Foundation Pair Kernel Physical Source Law Named Premises Do Not Force Candidate

What the named premises do not force

The declaration namedPremises_do_not_force_candidateA_bridge is a formal theorem inside the Recognition Science framework's machine-checked library of formal theorems. It states a negative result: the framework's named premises, the basic assumptions about cost, posting, and stationarity, do not by themselves force the existence of a physical source event that satisfies Candidate A, one of two competing laws for the absolute scale of a physical source. In plain language, the shared starting assumptions leave room for both candidate laws; the choice between them requires an additional, separate identification.

The theorem works by exhibiting a specific counterexample. The named premises hold for every real coupling value, including the value 2. The theorem then uses a separate result, oneActPhysicalDual_rejects_two, which shows that any event satisfying the physical dual condition must have scale 1, never 2. Therefore, if the named premises forced Candidate A for every coupling, they would force it for coupling 2, which is impossible. The contradiction proves the premises do not force Candidate A.

Candidate A itself is a definitional choice, not a derived consequence. It models the absolute source scale as the cost unit of a one-act recognition instrument, and the theorem oneActPhysicalDual_forces_scale_one shows that such an instrument must read scale 1. Candidate B, by contrast, sets the source magnitude to the reciprocal of the pi-free native action quantum, which evaluates to φ⁵, the fifth power of the golden ratio. The two candidates select distinct magnitudes: Candidate A gives 1, Candidate B gives φ⁵, and a theorem records that these are not equal.

What the declaration does not claim is just as important. It does not say Candidate A is false, nor that the named premises are inconsistent. It only says the premises alone are insufficient to single out Candidate A. Reaching a unique banked source scale remains an open target, tagged HYPOTHESIS, pending the attachment of the candidate predicate to the physical pair-kernel source object. The theorem is a precise boundary marker: it tells future work exactly where the current assumptions stop and where additional structure must begin.

theorem namedPremises_do_not_force_candidateA_bridge :
    ¬ (∀ coupling : ℝ,
      NamedPremises coupling →
        ∃ E : OneActDualEvent,
          E.scale = coupling ∧ CandidateA_Bridge E) := by
  intro hforce
  obtain ⟨E, hscale, hbridge⟩ := hforce 2 (namedPremises_all_couplings 2)
  exact oneActPhysicalDual_rejects_two E hbridge.1 hscale

/-- Named premises do not force Candidate B. -/
theorem namedPremises_do_not_force_nativeQuantumSourceLaw :
    ¬ (∀ coupling : ℝ,
      NamedPremises coupling → NativeQuantumSourceLaw coupling) := by
  intro hforce
  exact nativeQuantumSourceLaw_rejects_one
    (hforce 1 (namedPremises_all_couplings 1))

theorem oneActPhysicalDual_forces_scale_one
    (E : OneActDualEvent) (_hE : IsOneActPhysicalDual E) :
    E.scale = 1 := by
  have hunit : E.instrument.unit = 1 :=
    instrument_forces_canonical_unit E.instrument
  exact E.instrument_reads_scale.symm.trans hunit

theorem oneActPhysicalDual_rejects_two
    (E : OneActDualEvent) (hE : IsOneActPhysicalDual E) :
    E.scale ≠ 2 := by
  rw [oneActPhysicalDual_forces_scale_one E hE]
  norm_num

/-- Candidate A selects the unit ledger magnitude for the dual scale. -/
def candidateA_sourceMagnitudeExpr : LedgerExpr := .ofRat 1

theorem candidateB_sourceMagnitude_piFree :
    PiFree candidateB_sourceMagnitudeExpr.eval :=
  nativeActionQuantumInv_piFree

theorem candidateB_satisfies_nativeQuantumSourceLaw :
    NativeQuantumSourceLaw candidateB_sourceMagnitudeExpr.eval :=
  (nativeQuantumSourceLaw_iff_inv _).mpr rfl

What this page does not claim

This answer does not claim that Candidate A is false or that the named premises are inconsistent. This answer does not claim that the framework derives the fine-structure constant or any other coupling constant. This answer does not claim that the choice between Candidate A and Candidate B is settled; it remains open.

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/PairKernelPhysicalSourceLaw.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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