Foundation Pair Kernel Scale Covariant Observables S20 Consumer Action Quotient

The action quotient

In physics, action is a quantity that combines energy and time, and it appears in many fundamental laws. The Recognition Science framework, which models physical structure through a discrete record of recognition events called a ledger, proves a specific result about action. The theorem, named actionQuotient_consumer, establishes two facts. First, there exist two different descriptions of a physical system that contain the same recognition data, yet assign different action values to the same event. Second, for any two such descriptions that share recognition data, the product of their action values for two events is symmetric: swapping the events does not change the product.

The first fact is a statement about scale covariance, the idea that the laws of physics should look the same when you change your units of measurement. The theorem shows that the framework allows two observers, using different unit conventions, to genuinely disagree about the absolute action of a single event. This disagreement is not an error; it is a built-in feature of the model. The second fact is a consistency condition. It says that even though absolute action values differ between observers, the ratio of action values for two events is the same for both. This ratio is a scale-invariant quantity, and it survives the change of units.

The theorem does not claim that absolute action values are physically meaningless. It does not say that all observers must agree on every measurement. It also does not claim that the framework derives the value of the fine-structure constant or any other specific coupling. What it establishes is a precise boundary: dimensionless ratios of action survive a change of units, while equality between an absolute action and a fixed number does not. This distinction matters because it tells physicists which comparisons are reliable and which depend on the choice of units.

/-- The quotient result is intentionally asymmetric: dimensionless action
ratios survive, while equality between a physical event's absolute action and
the dimensionless RS `hbar = phi^-5` numeral does not survive the full
duration-energy unit quotient. The numeral and its D+2 exponent do survive. -/
theorem actionQuotient_consumer :
    (∃ left right : RecognitionPhysicalValuation3.{0} 3,
      SameRecognitionData3 left right ∧
        ∃ event : RealizedPostingEvent3 3,
          postingEventAction3 left.kinematics event.1 ≠
            postingEventAction3 right.kinematics event.1) ∧
      (∀ left right : RecognitionPhysicalValuation3.{0} 3,
        SameRecognitionData3 left right →
          ∀ first second : RealizedPostingEvent3 3,
            postingEventAction3 left.kinematics first.1 *
                postingEventAction3 right.kinematics second.1 =
              postingEventAction3 left.kinematics second.1 *
                postingEventAction3 right.kinematics first.1) :=
  ⟨absolute_eventAction_not_unit_invariant,
    fun _ _ hsame =>
      unitEquivalent_action_ratio_invariant hsame⟩

/-- The quotient result is intentionally asymmetric: dimensionless action
ratios survive, while equality between a physical event's absolute action and
the dimensionless RS `hbar = phi^-5` numeral does not survive the full
duration-energy unit quotient. The numeral and its D+2 exponent do survive. -/
theorem actionQuotient_consumer :
    (∃ left right : RecognitionPhysicalValuation3.{0} 3,
      SameRecognitionData3 left right ∧
        ∃ event : RealizedPostingEvent3 3,
          postingEventAction3 left.kinematics event.1 ≠
            postingEventAction3 right.kinematics event.1) ∧
      (∀ left right : RecognitionPhysicalValuation3.{0} 3,
        SameRecognitionData3 left right →
          ∀ first second : RealizedPostingEvent3 3,
            postingEventAction3 left.kinematics first.1 *
                postingEventAction3 right.kinematics second.1 =
              postingEventAction3 left.kinematics second.1 *
                postingEventAction3 right.kinematics first.1) :=
  ⟨absolute_eventAction_not_unit_invariant,
    fun _ _ hsame =>
      unitEquivalent_action_ratio_invariant hsame⟩

What this page does not claim

Absolute action values are physically meaningless. All observers must agree on every measurement. The framework derives the fine-structure constant from this theorem.

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/PairKernelScaleCovariantObservablesS20Consumer.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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