Foundation Pair Kernel Signed Posting Transport S11 Signed Graph Support Eq Reco

The collision repair

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition event is one account acknowledging another. The framework's library, a machine-checked collection of formal results, studies how such ledgers behave when the events repeat periodically on a three-dimensional torus, a space that wraps around on itself like a video game screen. At period two, a subtle problem arises: two events that happen in opposite directions on the same axis can land on the same ordered pair, so a simple record that stores only which pairs are connected forgets that two separate occurrences took place.

The declaration signedGraph_support_eq_recognitionGraph_support establishes that this collision does not change the underlying pattern of connections. It states that for every period and every relabeling of the three axes, a weighted graph that counts every forward and backward generator event has a nonzero connection between two sites exactly when the original, unrefined relation-set graph also has a nonzero connection there. In plain terms: the refined record and the simple record agree on which accounts are linked, even though they disagree on how many times the link was used. This result is verified in the machine-checked library with no unproved assumptions and no new axioms.

The declaration does not claim that the two graphs are identical. It says nothing about the multiplicity, the number of times a connection is recorded, which differs when collisions occur. The refined graph counts each occurrence, so a colliding pair has weight two, while the simple graph has weight one. The declaration also does not claim that the refined graph's Laplacian, an operator measuring how a field spreads across the graph, equals the framed six-neighbor operator for the simple graph; that stronger equality is a separate result in the same module, established for the refined graph alone. The support equality is exactly and only about which connections exist.

What this fact changes is the status of the framework's earlier countermodel. Before this declaration, the unrefined graph at period two had a known defect: its relation set forgot one occurrence. The support result shows that this defect is confined to multiplicity, not to the pattern of connections. The collision repair is therefore complete for the purpose of establishing which sites are linked, and the framework can move on to questions about the strength of those links without rechecking the connection pattern at every period.

/-- 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

/-- The signed multiset graph preserves exactly the S8 minimum-J support. -/
theorem signedRecognitionProductionGraph3_support_iff
    {N : ℕ} [NeZero N]
    (σ : Equiv.Perm (Fin 3))
    (i j : Fin (TorusCard3 N)) :
    (signedRecognitionProductionGraph3 N σ).weight i j ≠ 0 ↔
      (@encodedMinimumJRelation3 N _) i j := by
  change
    signedPostingWeight3 σ
        ((torusSiteEquivFin N).symm i)
        ((torusSiteEquivFin N).symm j) ≠ 0 ↔
      JMinimalGeneratedStep (Equiv.refl (Fin 3))
        ((torusSiteEquivFin N).symm i)
        ((torusSiteEquivFin N).symm j)
  rw [signedPostingWeight3_ne_zero_iff,
    jMinimalGeneratedStep_iff_framedAxisStep]
  exact
    framedAxisStep_frame_independent
      σ (Equiv.refl (Fin 3))
      ((torusSiteEquivFin N).symm i)
      ((torusSiteEquivFin N).symm j)

/-- 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 two graphs are identical in every respect. The refined graph's Laplacian equals the framed operator for the unrefined graph. The support equality holds for the unrefined S8 graph, which remains a countermodel for multiplicity.

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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