Foundation Pair Kernel Canonical Generator Source S9

The source scale

A ledger, a discrete record of events, needs a rule for how one event changes the state of the whole system. In Recognition Science, that rule is an action field, a quantity assigned to every possible pairing of sites. The module named foundation pair kernel canonical generator source s9 pins down the source side of that rule: it establishes, without fitting any number, that one elementary posting of magnitude 1 produces a source term of exactly 1/2.

The result follows from the normalization of two distinct units. One elementary ledger posting has divergence +1 at its source and -1 at its sink, a divergence being the net flow out of a point. The fixed pair action has first variation equal to 2 times the Laplacian, the operator that measures how a value at a point differs from its neighbors. Because the action's variation is twice the Laplacian, an action field paired with a posting of magnitude q solves the Green equation, the equation linking a response to its source, with right-hand coefficient q/2. For the primitive posting q=1, the canonical source scale is therefore 1/2.

The module proves this in a machine-checked library of formal theorems, with no unproved assumptions. The key equivalence, tagged as a theorem, states that the one-ledger law, the condition that a posting of magnitude 1 acts as a source, holds if and only if the scaled source equation holds with scale 1/2. A companion theorem shows the source scale is uniquely determined: if a symmetric action field satisfies both the posting-magnitude law and the scaled source equation, then the source scale must equal q/2. This uniqueness is what makes the 1/2 canonical rather than a convention.

The module also constructs the source explicitly on a three-dimensional torus, a finite grid with wrap-around edges. It defines a unit posting source as a dipole, a pair of opposite unit charges at the source and sink sites. Theorems confirm this source has total divergence zero, meaning nothing is created or destroyed, and that its value at the source is +1 and at the sink is -1. The action-normalized Green response, the field that solves the equation with this source, is shown to satisfy the framed torus Laplacian equation with the correct scale.

In plain terms, the module answers a question of bookkeeping: when the framework records one event, how strong is the push it exerts? The answer, 1/2, is not chosen to match data. It falls out of the two normalizations. A later physical conversion may assign a dimensional magnitude to one ledger posting; that magnitude stays an explicit parameter and is divided by two by the same theorem. No hydrogen or benchmark datum appears here, so the 1/2 is a structural fact of the framework, not an empirical fit.

/-- The existing unit `OneLedgerLaw` is exactly the half-scaled source
equation. -/
theorem oneLedgerLaw_iff_halfScaledSourceEquation
    {n : ℕ}
    (w : Fin n → Fin n → ℝ)
    (e : Fin n → ℝ)
    (a b : Fin n)
    (hsymm : ∀ i j, w i j = w j i) :
    OneLedgerLaw w e a b ↔
      ScaledSourceEquation w (dipole a b) (1 / 2) e := by
  simpa [PostingMagnitudeActionLaw, OneLedgerLaw,
    realGreenScaleFromPostingMagnitude] using
    postingMagnitudeActionLaw_iff_scaledSourceEquation
      1 w e a b hsymm

/-- 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_conserved
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N) :
    ∑ i : Fin (TorusCard3 N), unitPostingSource3 event i = 0 :=
  PairKernelDiscreteGauss.elementaryPosting_sum_div_zero
    event.1 event.2

What this page does not claim

This module does not assign a dimensional magnitude to a ledger posting; that magnitude is an explicit parameter. The 1/2 source scale is not fitted to any empirical datum. This module does not itself derive the physical constants of the framework.

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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