Prospective Local Geometry of Operator Generated Response Boundaries

@misc{lucis2026prospective,
  title = {{Prospective Local Geometry of Operator Generated Response Boundaries}},
  author = {Jun Lucis},
  howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
  year = {2026},
  month = sep,
  doi = {10.5281/zenodo.22893911},
  url = {https://doi.org/10.5281/zenodo.22893911},
  note = {CC BY},
}

TY  - UNPB
TI  - Prospective Local Geometry of Operator Generated Response Boundaries
AU  - Jun Lucis
PY  - 2026
DA  - 2026/09/22
PB  - Zenodo (CERN European Organization for Nuclear Research)
DO  - 10.5281/zenodo.22893911
UR  - https://doi.org/10.5281/zenodo.22893911
AB  - This preprint extends the Boundary Information Geometry program from local accessibility and finite-resolution kinematic closure to second-order local geometry and prospective geometric prediction. For the finite-window response F(T,P)=Z[U_T(P)]-Z[P], B22 measures the restricted Hessian, boundary normal, tangent-space shape operator, principal curvatures and directions, and temporal normal rotation in the exact normalized four-dimensional Family-C subspace inherited from B20.5 and B21. B22-E1 established two-scale numerical stability of the full local Hessian and three resolved principal modes at an already observed r10 anchor. B22-E2 retrospectively reconstructed temporal normal rotation and was explicitly excluded from prospective evidence. The claim-bearing tests used one reserved lower-epsilon r09 branch. E3 froze a root and future-normal prediction before any exact T1 trajectory was evaluated and passed all predeclared gates. E4 then completed the observed T1 Hessian descriptively. E5 froze an equal-step one-step extrapolation before any T2 trajectory was evaluated. The observed root error was 0.008789, the h2 normal-angle error was 0.37444 degrees, the h2 Hessian relative error was 0.008591, and the largest resolved principal-direction error was 0.379 degrees. Every predeclared gate passed, and the terminal stop rule was reached. The result is limited to one tested finite synthetic Family-C branch and two successive held-out short-time stages. E5 uses the observed T1 geometry and is not a two-step-ahead forecast from T0 alone. No continuum curvature theorem, invariant manifold, physical-space boundary law, universal boundary equation, or Navier–Stokes blow-up or regularity claim is made. The upload includes the authoritative English preprint, a Japanese reference translation, and a reproducibility archive containing the B22-E1 through B22-E5 notebooks, manuscript sources, figures, numerical summary tables, claim boundaries, citation metadata, 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 Repositoryhttps://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. Publication audit note (2026-10-19). Audit scope: Basically reproducible; an independent rerun audit remains incomplete. Preservation and checksum checks, algebraic checks of saved results, source/protocol comparisons, original-code reproduction, and independent PDE reruns are distinct. No new PDE computation was performed for this publication update. Original numerical values, formal classifications, frozen rules and stopping rules are retained. B22-E2 is retrospective. Its approximately 0.2205% normal-rate propagation scale is distinct from the B21 gradient-rate reference scale. Exact future-stage evaluations in E3 and E5 were held out, while coarse r09 branch summaries were already known; this is not a historically blind branch discovery. Existing chronology is supported by code assertions and stored timestamps, not an independent cryptographic proof. The saved-result audit does not establish a complete independently rerun source archive. Version 1.1 notes (19 October 2026): The English authoritative preprint and Japanese reference PDF are replaced by the revised manuscripts supplied by the author. Clarifications concern E2 uncertainty interpretation and the evidence scope and sequential nature of E3/E5. Original numerical results, formal classifications and stopping rules are retained. The original v1.0 reproducibility ZIP is retained unchanged as a historical archive; manuscript copies inside it remain the original edition. Basically reproducible; an independent rerun audit remains incomplete. No new PDE calculation or independent rerun was performed for this publication update.
ER  -