Foundation Pair Kernel Physical Source Covector Physical Source Covector Eq Impl

The source covector

A source in this framework is a discrete record of where something enters or leaves a system. The object in question, a physical source covector, is a rule that takes a field variation, a list of real numbers attached to the positions of a finite system, and returns a single real number. The framework's library, a machine-checked collection of formal theorems, proves that this rule has a very simple shape: it reads the variation at two named positions and returns the difference between those two values, multiplied by a scale factor. In symbols, if the rule is called φ and the scale is s, then φ(v) = s·(v(a) − v(b)).

The declaration named physicalSourceCovector_eq_implies_stationaryAtCoefficient proves the converse direction of an equivalence. It says: if a source pairing, another rule built from a weight function and an error function, equals the physical source covector for all field variations, then that pairing satisfies a condition called StationaryAtCoefficient. That condition means the pairing's response to a variation is exactly the scaled difference of the variation at the two source positions, with no extra terms. The theorem is the reverse of an earlier one, stationaryAtCoefficient_eq_physicalSourceCovector, which shows that the stationary condition implies the covector form. Together they establish that the two descriptions, one operational and one structural, pick out the same object.

What the theorem does not do is choose a numerical value for the scale s. The scale remains a free parameter, a degree of freedom in the framework. The theorem also does not identify this source covector with a Noether momentum-map covector, a different kind of object from a conserved quantity in a variational problem. The framework records that comparison as an open equality proposition, not a proved identity. It does not derive OneLedgerLaw, a separate statement about a single ledger law, and it does not involve the constants G or hbar. The theorem is axiom-clean, with no sorry and no new axioms, but the scale freedom and the Noether comparison remain outside its scope.

For a reader, the consequence is a precise trade-off. The shape of a physical source is forced by the discrete Gauss law, but its magnitude is not. That separation, shape forced, scale free, is what the theorem makes exact. It tells you where the framework has already closed a question and where it has deliberately left a parameter open for later identification.

/-- Conversely, if work-response equals the constructed covector at every
variation, then the field is stationary at that symbolic coupling. -/
theorem physicalSourceCovector_eq_implies_stationaryAtCoefficient {n : ℕ}
    [DecidableEq (Fin n)]
    (w : Fin n → Fin n → ℝ) (e : Fin n → ℝ) (a b : Fin n)
    (sourceScale : ℝ)
    (hcov :
      ∀ v, PairKernelSourceCoupling.physSourcePairing w e v =
        pairKernelPhysicalSourceCovector sourceScale a b v) :
    PairKernelSourceCoupling.StationaryAtCoefficient w e
      (PairKernelSourceVariation.dipole a b) sourceScale := by
  intro v
  have hdrop :
      (∑ i, v i * PairKernelSourceVariation.dipole a b i) = v a - v b :=
    PairKernelSourceVariation.sum_mul_dipole v a b
  calc
    PairKernelSourceCoupling.physSourcePairing w e v =
        pairKernelPhysicalSourceCovector sourceScale a b v := hcov v
    _ = sourceScale * (v a - v b) := rfl
    _ = sourceScale *
          ∑ i, v i * PairKernelSourceVariation.dipole a b i := by
            rw [hdrop]

/-- Operational evaluation as scaled potential drop. -/
theorem pairKernelPhysicalSourceCovector_apply {n : ℕ}
    (sourceScale : ℝ) (a b : Fin n) (v : Fin n → ℝ) :
    pairKernelPhysicalSourceCovector sourceScale a b v =
      sourceScale * (v a - v b) :=
  rfl

/-- 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 theorem does not identify the source covector with a Noether momentum-map covector. The theorem does not derive OneLedgerLaw. The theorem does not involve the constants G or hbar.

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)

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Derived articles

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