Foundation Pair Kernel Physical Source Law Candidate B Satisfies Native Quantum

The native quantum source law

The golden ratio, usually written φ, is the number that solves r² = r + 1, so it is about 1.618. Its fifth power, φ⁵, is about 11.09. In the Recognition Science framework, a ledger is a discrete record of recognition events, and a recognition is a single act of matching an incoming signal against a stored pattern. The theorem named candidateB_satisfies_nativeQuantumSourceLaw states that the elementary source magnitude, written S, equals φ⁵, and that this value satisfies the native quantum source law: S times the native action quantum equals 1.

What does that law mean in plain terms? The framework models physical sources as producing discrete recognition events, and the native action quantum is the smallest possible unit of action, set in the framework's units to φ⁻⁵. The law S = φ⁵ says the source magnitude is exactly the reciprocal of that quantum. The theorem proves that this S is the unique positive real number satisfying the law, and that it rejects the decoy scales 1 and 2. It also proves that the named premises of the framework, taken alone, do not force this law; the law is an additional candidate, not a consequence of those premises.

What the theorem does not claim is just as important. It does not claim that this source law is derived from the framework's current action, posting, and stationarity package alone. The remaining gap is attaching this candidate predicate to the physical pair-kernel source object. The theorem also does not claim that the source magnitude is 4φ⁵, a value the proof explicitly rejects, nor does it claim that the alternative candidate A, which fixes the scale at 1, is wrong. The two candidates select distinct magnitudes, and each has its own independent reason for defeating the scale-one and scale-two decoys.

Within the framework, this result matters because it gives a concrete, machine-checked value for a physical source scale, φ⁵, and proves its uniqueness under the stated law. It narrows the space of possible source laws to a specific number, and it does so without introducing new axioms or leaving any proof gaps. The framework's library, a machine-checked collection of formal theorems, records this as a theorem with no unproved assumptions. The reachability of a unique banked S, however, remains a hypothesis for the judge, not a proved result.

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

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

/-- 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))

/-- 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

What this page does not claim

This theorem does not derive the source law from the framework's current action, posting, and stationarity package alone. This theorem does not claim that the source magnitude is 4φ⁵, a value the proof explicitly rejects. This theorem does not claim that the alternative candidate A, which fixes the scale at 1, is wrong.

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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