Foundation Pair Kernel Physical Source Law Named Premises Do Not Force One Act P

The named premises and the one-act dual

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of registering one such event. The framework's foundational axioms, called the named premises, are meant to be the minimal assumptions about how recognition works. The theorem namedPremises_do_not_force_oneActPhysicalDual states that these named premises alone do not force the existence of a certain kind of physical source law, called a one-act physical dual. In plain terms: the basic rules of the framework are not enough, by themselves, to single out this particular physical law.

To see what this means, consider two candidate laws for the absolute scale of a physical source. Candidate A, the one-act dual lock, says that the scale is fixed to the value 1 by the curvature of a one-act recognition instrument. Candidate B, the native action-quantum reciprocal, says the scale equals the reciprocal of the pi-free native action quantum, which is the golden ratio raised to the fifth power, about 11.09. The theorem shows that the named premises do not force either candidate. It proves this by showing that a coupling value of 2, which the named premises allow, cannot be the scale of a one-act physical dual, because the one-act curvature lock forces the scale to be 1, not 2.

The theorem is not a claim that either candidate is true. It is a negative result: the named premises are too weak to decide between the two candidates. The framework's library, a machine-checked collection of formal theorems, proves this non-implication. The theorem does not say that no physical source law exists; it says that the named premises alone do not determine which one. The remaining work, attaching a candidate predicate to the physical pair-kernel source object, is left open. The reachability of a unique banked source magnitude is tagged as a hypothesis, not a proved theorem.

This result clarifies the structure of the framework. It shows that the named premises are not a complete physical theory by themselves. They are a starting point, and additional structure is needed to select a specific source law. The theorem also highlights the difference between the two candidates: they select distinct magnitudes, 1 and about 11.09, and the framework does not yet have a principle to choose between them. This is an honest statement of what the framework has and has not established.

/-- 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 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

/-- 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 claim that either candidate law is true or false. The theorem does not claim that no physical source law exists. The theorem does not claim that the named premises are inconsistent.

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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