Operator-Generated Response Boundaries in Finite-Window Vorticity Dynamics: Accessibility, Local Normal Geometry, and Prospective Crossing Tests

This work presents the B20 stage of Boundary Information Geometry (BIG), an operator-level numerical study of finite-window response boundaries in synthetic divergence-free vorticity dynamics.

We define the finite-window response

F_T(P) = Z[U_T(P)] - Z[P],

where P denotes a finite parameterization of the initial vorticity field, U_T is the finite-window viscous periodic evolution operator, and Z is the half-enstrophy readout. The associated response boundary is the zero set B_T = {P : F_T(P) = 0}.

B20.4 examines boundary accessibility in a frozen finite parameter family. The results show that accessibility is direction-dependent and cannot be reduced to proximity to the zero level alone. Multiple crossings occur in some parameter-time cells, so the response boundary is not generally representable as a single-valued scalar threshold.

B20.5 reconstructs local response gradients within a predeclared normalized four-dimensional parameter subspace. Gradient estimates are highly stable across two finite-difference scales. In prospective hold-out tests, predicted and observed directional-derivative signs agree in 24/24 comparisons, supporting the interpretation of the reconstructed gradient as a local response-boundary normal within the tested subspace.

B20.6 performs a prospective crossing test on a new frozen synthetic family. Local one-sided response information is acquired before the crossing scan and used to freeze linear crossing predictions. Of six independent parameter-space trajectories, only one informative crossing occurs within the predeclared scan domain. Its orientation is predicted correctly. The frozen crossing estimate is s_hat = 1.960017, while refinement gives a zero-crossing bracket [1.9921875, 1.994140625], with midpoint 1.9931641, corresponding to a location difference of approximately 1.66% of the refined midpoint.

Because the predeclared B20.6 PASS criterion required multiple independent informative crossings, B20.6 is classified as INCONCLUSIVE rather than PASS. No post-hoc domain extension, trajectory replacement, or success threshold is introduced.

The results support a limited operator-level picture in which finite-window response boundaries possess direction-dependent accessibility and measurable local response geometry that can be tested prospectively. All conclusions are restricted to the tested finite synthetic vorticity families and finite observation windows.

This work does not establish a Navier–Stokes singularity, regularity criterion, universal separatrix, invariant manifold, or physical-space refraction law.

The accompanying reproducibility archive contains audited numerical results, frozen predeclarations, parameter manifests, prospective prediction records, response tables, refinement records, claim boundaries, and SHA-256 checksums.

Project repository

For the current research map, related BIG studies, representative figures, terminology, and links to the broader Boundary Information Geometry (BIG) research program, see:

Boundary Information Geometry (BIG) — GitHub Repository
https://github.com/Jun-Lucis/BIG-theory

The Zenodo record remains the archival and citable source for this research output. The GitHub repository serves as a navigational overview of the continuing BIG research program.