Wev Hodge Decomp

World Extractable Value via Pair-Graph Hodge Decomposition

Status: 🧪 Draft — K₃/K₄/K₅ verified Type: Discrete cohomology / optimal transport / market microstructure Principle: WEV = non-exactness of the world, measured in L^p over H₁ Frame: Antisymmetric 1-cochain on pair graph → Hodge splits into coboundary (potential) + harmonic (residue) GF(3)/GF(5): Both required — GF(3) collapses uniform-sign RPS cycles; GF(5) witnesses orientation-H¹

Core Construction

Given pair graph G = (V, E) observed at epoch t with signed 1-cochain α : E → ℝ satisfying α(i,j) = −α(j,i):

1. Form incidence matrix B : ℝ^|E| × ℝ^|V|, B[e=(u,v), u] = −1, B[e, v] = +1.
2. Solve normal equations BᵀB φ = Bᵀα (Tikhonov-damped if graph has isolated components). L := BᵀB is the graph Laplacian on vertices.
3. Decompose α = δφ* + harmonic where δφ*(e=(u,v)) = φ*(v) − φ*(u).
4. WEV := ‖harmonic‖_p — edge-space p-norm. Common choices:
   • WEV_L¹ = min trade volume to close harmonic under uniform cost per trade.
   • WEV_L² = min trade energy under quadratic price-impact cost.

The coboundary part δφ* is explained by a vertex potential φ* — this is the Kantorovich dual / "type price" / clearing prices. The harmonic part is what no potential can explain — the extractable residue.

5-Instance Unification

Antisymmetric α required. Semantics splits into extractive (closure = transaction) vs diagnostic (observational):

Warning: symmetric edge measures (RSSI correlation, stamp Jaccard) violate antisymmetry and are the wrong shape. Use temporal precedence, not co-presence magnitude. Co-presence is the filter on which edges exist; precedence is the edge value.

Reference implementation (Guile, verified 2026-04-23)

Pure core — no goblins, no actors, no cells. Pair-graph Hodge in ~80 lines of Scheme:

;; solve BᵀBφ = Bᵀα via damped Gauss-Jordan
(define (hodge alpha E V)
  (let* ((B (incidence E V))
         (Bt (transpose B))
         (BtB (mat*mat Bt B))
         (Bta (mat*vec Bt alpha))
         (phi (solve BtB Bta 1e-6))
         (delta-phi (mat*vec B phi))
         (harmonic (vec-sub alpha delta-phi)))
    (values phi delta-phi harmonic)))

(define (WEV-L1 alpha E V)
  (call-with-values
    (lambda () (hodge alpha E V))
    (lambda (_ _ h) (vec-norm1 h))))

Forj/Clojure port drops naturally into propagator-nash.clj — the pair graph is already there from the Čech H¹ extension.

K₅ Worked Example (RPS tournament, σ=(R P S R P))

V = (a b c d e)
E = ((a b) (a c) (a d) (a e) (b c) (b d) (b e) (c d) (c e) (d e))
α = ( 1  -1   0   1   1  -1   0   1  -1   1)

Hodge:

• φ* = (R:−0.2, P:+0.2, S:0, R:−0.2, P:+0.2) — the strategy-type potential emerges automatically
• δφ* = (0.4, 0.2, 0, 0.4, -0.2, -0.4, 0, -0.2, 0.2, 0.4)
• harmonic = (0.6, -1.2, 0, 0.6, 1.2, -0.6, 0, 1.2, -1.2, 0.6)
• WEV_L¹ = 7.2, WEV_L² = 2.68
• Coboundary energy = 0.8

Basis-cycle sum was basis-dependent undercount: BFS-star fundamental cycles give Σ |⟨α,γ⟩| = 6, but edges in the harmonic carry non-zero mass on edges whose basis-cycle circulations cancel. Edge-L¹ (7.2) is the correct basis-free invariant.

Greedy cycle-by-cycle closure with uniform flow bleeds residue into previously-zero cycles because BFS-star cycles share edges. To close cleanly, subtract the L²-orthogonal projection onto span(γ), not the uniform-t projection.

Connections

• Kantorovich duality: duals exist ⇔ harmonic = 0 ⇔ α ∈ B¹ ⇔ clearing prices exist
• Martingale OT (Beiglböck–Henry-Labordère–Penkner): dynamic no-arb = sheaf-H¹ = 0 across time-epochs
• Sinkhorn / entropic OT: ^best-response-propagator's softmax at temperature 1/β IS the Schrödinger-bridge iteration; its convergence monitor is WEV → 0
• Kyle's λ: ∂WEV/∂(market depth) — Kyle is the marginal of WEV
• Tsao face-patch system (Chang-Tsao 2017, Bao-She-Tsao 2020): 6-patch face cover is a sheaf; prosopagnosia and Thatcher illusion predicted as non-trivial H¹ on the cover. fMRI pair-graph + Hodge = prosopagnosia index. StyleGAN alignment = sybil attack at the axis-code identity layer.

Open probes

• Weighted Hodge: replace L = BᵀB with BᵀWB where W weights edges by observation confidence / inverse-variance.
• Dynamic (sheaf over epochs): persistent harmonic across time → alert, transient → noise. Dynamic-WEV / static-WEV ratio as alert scalar.
• Cost-of-closure optimization: given per-edge costs c_ij, minimum-cost closure is an LP over edge flows; extractable fraction depends on cost structure.
• Directed-graph Hodge: for intrinsically asymmetric interactions (e.g. MM-vs-flow where MM doesn't attack trader back), use directed combinatorial Hodge (Jiang-Lim-Yao-Ye 2011 HodgeRank).

Verification trail

• Pure Guile K₃ (RPS): triangle-trit = 0 in GF(3) [collapse], = 3 in GF(5) [witness] ✓
• Pure Guile K₄ (σ_d = σ_a): only RPS sub-triangle circulation −3; others 0 ✓
• Pure Guile K₅ (σ = R P S R P): two active cycles ±3, WEV_L¹ = 7.2, WEV_L² = 2.68 ✓
• Memory trail: world-extractable-value.md, propagator-nash-cocycle.md

GF(3) triadic balance

• +1 Play: constructions (incidence, Laplacian, Hodge solve, WEV computation)
• 0 Witness: audit (antisymmetry check, basis-independence check, extract-vs-diagnose split)
• −1 Coplay: applications (MM harness, MAC observer, Passport sybil, face-patch fMRI)

Triadic sum over this skill = 0 (mod 3) ✓