Foundation Pair Kernel Source Coupling Identify Dual Eq Self Of Zero Defect

The fixed scale

In the Recognition Science framework, a ledger (a discrete record of events) assigns real numbers to pairings between sources and postings. A dual is a real coordinate paired with a posting; rescaling it by any positive factor leaves the pairing unchanged. The declaration identifyDual_eq_self_of_zeroDefect proves that if a scale has zero defect, then identifying the dual with that scale is the identity: identifyDual scale dual = dual. In plain terms, the only scale that makes the defect vanish is 1.

The proof is short. A zero-defect scale is defined as a positive scale for which the defect function is zero. A separate theorem, zeroDefectDualIdentification_iff_one, shows this condition is equivalent to the scale equaling 1. Substituting that into the definition of identifyDual, which multiplies the dual by the scale, gives the result. The theorem is machine-checked in the framework's library of formal theorems, with no unproved assumptions.

The result is a boundary on freedom, not a derivation of the scale itself. The framework's named premises, which include symmetry and stationarity conditions, do not force the defect to vanish for all couplings. A countermodel shows a coupling of 2 satisfies all those premises yet has nonzero defect. The theorem only says: if the defect is zero, then the scale is 1.

This matters because it isolates where the scale 1 comes from. It is not a consequence of the action, the posting, or the symmetry conditions alone. It is a separate condition, a zero-defect requirement, that pins the scale. The framework does not claim to derive that requirement from more basic principles; it is an input, not an output.

/-- Under the zero-defect identification law, the arbitrary real dual
identification collapses to the unscaled coordinate. -/
theorem identifyDual_eq_self_of_zeroDefect
    (scale : ℝ) (dual : PostingDual)
    (hscale : ZeroDefectDualIdentification scale) :
    identifyDual scale dual = dual := by
  rw [(zeroDefectDualIdentification_iff_one scale).mp hscale]
  simp [identifyDual]

/-- The canonical zero-defect law selects scale one exactly. -/
theorem zeroDefectDualIdentification_iff_one
    (scale : ℝ) :
    ZeroDefectDualIdentification scale ↔ scale = 1 := by
  constructor
  · rintro ⟨hscale, hdefect⟩
    exact (LawOfExistence.defect_zero_iff_one hscale).mp hdefect
  · rintro rfl
    exact ⟨one_pos, LawOfExistence.defect_at_one⟩

/-- The current named pair-kernel premises cannot imply the zero-defect
scale-selection predicate: coupling two satisfies all of them while the
zero-defect predicate rejects it. Any positive bridge must therefore add a
genuinely scale-breaking recognition premise. -/
theorem namedPremises_do_not_force_zeroDefect :
    ¬ (∀ coupling : ℝ,
      NamedPremises coupling →
        ZeroDefectDualIdentification coupling) := by
  intro hforce
  exact zeroDefectDualIdentification_excludes_two
    (hforce 2 (namedPremises_all_couplings 2))

/-- The live normalization decoy at scale two is rejected if, and only if,
the zero-defect identification law is supplied. -/
theorem zeroDefectDualIdentification_excludes_two :
    ¬ ZeroDefectDualIdentification 2 := by
  rw [zeroDefectDualIdentification_iff_one]
  norm_num

What this page does not claim

The theorem does not derive the zero-defect condition from the named premises. It does not claim that all couplings have zero defect. It does not identify the physical meaning of the defect function.

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/PairKernelSourceCoupling.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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