Foundation Pair Kernel Canonical Generator Source S9 Constructed Posting Source

The canonical source scale

A posting is the framework's name for a discrete record of an event: one unit of something appears at a source site, and one unit disappears at a sink site. In Recognition Science, the ledger (a discrete record of such events) carries a cost, and the framework's library of machine-checked theorems studies how fields respond to postings. The declaration constructedPostingSourceAttachment3_iff_halfScaledSourceEquation is an equivalence: it says that a field is attached to a primitive posting exactly when that field solves a scaled source equation with coefficient one half.

The number one half is not fitted. It comes from two unit conventions that the framework sets in advance. One elementary posting has divergence +1 at its source and -1 at its sink. Separately, the fixed pair action has first variation equal to twice the Laplacian. Because the action variation carries a factor of two, a posting of magnitude q must be divided by two to become the right-hand source term in the Green equation. For the primitive posting, whose magnitude is exactly one, the canonical Green source scale is therefore exactly 1/2. The theorem proves this scale is unique: any source scale that satisfies the equation with a given posting magnitude must equal q/2.

The result is relative to the ledger action unit. A later physical conversion may assign a dimensional magnitude to one posting, and that magnitude stays an explicit parameter, divided by two by the same theorem. No hydrogen atom, no benchmark datum, and no fitted constant appears in this setting. The declaration is a structural fact about the framework's own discrete geometry, not a statement about measured physics.

What the declaration does not claim is equally precise. It does not claim that any particular field exists; it states an equivalence between two conditions on a field, assuming a primitive posting is given. It does not assign a physical unit to the posting magnitude; that conversion is left as an explicit parameter. And it does not say that the scale 1/2 is the only scale at which any source can attach; other posting magnitudes yield other scales, each equal to q/2. The theorem's force is that the framework's unit conventions pin the primitive posting's source scale to one half, and that the equivalence holds for every symmetric production graph.

/-- Once the exact constructed-event attachment is supplied, no scale remains:
it is equivalent to the half-scaled equation on the S8 graph. -/
theorem constructedPostingSourceAttachment3_iff_halfScaledSourceEquation
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N)
    (hevent : event ∈ realizedPrimitivePostingPairs3 N)
    (field : Fin (TorusCard3 N) → ℝ) :
    ConstructedPostingSourceAttachment3 event field ↔
      ScaledSourceEquation
        (recognitionProductionGraph3 N).weight
        (dipole event.1 event.2) (1 / 2) field := by
  unfold ConstructedPostingSourceAttachment3
  rw [and_iff_right hevent]
  simpa [realGreenScaleFromPostingMagnitude] using
    postingMagnitudeActionLaw_iff_scaledSourceEquation
      1 (recognitionProductionGraph3 N).weight field
      event.1 event.2
      (recognitionProductionGraph3 N).weight_symm

/-- If the same nontrivial source field is presented with a free source
scale, action compatibility determines that scale uniquely as `q / 2`. -/
theorem sourceScale_eq_half_postingMagnitude
    {n : ℕ}
    (q sourceScale : ℝ)
    (w : Fin n → Fin n → ℝ)
    (e : Fin n → ℝ)
    (a b : Fin n)
    (hab : a ≠ b)
    (hsymm : ∀ i j, w i j = w j i)
    (haction : PostingMagnitudeActionLaw q w e a b)
    (hsource :
      ScaledSourceEquation w (dipole a b) sourceScale e) :
    sourceScale = realGreenScaleFromPostingMagnitude q := by
  unfold realGreenScaleFromPostingMagnitude
  have hhalf :=
    (postingMagnitudeActionLaw_iff_scaledSourceEquation
      q w e a b hsymm).mp haction
  have hs := hsource a
  have hh := hhalf a
  simp [dipole, hab, realGreenScaleFromPostingMagnitude] at hs hh
  exact hs.symm.trans hh

theorem unitPostingSource3_at_source
    {N : ℕ} [NeZero N]
    {event : PostingPair3 N}
    (hne : event.1 ≠ event.2) :
    unitPostingSource3 event event.1 = 1 :=
  PairKernelDiscreteGauss.elementaryPosting_div_source
    event.1 event.2 hne

theorem unitPostingSource3_at_sink
    {N : ℕ} [NeZero N]
    {event : PostingPair3 N}
    (hne : event.1 ≠ event.2) :
    unitPostingSource3 event event.2 = -1 :=
  PairKernelDiscreteGauss.elementaryPosting_div_sink
    event.1 event.2 hne

What this page does not claim

This theorem does not assign a physical unit or dimensional magnitude to a posting. This theorem does not prove that any particular field exists; it states an equivalence given a primitive posting. This theorem does not claim the scale 1/2 is the only scale at which any source can attach; other posting magnitudes yield other scales.

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/PairKernelCanonicalGeneratorSourceS9.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.

Derived articles

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