

What this page does not claim

The uniform scale theorem does not claim that the canonical energy readout is the unique valuation consistent with the recognition data. It does not claim that the price scale being one is a derived necessity rather than a definitional choice. It does not claim that the non-uniqueness result applies to the canonical choice itself, only that other choices exist.

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.

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: