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 CartanEinstein frame dialogue to a computable complex-frame algebra, through the ChernSimons (CS) currenthelicity 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 rst-class algebraic object \(E=(o, s, R) \in S E(3)\) yields machine-precision curvature via the commutator \([G_{\mu}, G_{\nu}]\) of the intrinsic gradient \(G_{\mu}\) .A \(U(3)\) complex frame eld \(U(x)\) supplies the phase layer; its three gauge freedoms (rotational →YangMills, scaling →Weyl metric, translational →CS torsion) produce the three interaction sectors. (2) Mathematical bridge. The local helicity density \(h(x)=u \cdot \omega\) is the hydrodynamic pullback of the Abelian CS 3-form under \(A=\kappa u^{b}-a\) 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 roundo), reaching 74.2%100% accuracy across noise, grid, and short NavierStokes evolution [1]. (b) In solar magnetograms (NOAA AR 11158), the CS-current projection produces a residual \(\Delta h\) concentrated at the polarity inversion line: core \(|\Delta h / h_{proxy }| ≈2 \%\) vs. PIL 122%, with PIL sign entropy \(\Lambda^{*}=0.999\) . (c) A 2562periodic DNS with a synthetic bipolar eld reproduces this pattern: core 4.6% vs. PIL 120% (26× contrast). (4) Candidate dynamics. The CS gradient \(\nabla_{(\mu} K_{\nu)}\) is a source, not an inertia term. The proposed topological inertia is a propagating constraint eld \(\sum _{\mu \nu}:(\square+m_{\sum }^{2}) \sum _{\mu \nu}=\frac{i \lambda}{32 \pi^{2}} \nabla_{(\mu} K_{\nu)}\) , with falsiable prediction \(m_{\sum } ~ 1 / L_{0}\) . (5) Global branch. Beyond the local eld equation, a projected topology-state free-energy landscape \(F_{top }(\xi)=-E_{0} \ln P(\xi)\) operates on reduced state coordinates \(\xi=\Phi[u]\) built from CS-derived observables. Near saddle-node bifurcations, the metastable gap scales as \(\Delta E \propto|\delta|^{3 / 2}\) . This is a phenomenological extension (T2T3), not a rst-principles consequence of the frame algebra. We state explicitly what is established (links 13) and what remains a candidate (links 45). 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.