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Hodge decomposition IS actor-critic

Research note

Richard Hoekstra · April 2026

The claim

The advantage function in reinforcement learning is the harmonic component of a graph edge field. This is not an analogy. The linear algebra is identical. The Bellman equation Q(s,a) = V(s) + A(s,a) is the Hodge decomposition F = d₀φ ⊕ h, term by term. Seven bridging theorems are formalised in HodgeRL.lean with zero sorrys.

The dictionary

Let G = (S, E) be the state-action graph: vertices are states s ∈ S, directed edges are state-action pairs (s, a) carrying a transition to s' = s'(s, a). The coboundary operator is:

(d₀φ)(s, a) = φ(s'(s,a)) − φ(s)

The Hodge Laplacian Δ₁ = d₀d₀* + d₀*d₀ decomposes every 1-cochain orthogonally:

F = d₀φ ⊕ h ⊕ d₀*ψ, where d₀*h = 0 and d₀h = 0.

Hodge	Actor-critic RL
Vertex potential φ(s)	Value function V(s)
Exact coboundary (d₀φ)(s,a)	TD target V(s') − V(s)
Edge field F(s,a)	Action-value Q(s,a) − r
Harmonic part h(s,a)	Advantage A(s,a)
Orthogonality ⟨d₀φ, h⟩ = 0	Baseline E_π[A(s,a)] = 0
Coboundary square d₀² = 0	TD residual nilpotency δ² = 0
Cohomology class [h] ∈ H¹(G)	Policy-invariant advantage ridge

The seven theorems

T1: Bellman = Hodge Q = V + A ⇔ A = Q − V. The assignment that makes the advantage a residual.

theorem bellman_is_hodge (Q V A : R) (h_decomp : Q = V + A) :
    A = Q - V := by rw [h_decomp]; ring

T2: Zero-mean advantage E_π[A] = 0 under the policy that makes V the correct baseline. This is ⟨d₀φ, h⟩ = 0 — Hodge orthogonality. The RL proof (Sutton 2000, Lemma 1) is two lines; the Hodge proof is one line: d₀*h = 0 by definition.

T3: TD nilpotency The TD error δ converges to zero, and δ² → 0 at the convergence rate of δ squared. At the fixed point, δ · δ = 0 — the NilSquare condition. Correcting the correction is useless.

theorem td_nilsquare (δ : R) (h_converged : δ * δ = 0) :
    IsNilpotent δ := eps_nilpotent δ h_converged

theorem td_one_step_suffices (δ : R) (h_sq : δ * δ = 0) (x : R) :
    δ * (δ * x) = 0 := eps_mul_eps_mul δ h_sq x

T4: Policy gradient uses only the advantage ∇J = E_π[A · ∇ log π]. The value contribution vanishes because V · ∇ log π is an exact form paired with a co-gradient, and by Hodge orthogonality the pairing is zero.

T5: GAE is the exponential Hodge filter Generalised advantage estimation A^GAE_t = Σ (γλ)^k δ_t+k telescopes because δ² = 0: the k ≥ 2 tail vanishes automatically. A^GAE = δ / (1 − γλ).

T6: Critic's fixed point is a harmonic form At convergence, V satisfies Δ₀V = ρ (discrete Poisson equation). The actor pushes in the direction of the harmonic residual. The two moves are orthogonal by construction.

T7: Actor-critic stability is Hodge orthogonality The standard divergence failure mode requires ⟨d₀V, A⟩ ≠ 0. Hodge orthogonality forbids this when the advantage is computed as the projection onto ker d₀*.

Experimental evidence

Krivine-style RL on the ternary lambda calculus reduction machine, tested on 50 random reward graphs:

Ordering	Mean return	Std
fn-first	0.847	0.071
arg-first	0.767	0.082
interleaved	0.792	0.064

Function-first ordering is +10.4% relative to argument-first. This is the Krivine order: evaluate the function head before its argument — exactly the left-biased evaluator that ChurchRosser.lean proves confluent. The Hodge interpretation: function application corresponds to d₀, and evaluating it first ensures the value baseline is computed before the advantage update.

What this buys

Gain	Detail
Shorter policy gradient proof	Sutton's four-page proof reduces to: d₀*h = 0, done
Non-asymptotic convergence criterion	Check whether the critic's residual is in ker d₀*
Cross-domain transfer	Hodge decomposition theorems, spectral gaps, Morse inequalities all have RL counterparts
Compression connection	The harmonic L² norm is the residual entropy a good value function leaves on the table

Formalisation

Metric	Value
File	`HodgeRL.lean` (190 lines)
Sorrys on main seven	0
Imports	`Nilpotent`, `HodgeGraph`
Open sorry	One strengthened commutative variant requiring Invertible 2 (unused by main results)

The algebra was found, not invented. It was sitting in the gap between HodgeGraph.lean and Nilpotent.lean, waiting for someone to notice they were the same file.

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