Local Helicity as a Chern-Simons Probe: Frame-Algebra Foundation, Controlled Numerical Discrimination, and a Candidate Propagating Constraint Field

This paper traces a research line from the Cartan-Einstein frame dialogue to a computable complex-frame algebra, through the Chern-Simons (CS) current-helicity identity, to controlled numerical and observational evidence for a projected topological sector. The argument has four links: (1) Frame algebra. Promoting the frame to a first-class algebraic object E = (o, s, R) ∈ SE(3) yields machine-precision curvature via the commutator [G_μ, G_ν] of the intrinsic gradient G_μ. A U(3) complex frame field 𝕌(x) supplies the phase layer; its three gauge freedoms (rotational → Yang-Mills, scaling → Weyl metric, translational → CS torsion) produce the three interaction sectors. (2) Mathematical bridge. The local helicity density h(x) = u · ω is the hydrodynamic pullback of the Abelian CS 3-form under A = κu³ → a pointwise identity, not a global integral. (3) Evidence. (a) In controlled incompressible vortex tubes, the local helicity distribution distinguishes states degenerate under core geometry (33.3% baseline) and global helicity (33.3% matched to roundoff), reaching 74.2%–100% accuracy across noise, grid, and short Navier-Stokes evolution [1]. (b) In solar magnetograms (NOAA AR 11158), the CS-current projection produces a residual Δh concentrated at the polarity inversion line: core |Δh/h_proxy| ≈ 2% vs. PIL 122%, with PIL sign entropy Λ* = 0.999. (c) A 256² periodic DNS with a synthetic bipolar field reproduces this pattern: core 4.6% vs. PIL 120% (26× contrast). (4) Candidate dynamics. The CS gradient ∇(μK_ν) is a source, not an inertia term. The proposed topological inertia is a propagating constraint field Σ_μν: (□ + m_Σ²)Σ_μν = (iλ/32π²)∇(μK_ν), with falsifiable prediction m_Σ ∼ 1/L₀. (5) Global branch. Beyond the local field equation, a projected topology-state free-energy landscape F_top(ξ) = −E₀ ln P(ξ) operates on reduced state coordinates ξ = Φ[u] built from CS-derived observables. Near saddle-node bifurcations, the metastable gap scales as ΔE ∝ |δ|^(3/2). This is a phenomenological extension (T2-T3), not a first-principles consequence of the frame algebra. We state explicitly what is established (links 1–3) and what remains a candidate (links 4–5). The Christmas equation form from earlier work is retained for continuity but corrected: the CS gradient belongs on the RHS as a boundary-source input, not on the LHS as a bulk inertial term.