Foundation Pair Kernel Physical Source Covector Pair Kernel Physical Source Cove

The source pairing

A physical source, in the Recognition Science framework, is a rule that reads a field and returns a number: the source's response to that field. The declaration pairKernelPhysicalSourceCovector_eq_source_pairing proves that a specific such rule, built from two marked points a and b, equals a weighted sum over all field values. Concretely, for any field v and any scale s, the rule's output is s times the difference v(a) minus v(b). The theorem states this is the same as summing, over every point i, the field value v(i) multiplied by a density that is nonzero only at the two marked points.

The identity is a theorem in the machine-checked library of formal theorems, meaning it is proved from definitions with no unproved assumptions. It shows the source is a linear functional: it respects addition and scaling of fields. It also shows the source's action is local, depending only on the field at a and b, not on distant values. The density itself is a scaled unit dipole, a pair of opposite unit charges at the two points, which is the discrete analogue of a point dipole in continuous physics.

What the theorem does not claim is as important as what it proves. It does not select a numerical value for the scale s; that freedom remains a parameter of the construction. It does not derive the OneLedgerLaw, a separate statement about the ledger's accounting. It does not identify this source with a Noether momentum-map covector, a different object from a conserved-quantity construction. The theorem is a bridge between a discrete combinatorial object and a real-valued physical carrier, not a claim about which scale nature chooses.

/-- Operational characterization: the covector is the pairing against the
scaled realified elementary Gauss source. -/
theorem pairKernelPhysicalSourceCovector_eq_source_pairing {n : ℕ}
    (sourceScale : ℝ) (a b : Fin n) (v : Fin n → ℝ) :
    pairKernelPhysicalSourceCovector sourceScale a b v =
      ∑ i, v i * pairKernelPhysicalSourceDensity sourceScale a b i := by
  have hpair :
      (∑ i, v i * unitDipole a b i) = v a - v b :=
    sum_mul_unitDipole v a b
  calc
    pairKernelPhysicalSourceCovector sourceScale a b v =
        sourceScale * (v a - v b) := rfl
    _ = sourceScale * ∑ i, v i * unitDipole a b i := by rw [← hpair]
    _ = ∑ i, v i * (sourceScale * unitDipole a b i) := by
          rw [Finset.mul_sum]
          apply Finset.sum_congr rfl
          intro i _
          ring
    _ = ∑ i, v i * pairKernelPhysicalSourceDensity sourceScale a b i := by
          apply Finset.sum_congr rfl
          intro i _
          rw [pairKernelPhysicalSourceDensity_eq_scale_unitDipole]

/-- **MODEL.** Real covector on posting/field variations for a realized
elementary posting, at an explicit symbolic source scale.

Evaluation equals `sourceScale * (v a - v b)`, equivalently the pairing of `v`
against `sourceScale · divF(elementaryPosting a b)`. -/
def pairKernelPhysicalSourceCovector {n : ℕ}
    (sourceScale : ℝ) (a b : Fin n) : PhysicalSourceCarrier n where
  toFun v := sourceScale * (v a - v b)
  map_add' v w := by
    simp only [Pi.add_apply]
    ring
  map_smul' c v := by
    simp only [Pi.smul_apply, RingHom.id_apply, smul_eq_mul]
    ring

theorem pairKernelPhysicalSourceDensity_eq_scale_unitDipole {n : ℕ}
    (sourceScale : ℝ) (a b i : Fin n) :
    pairKernelPhysicalSourceDensity sourceScale a b i =
      sourceScale * unitDipole a b i := by
  simp [pairKernelPhysicalSourceDensity,
    realifiedElementaryGaussSource_eq_unitDipole]

What this page does not claim

The theorem does not fix the numerical value of the source scale. The theorem does not derive the OneLedgerLaw. The theorem does not identify the source with a Noether momentum-map covector.

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/PairKernelPhysicalSourceCovector.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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