Foundation Pair Kernel Gap2a Production Side Invariant Residual Doubled Coherent

The doubled action theorem

A coherence event is a discrete record of one primitive posting carrying one unit of coherence energy for one fundamental Tick. The declaration doubledCoherentAction_phase_action_eq_two_mul proves, for every one of the eight phases, that the action of a posting whose energy is doubled is exactly twice the action of the original posting. This is a linear scaling fact: action scales with energy, phase by phase, with no exception among the eight.

The theorem is one link in a larger chain. The framework's machine-checked library of formal theorems also proves that the doubled-energy posting shares the same recognition-side carrier, survival, and price data as the original, and therefore falls into the same observable class. Yet the doubled posting does not satisfy the coherence-event invariant: it is not one primitive posting carrying one unit of coherence energy for one Tick. The invariant rejects it even though all observable data match. This is the sharp content of the declaration: it pins down what doubling does, and what it does not do.

In Recognition Science, the result establishes that the coherence-event invariant forces the native primitive action assignment. The invariant is a production-side predicate, meaning it lives on the posting kinematics rather than on any quotient observable. The theorem productionSideCoherenceInvariant_not_S20_descended shows the invariant cannot descend through the S20 observable quotient: two valuations with identical observable class can differ on the invariant. This is exactly why the doubled posting, despite sharing all recognition-side data, fails the invariant.

What the declaration does not claim is equally precise. It does not identify the pair-kernel source coordinate as the positive action-dual source. The invariant has no source-coordinate field, so two distinct positive source maps can coexist with the same invariant valuation. It also does not identify a Lane-C-style continuum source with the complete eight-phase transaction aggregate. Both joins remain open until further source laws are added. The theorem proves a scaling fact and a rejection, not a full source identification.

theorem doubledCoherentAction_phase_action_eq_two_mul
    (phase : Fin 8) :
    postingEventAction3 doubledCoherentActionValuation3.kinematics
        (phaseBearingPostingEvent3 phase).1 =
      2 *
        postingEventAction3 coherentEventValuation3.kinematics
          (phaseBearingPostingEvent3 phase).1 := by
  unfold doubledCoherentActionValuation3
  simpa using
    (scalePhysicalValuation_eventAction
      2 1 (by norm_num) (by norm_num)
      coherentEventValuation3
      (phaseBearingPostingEvent3 phase).1)

theorem coherent_doubled_sameObservableClass :
    physicalObservableClass3 coherentEventValuation3 =
      physicalObservableClass3 doubledCoherentActionValuation3 :=
  sameRecognitionData_sameObservableClass
    coherent_doubled_sameRecognitionData

theorem doubledCoherentAction_not_invariant :
    ¬ ProductionSideCoherenceEventInvariant3
      doubledCoherentActionValuation3 := by
  intro hinvariant
  obtain ⟨event, hevent⟩ :=
    realizedPrimitivePostingPair3_exists
  have hduration := hinvariant.2 event hevent
  change
    2 * coherentEventValuation3.kinematics.duration event =
      Constants.tau0 at hduration
  rw [(coherentEventValuation_satisfies_invariant.2
    event hevent)] at hduration
  linarith [Constants.tau0_pos]

/-- The coherence-event invariant is exactly the kind of production-side
premise the previous residual proved was necessary: it cannot descend through
the S20 observable quotient. -/
theorem productionSideCoherenceInvariant_not_S20_descended :
    ¬ PhaseProductionPremiseDescendsThroughS20
      ProductionSideCoherenceEventInvariant3 := by
  intro hdescends
  exact doubledCoherentAction_not_invariant
    (hdescends coherentEventValuation3
      doubledCoherentActionValuation3
      coherent_doubled_sameRecognitionData
      coherentEventValuation_satisfies_invariant)

What this page does not claim

The theorem does not identify the pair-kernel source coordinate as the positive action-dual source. The theorem does not identify a Lane-C-style continuum source with the complete eight-phase transaction aggregate. The theorem does not prove that all action scales linearly, only for the doubled coherent posting.

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/PairKernelGap2aProductionSideInvariantResidual.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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