Information Helmholtz Decomposition Helmholtz Decomposition Cert Holds

The decomposition certificate

The Helmholtz decomposition is a classical result in vector calculus: under mild conditions, any smooth vector field on a bounded domain can be written uniquely as the sum of a gradient of a scalar potential and a divergence-free (solenoidal) component. The classical theorem, named after Hermann von Helmholtz, underpins fluid dynamics, electromagnetism, and the study of conservative versus rotational forces. In finite dimensions, the same idea applies to a vector field on a discrete set of states: the field decomposes into a part that flows downhill along a potential and a part that circulates without accumulating.

In Recognition Science, the framework models a nonequilibrium steady state (NESS) vector field as a structure with three components: the actual flow v, a stationary density rho, and a free-energy gradient. The framework defines the gradient part as the negative of that free-energy gradient, and the circulating part as the remainder that makes the split exact. The machine-checked library of formal theorems proves that for any finite index type, the split holds identically: at every state, the flow equals the gradient part plus the circulating part. The proof is a direct algebraic identity, closed by the ring tactic, with no hidden assumptions beyond the structure definitions.

The certificate structure adds one further condition: if the circulating part is divergence-free, in the sense that the sum of its components over all states is zero, then that sum is indeed zero. The theorem helmholtzDecompositionCert_holds packages both the split and this divergence-free implication into a single certificate object. In plain terms, the declaration establishes that the decomposition is not merely a definitional choice but a proved fact, and that the divergence-free condition, when it holds, is respected by the construction.

What the declaration does not claim is broader than what it proves. It does not assert that every NESS vector field is divergence-free; the condition is an implication, not a universal property. It does not claim uniqueness of the decomposition, a feature of the classical theorem that requires additional boundary or regularity conditions. It does not establish any physical law about real systems; the structure is a formal model, and the theorem is about the model's internal consistency. The certificate is a building block for the framework's free-energy principle bridge, not a statement about any particular physical system.

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

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

What this page does not claim

The theorem does not claim that every NESS vector field is divergence-free. The theorem does not claim uniqueness of the decomposition. The theorem does not establish any physical law about real systems.

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)

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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