Foundation Pair Kernel Source Coupling

Source coupling and its scale freedom

In Recognition Science, a ledger is a discrete record of events, and a recognition is a cost assigned to a pair of events. The module PairKernelSourceCoupling studies how the source of a field, the thing that generates it, pairs with the field itself. The central question is whether a proposed law of work and response, called the One-Ledger law, uniquely fixes the relationship between a source and its field.

The module proves that the One-Ledger law selects the doubled Laplacian branch: if a symmetric weight function w and a field e satisfy the law, then for every index i, 2 * laplacian(w, e, i) equals the dipole source at i. This is a theorem, not a guess. The proof is machine-checked in the framework's library of formal theorems, with no gaps and no extra axioms.

But the module also shows a limit. The One-Ledger law does not fix the scale of the source. A primitive posting, the unit of source, fixes the integral generator, but the real coordinate paired with that generator still admits an arbitrary identification scale. The theorem namedPremises_do_not_force_zeroDefect proves that the named premises do not force the scale to be defect-free; a two-site model realizes the law at every real coupling while preserving the elementary posting fixed.

This is the plain-language result: the law of work and response pins down the form of the field equation, the doubled Laplacian, but it leaves the normalization of the source free. The scale freedom is not a failure of the framework; it is a precise statement of what the law does and does not determine. The module isolates the decisive point with a rank-one integral model, showing exactly where the freedom lives.

/-- Conditional branch selection: the One-Ledger law implies the doubled
source equation. This theorem does not derive the One-Ledger law. -/
theorem source_law_selects_doubled_branch {n : ℕ}
    (w : Fin n → Fin n → ℝ) (e : Fin n → ℝ) (a b : Fin n)
    (hsymm : ∀ i j, w i j = w j i)
    (hledger : OneLedgerLaw w e a b) :
    ∀ i, 2 * laplacian w e i = dipole a b i := by
  have hstationary :
      StationaryAtCoefficient w e (dipole a b) 1 := by
    intro v
    rw [one_mul]
    calc
      physSourcePairing w e v =
          ∑ i, v i * postingDivergence (elementaryPosting a b) i :=
        hledger v
      _ = ∑ i, v i * dipole a b i := by
            apply Finset.sum_congr rfl
            intro i _
            rw [elementaryPosting_divergence_eq_dipole]
  simpa using
    (stationaryAtCoefficient_iff w e (dipole a b) 1 hsymm).1
      hstationary

/-- 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))

theorem namedPremises_all_couplings
    (coupling : ℝ) : NamedPremises coupling where
  primitive_generator := primitivePosting_isPrimitive
  action_symmetric := twoSiteWeight_symmetric
  posting_antisymmetric := elementaryPosting_antisymmetric (0 : Fin 2) 1
  posting_is_unit_source := elementaryPosting_divergence_eq_dipole (0 : Fin 2) 1
  work_stationary := twoSite_stationary_at_every_coefficient coupling

theorem primitivePosting_isPrimitive : IsPrimitive primitivePosting := by
  norm_num [IsPrimitive, primitivePosting]

What this page does not claim

The module does not claim that the One-Ledger law is derived from the current action, Gauss, posting-integrality, and primitivity package. The module does not claim that the source scale is fixed to one; it explicitly proves the opposite. The module does not claim that the two-site model is the only model satisfying the named premises.

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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Derived articles

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