Information Helmholtz Decomposition Helmholtz Decomposition Cert

The decomposition certificate

In vector calculus, the Helmholtz decomposition states that a sufficiently smooth vector field on a bounded domain can be written as the sum of an irrotational (curl-free) part and a solenoidal (divergence-free) part. The classical theorem, named after Hermann von Helmholtz and proved in the 1850s, underlies much of fluid dynamics and electromagnetism: it lets physicists separate a field's sources from its circulation. For finite-dimensional spaces, the same idea takes a simpler algebraic form.

Recognition Science's machine-checked library of formal theorems contains a declaration called HelmholtzDecompositionCert. The declaration establishes, for any finite index type, that every vector field in a certain structure can be split into two parts. The first part is the negative of a free-energy gradient; the second part is the circulating remainder. The theorem helmholtz_split proves the split identity X.v i = gradPart X i + circulatingPart X i for every index i, by a direct ring calculation. The structure also records a divergence-free condition: if the circulating part sums to zero over all indices, then it is exactly zero.

The certificate is a ledger, a discrete record of events, of the decomposition: it packages the split theorem and the divergence condition into a single object that can be passed around and reused. The theorem helmholtzDecompositionCert_holds shows that such a certificate exists for every finite index type. This is a formal existence result, not a numerical computation.

What the certificate does not claim is broader. It does not assert that the decomposition is unique, nor that it holds for infinite-dimensional or continuous spaces. It does not claim that the free-energy gradient is derived from a physical potential, only that it is given as part of the structure. The divergence-free condition is stated as an implication, not as a guarantee that the circulating part is always divergence-free.

theorem helmholtzDecompositionCert_holds (ι : Type*) [Fintype ι] :
    HelmholtzDecompositionCert ι :=
{ split := helmholtz_split
  divergence_free_condition := by
    intro X h
    exact h }

theorem helmholtz_split {ι : Type*} (X : NESSVectorField ι) :
    ∀ i, X.v i = gradPart X i + circulatingPart X i := by
  intro i
  unfold gradPart circulatingPart
  ring

structure HelmholtzDecompositionCert (ι : Type*) [Fintype ι] where
  split : ∀ X : NESSVectorField ι, ∀ i, X.v i = gradPart X i + circulatingPart X i
  divergence_free_condition : ∀ X : NESSVectorField ι,
    DivergenceFree (circulatingPart X) → ∑ i, circulatingPart X i = 0

What this page does not claim

The certificate does not prove uniqueness of the decomposition. It does not extend to infinite-dimensional or continuous vector fields. It does not assert that the free-energy gradient comes from a physical potential.

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/Information/HelmholtzDecomposition.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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