Foundation Pair Kernel Signed Posting Transport S11 Signed Real Green Field3 Sca

The signed Green field

A recognition event, in this framework, is a discrete record of an interaction between two sites on a finite torus. The framework's ledger, a discrete record of such events, must count each occurrence. A naive graph that stores only which pairs have interacted forgets when a forward and a backward event land on the same pair, a collision that loses one occurrence. The declaration signedRealGreenField3_scaledSourceEquation proves that a refined graph, one that counts every occurrence, supports a real-valued field that exactly solves a scaled source equation.

The field itself, signedRealGreenField3, is the real part of a canonical complex Green response, pulled back to the finite graph. The theorem states that this real field satisfies a ScaledSourceEquation with a source that is a dipole, scaled by q/2, on the signed graph. The proof is machine-checked, with no sorry and no new axiom. The result holds for every nondegenerate period, meaning every torus size N with N at least 2, and for every posting event in the realized primitive set.

This theorem is the collision repair. The earlier S8 graph, which stored only a relation set, had a countermodel at period two where it failed to represent the doubled occurrence. The signed graph fixes that by preserving multiplicity. The theorem shows the real Green field is the exact solution on this repaired graph, closing the loop for the real posting-source attachment. It is a precise, internal consistency result within the framework's construction, not a claim about any physical measurement.

The declaration does not claim that this real field is the only solution, nor that it minimizes any cost. It does not claim the complex Green response itself is real, only that its real part solves the real equation. It does not assert that the signed graph is the unique correct construction, only that this one repairs the specific collision. The theorem is a step in a larger formal development, not a standalone physical law.

/-- The real canonical Green field solves the signed graph source equation at
the exact S9 scale `q/2`. -/
theorem signedRealGreenField3_scaledSourceEquation
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (σ : Equiv.Perm (Fin 3))
    (q : ℝ)
    (event : PostingPair3 N) :
    ScaledSourceEquation
      (signedRecognitionProductionGraph3 N σ).weight
      (dipole event.1 event.2) (q / 2)
      (signedRealGreenField3 q event) := by
  intro i
  let p : TorusSite3 N := (torusSiteEquivFin N).symm i
  let ψ : TorusSite3 N → ℂ :=
    actionNormalizedGreenResponse3 (q : ℂ) event
  let realψ : TorusSite3 N → ℝ := fun x => (ψ x).re
  have hbridge :=
    pulledBackSignedPostingLaplacian3_eq_framed
      σ realψ p
  have hcommute :=
    framedTorusLaplacian_ofReal_re σ ψ p
  have hpull :
      pulledBackSignedPostingLaplacian3 σ realψ p =
        (framedTorusLaplacian σ ψ p).re := by
    have hre := congrArg Complex.re (hbridge.trans hcommute)
    simpa using hre
  have hgreen :=
    actionNormalizedGreenResponse3_solves
      hN (q : ℂ) event σ p
  rw [hgreen] at hpull
  have hsource :=
    actionNormalizedGreenSource3_re_encoded q event i
  unfold pulledBackSignedPostingLaplacian3 at hpull
  simpa [signedRealGreenField3, ψ, realψ, p] using
    hpull.trans hsource

/-- Signed generator-event multiplicity between two torus sites. Each axis has
one forward and one backward occurrence, even when periodicity makes their
targets coincide. -/
def signedPostingWeight3
    {N : ℕ}
    (σ : Equiv.Perm (Fin 3))
    (p q : TorusSite3 N) : ℝ :=
  ∑ k : Fin 3,
    ((if q = framedForward σ p k then 1 else 0) +
      (if q = framedBackward σ p k then 1 else 0))

/-- The real part of the complex source is the real half-scaled dipole in the
finite enumeration. -/
theorem actionNormalizedGreenSource3_re_encoded
    {N : ℕ} [NeZero N]
    (q : ℝ)
    (event : PostingPair3 N)
    (i : Fin (TorusCard3 N)) :
    (actionNormalizedGreenSource3 (q : ℂ) event
      ((torusSiteEquivFin N).symm i)).re =
      (q / 2) * dipole event.1 event.2 i := by
  classical
  unfold actionNormalizedGreenSource3 greenScaleFromPostingMagnitude
    scaledTorusDipole postingEventSourceSite3 postingEventSinkSite3 dipole
  by_cases hsource : i = event.1 <;>
    by_cases hsink : i = event.2 <;>
    by_cases heq : event.1 = event.2 <;>
    simp [hsource, hsink, heq, eq_comm]

What this page does not claim

The theorem does not claim the real Green field is unique or minimizes any cost. It does not claim the complex Green response itself is real-valued. It does not claim the signed graph is the only construction that repairs the collision.

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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