Foundation Pair Kernel Physical Source Covector Every Source Scale Admits Physic

The source covector carrier

A ledger, a discrete record of events, must connect to continuous field variations if it is to describe physics. The declaration every_sourceScale_admits_physicalSourceCovector proves that for any chosen real scale, a unique real-valued linear functional exists that realizes the Gauss/source-coupling law for an elementary posting. In plain terms: pick any strength for the source, and there is exactly one way to turn the ledger's discrete divergence into a real physical source that responds to field variations.

The carrier is the pairKernelPhysicalSourceCovector, defined by its action on a field variation v as sourceScale * (v a - v b). This is a dipole: it reads the difference between the field's values at two ledger positions a and b, scaled by the chosen sourceScale. The declaration's proof is direct: it shows this covector satisfies the defining equation φ v = sourceScale * ∑ i, v i * divF (elementaryPosting a b) i, where divF is the discrete Gauss divergence of the posting. The uniqueness theorem pairKernelPhysicalSourceCovector_unique then confirms that no other covector realizes the same coupling at that scale.

In Recognition Science, this bridges a gap: the integer Gauss source of a posting was already known to be a unit dipole, but that integer object was not yet a real physical source. The new carrier realifies it, coupling to field variations at an explicit symbolic positive scale. The scale parameter is free; the Gauss law fixes the dipole shape, not the coupling magnitude. The injectivity theorem pairKernelPhysicalSourceCovector_scale_injective proves that different scales give different covectors on a nontrivial posting, so the scale freedom is genuine.

What this declaration does not do is as important as what it proves. It does not select a numerical value for the source scale; that remains a separate choice. It does not derive OneLedgerLaw, the framework's law of forced composition. It does not identify this carrier with a Noether momentum-map covector; that comparison is recorded as an ordinary equality proposition, not a theorem. The declaration is a construction and its uniqueness proof, not a claim about which scale physics actually uses.

/-- Every real scale admits a Gauss-law-realizing physical source covector.
No uniqueness of scale is claimed. -/
theorem every_sourceScale_admits_physicalSourceCovector {n : ℕ}
    (sourceScale : ℝ) (a b : Fin n) :
    RealizesPairKernelGaussSourceCoupling
      (pairKernelPhysicalSourceCovector sourceScale a b) sourceScale a b :=
  pairKernelPhysicalSourceCovector_realizes sourceScale a b

/-- Uniqueness: any covector realizing the same Gauss/source-coupling law at
the same symbolic scale equals the constructed carrier. -/
theorem pairKernelPhysicalSourceCovector_unique {n : ℕ}
    (φ : PhysicalSourceCarrier n) (sourceScale : ℝ) (a b : Fin n)
    (hφ : RealizesPairKernelGaussSourceCoupling φ sourceScale a b) :
    φ = pairKernelPhysicalSourceCovector sourceScale a b := by
  apply LinearMap.ext
  intro v
  calc
    φ v =
        sourceScale *
          ∑ i, v i *
            PairKernelDiscreteGauss.divF
              (PairKernelDiscreteGauss.elementaryPosting a b) i :=
      hφ v
    _ = pairKernelPhysicalSourceCovector sourceScale a b v :=
          (pairKernelPhysicalSourceCovector_eq_divF_pairing
            sourceScale a b v).symm

/-- **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

/-- Distinct symbolic scales give distinct covectors on a nontrivial posting.
This is the precise sense in which scale freedom lives in the constructor:
Gauss fixes shape; scale remains a free real parameter. -/
theorem pairKernelPhysicalSourceCovector_scale_injective {n : ℕ}
    {a b : Fin n} (hab : a ≠ b) {s₁ s₂ : ℝ}
    (h : pairKernelPhysicalSourceCovector s₁ a b =
      pairKernelPhysicalSourceCovector s₂ a b) :
    s₁ = s₂ := by
  have hs :=
    congrArg
      (fun φ : PhysicalSourceCarrier n =>
        φ (fun i => if i = a then (1 : ℝ) else 0)) h
  have h1 := pairKernelPhysicalSourceCovector_source_basis s₁ a b hab
  have h2 := pairKernelPhysicalSourceCovector_source_basis s₂ a b hab
  simp only [h1, h2] at hs
  exact hs

What this page does not claim

The declaration does not select a numerical value for the source scale. The declaration does not prove OneLedgerLaw. The declaration does not identify the carrier 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.

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