Foundation Pair Kernel Gap2a Production Orbit Residual Production Response Orbit

The aggregate scale

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of registering one event against another. The theorem productionResponseOrbitAggregateGreenSourceScale_eq concerns a closed cycle of such acts: a finite, repeating schedule in which every committed response is covered by a posting event. The declaration states that for any such cycle, the aggregate source scale is simply the cycle's period multiplied by the posting magnitude divided by two. In symbols: if the cycle has period p and the posting magnitude is m, then the aggregate scale equals p × m / 2.

This is a proved theorem in the framework's machine-checked library of formal theorems, not a definitional choice. It follows from the definition of the aggregate scale and the algebraic identity that the real-valued green scale from a posting magnitude is half that magnitude. The theorem holds for every closed production-response orbit, regardless of its period. It is a structural fact about how the framework's source scale accumulates over a complete cycle.

The theorem does not claim that the cycle's period is fixed at five or eight. A separate result shows that a response-simple orbit, one where each phase reads a distinct response, must have period five. Another construction yields a complete period-eight orbit, but that orbit necessarily repeats an observable response. Both orbits exist as model witnesses, and the aggregate scale theorem applies to both. The theorem also does not claim that the aggregate scale distinguishes between the five and eight cycles; a further theorem shows that for a positive posting magnitude, the two canonical orbits do produce different aggregate scales, but that is a separate statement.

What the theorem establishes is a clean computational law: once a closed cycle and a posting magnitude are given, the aggregate source scale is determined. This matters because it shows how the framework's source transport behaves at the level of whole cycles, independent of the cycle's internal phase structure. The scale depends only on the period and the magnitude, not on which responses appear in which order.

theorem productionResponseOrbitAggregateGreenSourceScale_eq
    (orbit : ClosedProductionResponseOrbit3)
    (postingMagnitude : ℝ) :
    productionResponseOrbitAggregateGreenSourceScale
        orbit postingMagnitude =
      (orbit.period : ℝ) * postingMagnitude / 2 := by
  unfold productionResponseOrbitAggregateGreenSourceScale
    realGreenScaleFromPostingMagnitude
  ring

/-- Every response-simple complete production orbit has period exactly five.
This is the intrinsic closure count supplied by S24's response catalog. -/
theorem simpleClosedProductionResponseOrbit_period_eq_five
    (orbit : ClosedProductionResponseOrbit3)
    (hsimple : ProductionResponseOrbitIsSimple3 orbit) :
    orbit.period = 5 := by
  let read :
      Fin orbit.period → RecognitionParentResponse3 :=
    fun phase =>
      productionEventResponseAct_read
        (orbit.phaseAct phase)
  have hcard :
      Fintype.card (Fin orbit.period) =
        Fintype.card RecognitionParentResponse3 :=
    Fintype.card_congr
      (Equiv.ofBijective read
        ⟨hsimple, orbit.responseComplete⟩)
  calc
    orbit.period =
        Fintype.card (Fin orbit.period) := by simp
    _ = Fintype.card RecognitionParentResponse3 := hcard
    _ = 5 := recognitionParentResponse_card_eq_five

theorem canonicalEightProductionResponseOrbit_not_simple :
    ¬ ProductionResponseOrbitIsSimple3
      canonicalEightProductionResponseOrbit3 := by
  intro hsimple
  have hperiod :=
    simpleClosedProductionResponseOrbit_period_eq_five
      canonicalEightProductionResponseOrbit3 hsimple
  norm_num [canonicalEightProductionResponseOrbit3] at hperiod

/-- Actual response-complete closed production orbits admit distinct
five-phase and eight-phase linear source aggregates at every positive posting
magnitude.  Thus response completeness does not select a unique window under
the already-banked linear per-posting Green aggregation law. -/
theorem completeProductionResponseOrbits_admit_distinct_five_and_eight_sources
    (postingMagnitude : ℝ)
    (hposting : 0 < postingMagnitude) :
    canonicalFiveProductionResponseOrbit3.period = 5 ∧
      canonicalEightProductionResponseOrbit3.period = 8 ∧
      productionResponseOrbitAggregateGreenSourceScale
          canonicalFiveProductionResponseOrbit3 postingMagnitude ≠
        productionResponseOrbitAggregateGreenSourceScale
          canonicalEightProductionResponseOrbit3 postingMagnitude := by
  refine ⟨rfl, rfl, ?_⟩
  rw [productionResponseOrbitAggregateGreenSourceScale_eq,
    productionResponseOrbitAggregateGreenSourceScale_eq]
  change
    (5 : ℝ) * postingMagnitude / 2 ≠
      (8 : ℝ) * postingMagnitude / 2
  nlinarith

What this page does not claim

The theorem does not claim that a closed cycle's period is fixed at five or eight. The theorem does not claim that the aggregate scale is the same for all cycles of different periods. The theorem does not claim that the aggregate scale formula applies outside the framework's closed production-response orbit model.

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/PairKernelGap2aProductionOrbitResidual.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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