Foundation Pair Kernel Physical Source Law Candidate B Not Four Mul Phi Pow Five

The source magnitude

In the Recognition Science framework, physical structure is built from a ledger, a discrete record of recognition events. A central question is the absolute scale of an elementary source: what number fixes the strength of a single source event. The framework's machine-checked library of formal theorems contains a candidate answer, expressed as a recognition source magnitude, the number assigned to one elementary source event in ledger units.

One candidate, called Candidate B, sets this source magnitude to the reciprocal of a native action quantum, which works out to the fifth power of the golden ratio, written as φ⁵. The theorem candidateB_not_four_mul_phi_pow_five proves a negative fact about this candidate: its value is not equal to four times φ⁵. The proof is short and relies on the definition of the candidate and the positivity of φ⁵; it does not depend on any additional physical assumptions.

The theorem also establishes that Candidate B is distinct from another candidate, Candidate A, which sets the source magnitude to 1. The two candidates therefore select different magnitudes, and the framework does not conflate them. Each candidate defeats a decoy scale of 1 or 2 for an independent reason, and neither is implied by the framework's named premises alone.

What the theorem does not claim is that Candidate B is the correct physical source law. The framework explicitly tags the reachability of a unique banked source magnitude as a HYPOTHESIS. The remaining work is attaching the candidate predicate to the physical pair-kernel source object, a step that is not yet derived. The theorem is a precise statement about a candidate's value, not a proof that the candidate is realized in nature.

/-- Endpoint non-fit: Candidate B banks `φ⁵`, not the integration target
`4 · φ⁵`. The factor-four comparison is Lane C work and is deliberately
absent as a fitting step. -/
theorem candidateB_not_four_mul_phi_pow_five :
    candidateB_sourceMagnitudeExpr.eval ≠
      (4 : ℝ) * goldenRatio ^ (5 : ℕ) := by
  change nativeActionQuantumInv ≠ 4 * goldenRatio ^ (5 : ℕ)
  rw [nativeActionQuantumInv_eq_phi_pow_five]
  have hphi : (0 : ℝ) < goldenRatio ^ (5 : ℕ) := pow_pos goldenRatio_pos 5
  intro h
  have : (1 : ℝ) = 4 := by
    have h' := congrArg (fun x : ℝ => x / goldenRatio ^ (5 : ℕ)) h
    field_simp [ne_of_gt hphi] at h'
    exact h'
  norm_num at this

/-- 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

/-- 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))

What this page does not claim

The theorem does not prove that Candidate B is the correct physical source law. The theorem does not derive the value of the source magnitude from the framework's named premises alone. The theorem does not claim that the golden ratio itself is derived from the framework.

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)

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Derived articles

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