Foundation Pair Kernel Source Coupling Elementary Posting Divergence Eq Dipole

The elementary posting

In the Recognition Science framework, a ledger is a discrete record of events. The most basic event is an elementary posting: a single unit of directed flow from one site to another. The declaration elementaryPosting_divergence_eq_dipole proves a simple identity: the total outflow from any site, computed by summing the posting's entries, equals the value of a dipole at that site. A dipole, in this context, is a pair of equal and opposite unit sources. The theorem states that one posting between two sites is indistinguishable, at the level of divergence, from a dipole centered on those sites.

This is not a physical law. It is a bookkeeping fact, a direct consequence of how the ledger and the posting are defined. The proof is immediate from the definitions; it does not rely on any additional assumptions. The identity is used as a premise in a larger, separate investigation into whether a proposed law of work and response can be derived from the framework's core principles. That investigation, documented in the same module, reaches a negative conclusion: the current set of axioms does not force the law. A two-site model satisfies all the stated conditions for any real value of the coupling constant, not just one.

The identity itself, however, is reliable. It holds for any number of sites and for any choice of the two sites involved. It is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified by a computer. This is a necessary foundation, but it is not a sufficient one. The fact that a posting is a dipole does not, by itself, determine the strength of the interaction between sites. That strength, the coupling, remains a free parameter in the current framework.

What this means for the broader theory is a clear boundary. The elementary posting is a well-defined, proven object. It can serve as a building block. But the framework does not yet claim to derive the specific value of the coupling from its first principles. That derivation remains an open target. The identity is a clean, verified step, not the end of the journey.

/-- The elementary integral posting has exactly the unit dipole divergence.
This reproduces the source fact needed here without importing any physical
constant module. -/
theorem elementaryPosting_divergence_eq_dipole {n : ℕ}
    (a b i : Fin n) :
    postingDivergence (elementaryPosting a b) i = dipole a b i := by
  unfold postingDivergence elementaryPosting dipole
  have hforward :
      (∑ j : Fin n, if i = a ∧ j = b then (1 : ℝ) else 0) =
        if i = a then 1 else 0 := by
    by_cases hia : i = a
    · subst i
      simp
    · simp [hia]
  have hbackward :
      (∑ j : Fin n, if i = b ∧ j = a then (1 : ℝ) else 0) =
        if i = b then 1 else 0 := by
    by_cases hib : i = b
    · subst i
      simp
    · simp [hib]
  rw [Finset.sum_sub_distrib, hforward, hbackward]

/-- 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 determine the value of any coupling constant. The theorem does not prove that the One-Ledger work-response law is true. The theorem does not claim that the physical world is a two-site system.

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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