Foundation Pair Kernel Physical Source Law Named Premises Do Not Force Native Qu

What the premises do not force

In the Recognition Science framework, the question of what sets the absolute scale of a physical source is open. The framework's ledger (a discrete record of events) and its recognition (the forced cost of registering an event) give structure, but they do not by themselves pick a number. The declaration namedPremises_do_not_force_nativeQuantumSourceLaw is a formal theorem, checked in the framework's machine-checked library, that makes this boundary exact.

The theorem states that the named premises do not imply the NativeQuantumSourceLaw, the proposition that a source magnitude S is positive and satisfies S * nativeActionQuantum = 1. In plain terms, the premises alone do not force the source magnitude to be the reciprocal of the native action quantum. The proof is a counterexample: the premises hold for a coupling of 1, but the native quantum source law rejects 1, because the native action quantum is the golden ratio to the fifth power, which is greater than 1. So the premises are consistent with a coupling that the law rejects.

This is a negative result with a positive use. It separates what is derived from what is chosen. Two candidate laws for the source scale exist in the framework. Candidate A sets the source magnitude to 1, using a one-act dual lock that forces the instrument unit to 1 by curvature. Candidate B sets the source magnitude to the reciprocal of the pi-free native action quantum, so S = φ⁵ as a ledger expression. Each candidate defeats the scale-one and scale-two decoys for an independent reason, and neither is implied by the named premises. The theorem proves the premises do not force Candidate B; a companion theorem proves they do not force Candidate A either.

What remains is the attachment of a candidate predicate to the physical pair-kernel source object. That step is tagged HYPOTHESIS, not theorem. The uniqueness of a banked S is a prediction with a named falsifier: a coupling that satisfies the premises but violates the candidate law. The dependency closure stays on the source-side grammar, coupling, and one-act calibration surfaces, with no new axiom and no sorry. The theorem's value is that it tells a reader exactly where the framework stops deriving and starts choosing.

/-- 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))

private theorem goldenRatio_pow_five_gt_one :
    (1 : ℝ) < goldenRatio ^ (5 : ℕ) := by
  have hphi : (1 : ℝ) < goldenRatio := one_lt_goldenRatio
  have hone : (1 : ℝ) = (1 : ℝ) ^ (5 : ℕ) := by norm_num
  rw [hone]
  exact pow_lt_pow_left₀ hphi (by norm_num) (by decide)

/-- 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 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

/-- 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

The theorem does not prove that the named premises are consistent with any particular source law, only that they do not force one. The theorem does not establish that Candidate B is false; it only shows the premises do not imply it. The theorem does not claim that the source scale is unconstrained, only that the named premises alone are insufficient.

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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