Foundation Pair Kernel Source Coupling Primitive Dual Pairing Scale One Ne Two

The scale of a dual coordinate

In the Recognition Science framework, a ledger (a discrete record of events) tracks postings, which are elementary transfers between two sites. Each posting has a dual coordinate, a real number that pairs with it to form an inner product. The declaration primitive_dual_pairing_scale_one_ne_two proves a simple fact: if you take a nonzero dual coordinate and scale it by 1, the resulting pairing with the primitive posting (the posting of size 1) differs from the pairing you get when you scale the same coordinate by 2. In plain terms, the theorem states that rescaling a dual coordinate by a factor of two changes the value of the pairing; the two scales are not interchangeable.

This result is a theorem in the framework's machine-checked library of formal theorems. It is proved from the definitions of the pairing, the primitive posting, and the scaling operation, with no additional axioms. The theorem itself is narrow: it establishes an inequality between two real numbers. It does not, by itself, say anything about which scale is physically correct. The broader context in the same module shows why that matters. A separate theorem, namedPremises_do_not_force_zeroDefect, proves that the framework's named premises, the conditions it currently takes as given, do not force the scale to be 1. A two-site model satisfies all those premises at every real coupling value, including 2, while still preserving the elementary posting fixed.

The upshot is a precise statement about the current state of the framework. The primitive posting fixes the integral generator of the source, but the real coordinate paired with that generator still carries an arbitrary identification scale. The theorem primitive_dual_pairing_scale_one_ne_two shows that this freedom is real: different scales give different pairings. It does not resolve which scale the framework will ultimately select. That selection, if it comes, will require additional structure beyond the premises currently named in the module.

/-- Concrete discriminator: the primitive posting remains fixed while scale
one and scale two give distinct work values for every nonzero dual coordinate.
-/
theorem primitive_dual_pairing_scale_one_ne_two
    (dual : PostingDual) (hdual : dual ≠ 0) :
    dualPairing (identifyDual 1 dual) primitivePosting ≠
      dualPairing (identifyDual 2 dual) primitivePosting := by
  simp only [dualPairing_identifyDual]
  norm_num [dualPairing, primitivePosting, identifyDual]
  intro h
  apply hdual
  linarith

/-- 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

What this page does not claim

The theorem does not claim that scale 1 is the physically correct scale. The theorem does not claim that the framework's named premises are complete. The theorem does not claim that the two-site model is physically realistic.

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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