Foundation Pair Kernel Physical Source Law

Source scale candidates

In Recognition Science, a ledger (a discrete record of recognition events) needs a scale for the magnitude of physical sources. The pair kernel (the core interaction structure between two entities) requires this scale to be fixed, not chosen freely. The module PairKernelPhysicalSourceLaw examines two mechanistically distinct candidate laws for this absolute source scale, and proves formal properties about each.

Candidate A (one-act dual lock). This candidate identifies the dual-identification scale with the cost unit of a physical one-act recognition instrument. The one-act curvature lock forces that unit to 1, rejecting scale 2 by continuum curvature rather than by applying a defect to a free coordinate. The module proves that any one-act physical dual event has scale 1, and therefore rejects scale 2. It also proves this candidate is not implied by the named premises alone.

Candidate B (native action-quantum reciprocal). This candidate sets the elementary recognition source magnitude equal to the reciprocal of the pi-free native action quantum φ⁻⁵, so S = φ⁵ as a ledger expression. The module proves this candidate satisfies the native quantum source law, is pi-free, and uniquely determines the source magnitude. It also proves this candidate rejects both scale 1 and scale 2, since φ⁵ is greater than both.

The two candidates select distinct magnitudes: Candidate A gives source magnitude 1, Candidate B gives φ⁵. Each defeats the scale-one / scale-two decoy for an independent reason. Neither candidate is claimed as derived from the current action+posting+stationarity package alone. The remaining residual is the attachment of the candidate predicate to the physical pair-kernel source object. Reachability of a unique banked S is tagged HYPOTHESIS for the judge, with no sorry and no new axiom.

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

/-- Named premises do not construct a one-act dual lock: coupling `2`
satisfies every named premise while Candidate A rejects scale `2`. -/
theorem namedPremises_do_not_force_oneActPhysicalDual :
    ¬ (∀ coupling : ℝ,
      NamedPremises coupling →
        ∃ E : OneActDualEvent,
          E.scale = coupling ∧ IsOneActPhysicalDual E) := by
  intro hforce
  obtain ⟨E, hscale, hE⟩ := hforce 2 (namedPremises_all_couplings 2)
  exact oneActPhysicalDual_rejects_two E hE hscale

theorem nativeQuantumSourceLaw_unique
    (S₁ S₂ : ℝ)
    (h₁ : NativeQuantumSourceLaw S₁)
    (h₂ : NativeQuantumSourceLaw S₂) :
    S₁ = S₂ := by
  rw [(nativeQuantumSourceLaw_iff_inv S₁).mp h₁,
      (nativeQuantumSourceLaw_iff_inv S₂).mp h₂]

/-- Scale-one decoy fails Candidate B for an independent reason. -/
theorem nativeQuantumSourceLaw_rejects_one :
    ¬ NativeQuantumSourceLaw 1 := by
  intro h
  have hS := (nativeQuantumSourceLaw_iff_inv 1).mp h
  rw [nativeActionQuantumInv_eq_phi_pow_five] at hS
  exact (ne_of_gt goldenRatio_pow_five_gt_one) hS.symm

/-- Scale-two decoy fails Candidate B. -/
theorem nativeQuantumSourceLaw_rejects_two :
    ¬ NativeQuantumSourceLaw 2 := by
  intro h
  have hS := (nativeQuantumSourceLaw_iff_inv 2).mp h
  rw [nativeActionQuantumInv_eq_phi_pow_five] at hS
  exact (ne_of_gt goldenRatio_pow_five_gt_two) hS.symm

/-- The two candidates select different magnitudes. -/
theorem candidates_select_distinct_magnitudes :
    candidateA_sourceMagnitudeExpr.eval ≠
      candidateB_sourceMagnitudeExpr.eval := by
  rw [candidateA_sourceMagnitude_eq_one]
  change (1 : ℝ) ≠ nativeActionQuantumInv
  rw [nativeActionQuantumInv_eq_phi_pow_five]
  exact ne_of_lt goldenRatio_pow_five_gt_one

What this page does not claim

Neither candidate is derived from the action+posting+stationarity package alone. The attachment of either candidate to the physical pair-kernel source object is not yet established. Reachability of a unique banked S is a hypothesis, not a 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/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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