plurigrid de Rham Cohomology Skill: Differential Forms on Signal Manifolds
Status: Production Ready Trit: -1 (MINUS - validator) Color: #D826E8 (Magenta) Principle: de Rham cohomology bridges differential geometry and topology Frame: Exterior algebra with Hodge decomposition
Overview
de Rham Cohomology implements differential forms on signal manifolds. Implements:
- Exterior algebra: Wedge product alpha ^ beta with antisymmetry
- de Rham complex: d: Omega^k -> Omega^(k+1) with d^2 = 0
- Hodge star: *: Omega^k -> Omega^(n-k) using metric duality
- Codifferential: delta = (-1)^k * d (adjoint of d)
- Hodge-Laplacian: Delta = ddelta + deltad (harmonic analysis)
- Cohomology classification: closed/exact/harmonic form identification
Correct by construction: d^2 = 0 is verified computationally (||d^2|| = 0.000000).
Core Formulae
de Rham complex:
0 -> Omega^0 -d-> Omega^1 -d-> Omega^2 -d-> ... -d-> Omega^n -> 0
Exterior derivative:
d: Omega^k -> Omega^(k+1)
d^2 = 0 (fundamental property)
Wedge product:
alpha ^ beta = (-1)^(kl) beta ^ alpha (graded antisymmetry)
alpha ^ alpha = 0 (for odd-degree forms)
Hodge star:
*: Omega^k -> Omega^(n-k)
** = (-1)^(k(n-k)) on k-forms
Hodge decomposition:
Omega^k = H^k ⊕ d(Omega^(k-1)) ⊕ delta(Omega^(k+1))
H^k = ker(Delta) = harmonic forms
de Rham isomorphism:
H^k_dR(M) ≅ H^k_sing(M; R)
Stokes' theorem:
integral_M d(omega) = integral_(dM) omega
Gadgets
1. DifferentialFormEngine
Represent and manipulate k-forms:
;; A k-form: map from k-index sets to coefficients
;; 0-form: {[] 3.0}
;; 1-form: {[0] 2.0, [1] -1.0, [2] 0.5}
;; 2-form: {[0 1] 1.5, [0 2] -0.3, [1 2] 0.7}
(defn form-degree [form]
(count (first (keys form))))
;; BCI signal -> 1-form
(defn signal-to-1form [signal-values]
(into {} (map-indexed (fn [i v] [[i] (double v)]) signal-values)))
;; Signal curvature -> 2-form (field strength F = dA)
(defn signal-curvature-2form [signal-values]
(exterior-derivative-1form (signal-to-1form signal-values) (count signal-values)))
2. WedgeProduct
Antisymmetric multiplication of forms:
(defn wedge-product [form1 form2 dim]
"alpha ^ beta: (k-form) ^ (l-form) -> (k+l)-form"
(reduce (fn [result [idx1 coeff1]]
(reduce (fn [acc [idx2 coeff2]]
(let [combined (vec (concat idx1 idx2))
distinct? (= (count (set combined)) (count combined))]
(if distinct?
(let [[sorted sign] (sort-index-with-sign combined)]
(update acc sorted (fnil + 0.0) (* sign coeff1 coeff2)))
acc)))
result form2))
{} form1))
;; Antisymmetry verified:
;; alpha^beta[0,2] = 0.7000
;; beta^alpha[0,2] = -0.7000
;; sum = 0.000000
3. ExteriorDerivative
The d operator on each degree:
(defn exterior-derivative [form dim]
(case (form-degree form)
0 (exterior-derivative-0form form dim) ;; gradient
1 (exterior-derivative-1form form dim) ;; curl-like
2 (exterior-derivative-2form form dim) ;; divergence-like
{})) ;; d of top-form is zero
;; Key verification: d^2 = 0
;; d(f) -> 1-form, d(df) -> 2-form, d(d(df)) -> 3-form
;; ||d^2|| = 0.000000 VERIFIED
4. HodgeStar
Metric duality operator:
(defn hodge-star [form dim]
"*: Omega^k -> Omega^(n-k)"
(reduce (fn [result [indices coeff]]
(let [complement (sort (vec (set/difference (set (range dim)) (set indices))))
sign (levi-civita-sign-of-permutation ...)]
(assoc result (vec complement) (* sign coeff))))
{} form))
;; *(1-form) -> 3-form in R^4
;; *(2-form) -> 2-form in R^4 (self-dual!)
5. HodgeLaplacian
Harmonic analysis on forms:
(defn laplace-beltrami [form dim]
"Delta = d*delta + delta*d"
(let [d-form (exterior-derivative form dim)
delta-form (codifferential form dim)
delta-d (codifferential d-form dim)
d-delta (exterior-derivative delta-form dim)]
(merge-with + delta-d d-delta)))
;; Harmonic forms: Delta(omega) = 0
;; These represent cohomology classes
6. CohomologyClassifier
Classify forms into cohomological types:
(defn de-rham-cohomology-class [form dim]
(let [closed-info (is-closed? form dim 0.01)
harmonic (laplace-beltrami form dim)
harmonic-norm (norm harmonic)]
{:cohomology-class
(cond
(< harmonic-norm 0.1) :harmonic-representative
(:closed closed-info) :closed-not-exact
:else :not-closed)}))
BCI Integration (Layer 17)
Part of the 17-layer BCI orchestration pipeline:
Layer 8 (Persistent Homology) → Betti numbers from simplicial complex
Layer 13 (Relative Homology) → signal/noise separation
Layer 14 (Cohomology Ring) → cup product structure
Layer 17 (de Rham Cohomology) → differential form representation
Cross-Layer Connections
- L14 Cohomology Ring: Cup product = wedge product (de Rham isomorphism)
- L16 Spectral Methods: Laplacian eigenforms = harmonic forms (Hodge theory)
- L5 Riemannian Manifolds: Metric defines Hodge star operator
- L8 Persistent Homology: Integration of forms over cycles = homology pairing
Topology Chain
L8 (Persistent Homology)
→ L13 (Relative Homology)
→ L14 (Cohomology Ring)
→ L17 (de Rham Cohomology)
de Rham isomorphism: H^k_dR(M) ≅ H^k_sing(M; R)
This chain connects all topological layers through one theorem.
Mathematical Foundation
Exterior Algebra
Wedge product properties:
alpha ^ beta = (-1)^(deg(alpha)*deg(beta)) beta ^ alpha
alpha ^ alpha = 0 (odd degree)
(alpha ^ beta) ^ gamma = alpha ^ (beta ^ gamma)
d(alpha ^ beta) = d(alpha) ^ beta + (-1)^k alpha ^ d(beta)
de Rham Complex
0 -> C^inf(M) -d-> Omega^1(M) -d-> Omega^2(M) -d-> ... -d-> Omega^n(M) -> 0
H^k_dR(M) = ker(d: Omega^k -> Omega^(k+1)) / im(d: Omega^(k-1) -> Omega^k)
= {closed k-forms} / {exact k-forms}
Hodge Theory
Hodge decomposition:
Omega^k = H^k ⊕ im(d) ⊕ im(delta)
Every cohomology class has unique harmonic representative.
dim(H^k) = beta_k (Betti number)
Hodge-Laplacian: Delta = d delta + delta d
delta = (-1)^(nk+n+1) * d * (codifferential)
Stokes' Theorem
integral_M d(omega) = integral_(partial M) omega
Generalizes:
- Fundamental theorem of calculus (0-forms on R)
- Green's theorem (1-forms on R^2)
- Divergence theorem (2-forms on R^3)
- Stokes' theorem (k-forms on M)
Example Output
de Rham Complex: 0 -> Omega^0 -d-> Omega^1 -d-> Omega^2 -d-> Omega^3 -d-> Omega^4 -> 0
1-Forms (BCI signal):
omega = 0.8*dx0 - 0.3*dx1 + 0.5*dx2 + 0.1*dx3
d(omega): 6 components (curvature 2-form)
Wedge Products:
alpha ^ beta [0,2]: 0.7000
beta ^ alpha [0,2]: -0.7000
Antisymmetry sum: 0.000000
Hodge Star:
*(1-form) -> 3-form in R^4
*(2-form) -> 2-form (self-dual in R^4)
Fundamental Property:
||d^2|| = 0.000000
d^2 = 0: VERIFIED
Multi-World de Rham:
world-a: 1-form class = harmonic-representative
world-b: 1-form class = harmonic-representative
world-c: 1-form class = harmonic-representative
GF(3): +1 + 0 - 1 = 0 [check]
DuckDB Schema
CREATE TABLE differential_forms (
form_id UUID PRIMARY KEY,
degree INTEGER,
dimension INTEGER,
n_components INTEGER,
is_closed BOOLEAN,
d_norm FLOAT,
world_name VARCHAR,
recorded_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP
);
CREATE TABLE cohomology_classes (
class_id UUID PRIMARY KEY,
degree INTEGER,
cohomology_type VARCHAR,
harmonic_norm FLOAT,
is_harmonic BOOLEAN,
world_name VARCHAR
);
Skill Name: derham-cohomology Type: Differential Forms / Exterior Calculus / Hodge Theory Trit: -1 (MINUS) Color: #D826E8 (Magenta) GF(3): Forms valid triads with ERGODIC + PLUS skills
Integration with GF(3) Triads
stochastic-resonance (+1) ⊗ spectral-methods (0) ⊗ derham-cohomology (-1) = 0 ✓
gay-mcp (+1) ⊗ lyapunov-stability (0) ⊗ derham-cohomology (-1) = 0 ✓
nats-color-stream (+1) ⊗ acsets (0) ⊗ derham-cohomology (-1) = 0 ✓
The Perfect Triad: Layers 15-16-17
stochastic-resonance (+1, GENERATOR)
× spectral-methods (0, ERGODIC)
× derham-cohomology (-1, VALIDATOR)
= 0 ✓ [GF(3) conserved]
This triad forms a complete analytical framework:
+1: Noise injection and enhancement (stochastic)
0: Frequency decomposition and coordination (spectral)
-1: Differential form validation and cohomology (de Rham)
CT lattice atlas
Part of: para-mensch-commons (CT lattice family).