Foundation Pair Kernel Signed Posting Transport S11 Signed Posting Occurrence Di

The unit dipole source

In the Recognition Science framework, a ledger is a discrete record of events, and a posting event is the basic act of recording one occurrence. The declaration in question proves a precise statement about the divergence of such an event. Divergence here measures the net outflow of a field from a point, and the theorem says that the divergence of a single posting event equals a unit dipole, a pair of equal and opposite unit sources separated by one step. The proof is a direct invocation of an earlier result, elementaryPosting_divergence_eq_dipole, so the claim is not new ground but a restatement in the signed posting context.

The setting matters. The framework stores realized ordered pairs as a relation set, but at period two, forward and backward events on one axis land on the same ordered pair, so the set forgets one occurrence. The signed posting transport keeps both occurrences through the graph fold, producing a weighted graph that counts every forward and backward generator event. The theorem says that this refinement, which changes multiplicities, does not change the fundamental source character of a posting: each oriented posting occurrence still has unit dipole source. Collisions change multiplicity, not support, and the divergence identity holds for every posting event and every site on the torus.

What the declaration does not claim is broader. It does not assert that the signed graph equals the original relation-set graph; the two have identical support but differ in collision multiplicity. It does not claim that the divergence identity itself is new, since it inherits from an earlier theorem. It does not claim anything about the physical interpretation of a dipole in three-dimensional space, nor does it assert that the signed graph is the unique refinement that repairs the collision. The theorem is one link in a chain that shows the signed graph's Laplacian equals the framed six-neighbor operator, but this declaration alone establishes only the source identity.

/-- Refining pair multiplicity does not change double-entry source
divergence: each oriented posting occurrence still has unit dipole source. -/
theorem signedPostingOccurrence_divergence_eq_dipole
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N)
    (i : Fin (TorusCard3 N)) :
    postingDivergence
        (PairKernelSourceCoupling.elementaryPosting event.1 event.2) i =
      dipole event.1 event.2 i :=
  elementaryPosting_divergence_eq_dipole event.1 event.2 i

/-- Signed multiplicity is nonzero exactly on the minimum-J axis-step support.
Collisions change multiplicity, not support. -/
theorem signedPostingWeight3_ne_zero_iff
    {N : ℕ}
    (σ : Equiv.Perm (Fin 3))
    (p q : TorusSite3 N) :
    signedPostingWeight3 σ p q ≠ 0 ↔
      FramedAxisStep σ p q := by
  constructor
  · intro hne
    by_contra hstep
    have hfwd :
        ∀ k : Fin 3, q ≠ framedForward σ p k := by
      intro k h
      exact hstep ⟨k, Or.inl h⟩
    have hbwd :
        ∀ k : Fin 3, q ≠ framedBackward σ p k := by
      intro k h
      exact hstep ⟨k, Or.inr h⟩
    simp [signedPostingWeight3, hfwd, hbwd] at hne
  · rintro ⟨k, hk | hk⟩
    · apply ne_of_gt
      unfold signedPostingWeight3
      refine Finset.sum_pos'
        (fun i _ => by split_ifs <;> norm_num) ?_
      refine ⟨k, Finset.mem_univ k, ?_⟩
      simp [hk]
      split_ifs <;> norm_num
    · apply ne_of_gt
      unfold signedPostingWeight3
      refine Finset.sum_pos'
        (fun i _ => by split_ifs <;> norm_num) ?_
      refine ⟨k, Finset.mem_univ k, ?_⟩
      simp [hk]
      split_ifs <;> norm_num

/-- The refined signed graph and S8 relation-set graph have identical support
at every period. Only collision multiplicity differs. -/
theorem signedGraph_support_eq_recognitionGraph_support
    {N : ℕ} [NeZero N]
    (σ : Equiv.Perm (Fin 3))
    (i j : Fin (TorusCard3 N)) :
    (signedRecognitionProductionGraph3 N σ).weight i j ≠ 0 ↔
      (recognitionProductionGraph3 N).weight i j ≠ 0 := by
  rw [signedRecognitionProductionGraph3_support_iff]
  constructor
  · intro hrel
    have hmem :
        (i, j) ∈ realizedPrimitivePostingPairs3 N := by
      simpa using hrel
    rw [recognitionProductionGraph3_weight_of_mem hmem]
    norm_num
  · intro hweight
    by_contra hrel
    have hnot :
        (i, j) ∉ realizedPrimitivePostingPairs3 N := by
      simpa using hrel
    rw [recognitionProductionGraph3_weight_of_not_mem hnot] at hweight
    exact hweight rfl

What this page does not claim

The signed graph is not the same as the original relation-set graph; they differ in collision multiplicity. The divergence identity is not new; it inherits from an earlier theorem. The theorem does not establish any physical interpretation of a dipole in three-dimensional space.

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/PairKernelSignedPostingTransportS11.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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