TNFR–Hodge Discrete Cochain Research Notes

Status: Pre-registered research programme; HC-1 diagnostic implemented; obstruction classified Branch B3-leaning (strong negative — no discrete TNFR closure of the actual conjecture) Date: 2026-06-13 Scope: TNFR-internal discrete/combinatorial Hodge theory on the tetrad cochain tower; not a proof or attack on the Clay Hodge conjecture Primary anchors: nodal equation ∂EPI/∂t = νf · ΔNFR(t), structural field tetrad (Φ_s, |∇φ|, K_φ, ξ_C), the k=1 Helmholtz–Hodge decomposition already shipped in examples/107

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0. Terminology Discipline

This programme is formulated in TNFR language only. References to the Hodge conjecture are treated as an external comparison target.

No claim in this document should be read as a solution of the Clay Millennium Problem. The Clay Hodge conjecture asserts: on a non-singular complex projective variety, every Hodge class (a rational cohomology class of type (p,p)) is a rational combination of cohomology classes of algebraic cycles (subvarieties cut out by polynomial equations). Nothing here establishes that statement; this programme delivers an honest strong negative about the reach of the discrete/structural setting.

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1. TNFR-Native Object: the Tetrad Cochain Tower

Example 107 established the k=1 Helmholtz–Hodge decomposition of the phase field on a graph (gradient ⊕ cycle). Extending the field to a 2-complex (triangles) gives the full discrete Hodge decomposition. The tetrad supplies the natural cochain degrees:

Degree	Tetrad object	Cochain
0	phase value	vertex 0-cochain
1	phase gradient `	∇φ
2	phase curvature K_φ	triangle 2-cochain (discrete curl / holonomy)

With simplicial boundary maps d1 (edges → vertices) and d2 (triangles → edges), the combinatorial Hodge Laplacians are

$$L_0 = d_1 d_1^{\mathsf T},\quad L_1 = d_1^{\mathsf T} d_1 + d_2 d_2^{\mathsf T},\quad L_2 = d_2^{\mathsf T} d_2 .$$

Eckmann's theorem (1944): harmonic k-cochains ≅ homology H_k, so dim ker L_k = b_k (the k-th Betti number).

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2. HC-1 Result (DONE)

Reproduced in examples/09_millennium/111_hodge_discrete_and_honest_gap.py.

• Chain complex. d1 d2 = 0 to machine precision (the tetrad cochain tower is a genuine complex).
• Eckmann, exact. On a triangulated torus (|V|=25, |E|=75, |T|=50, Euler 0): harmonic dimensions (dim ker L_0, L_1, L_2) = (1, 2, 1) = Betti (1, 2, 1). The 2 harmonic 1-forms are closed (|d1 h| ~ 1e-16) and co-closed (|d2^T h| ~ 1e-15).
• Topology tracking. Octahedron (sphere) gives harmonic dims (1, 0, 1) = Betti (1, 0, 1); the torus gives (1, 2, 1). The harmonic count is a faithful topological invariant: the sphere has no 1-loops, the torus has 2 (the loops a TNFR phase field can wind around; cf. examples/107).

HC-1 verdict: the TNFR cochain tower carries a complete discrete Hodge decomposition; harmonic = homology exactly, across spaces.

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3. The Honest Gap (why this is NOT the Hodge conjecture)

Two features constitute the difficulty of the Hodge conjecture, and the discrete TNFR setting has neither:

• A. Complex (p,p) bigrading. The conjecture lives in the Hodge decomposition H^k = ⊕_{p+q=k} H^{p,q} of a Kähler manifold, requiring a complex structure. The real combinatorial Laplacian L_k has no (p,q) bigrading — only one real harmonic space per degree.
• B. Algebraicity. An "algebraic cycle" is cut out by polynomial equations — strictly stronger than an "integer topological cycle." In the combinatorial setting every harmonic class is already an integer simplicial cycle (Eckmann), so the discrete analogue of the conjecture is trivially true — precisely because the discrete setting cannot even express the algebraicity distinction that is the whole difficulty.

So TNFR's discrete Hodge captures the topological half (harmonic = homology) exactly and is structurally blind to the complex-algebraic content. This is the honest analogue of the Riemann result that the emergent substrate is "blind" to arithmetic content.

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4. Honest Obstruction Classification

• Branch A (closure inside the catalog) — not applicable; the conjecture cannot even be posed discretely.
• Branch B (open attack surface) — not the right classification here. Unlike P-vs-NP (PNP-2) or BSD (BSD-2), there is no concrete discrete next milestone toward the conjecture: bridging to (p,p) bigrading and algebraicity requires leaving the discrete/structural setting entirely.
• Branch B3-leaning (no TNFR closure) — current classification. The discrete TNFR setting is structurally blind to the actual conjecture; the value delivered is a precise localisation of why Hodge is hard, not an attack surface on it.

This is the strongest negative of the TNFR Millennium programs. Where the Riemann S(T) residual, the NS cascade, the Yang–Mills continuum gap, and the P-vs-NP trapping are open obstructions with attack surfaces, the Hodge gap is a qualitative blindness: the discrete cochain tower cannot represent the algebraic-complex structure at all.

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5. Milestone Roadmap

HC	Title	Status
HC-1	Discrete Hodge on the tetrad cochain tower (Eckmann); honest gap	DONE (examples/111)
HC-2	Whether any TNFR-native complex structure induces a (p,p) bigrading	open, expected negative
HC-3	Whether algebraicity has any structural (non-topological) TNFR analogue	open, expected negative

HC-2 and HC-3 are recorded for completeness; the honest a-priori expectation is that the discrete/structural setting cannot supply either ingredient.

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6. What This Program Does and Does Not Do

Does: provide a TNFR-native discrete Hodge decomposition on the tetrad cochain tower; verify Eckmann (harmonic = homology) exactly; localise precisely the two features (complex (p,p) bigrading, algebraicity) that the discrete setting cannot represent; classify the obstruction honestly as the strongest negative (Branch B3-leaning).

Does not: prove, disprove, or even attack the Hodge conjecture; claim any bridge from discrete harmonic classes to algebraic cycles; introduce a complex or Kähler structure. The TNFR value-add is a precise honest delimitation of the reach of the structural setting — consistent with the disciplined pattern of the Riemann, Navier–Stokes, Yang–Mills, P-vs-NP, and BSD programs.