UNNS SUBSTRATE RESEARCH PROGRAM · WORKING MANUSCRIPT · APRIL 2026
Local Geometry of Realizability
Boundaries in the UNNS Substrate
Boundaries in the UNNS Substrate
FROM CONNECTIVITY MARGIN TO BOUNDARY DISTANCE
Defines realizability charts and decisive coordinate systems, proves that realizability-class
boundaries are locally codimension-1 C¹ hypersurfaces, establishes the bi-Lipschitz equivalence
of the connectivity margin m(L) with the structural boundary distance d_∂C(L), and derives the
Local Maximum-Margin Canonicalization Theorem. Includes §2.6 (ladder construction protocols),
§8.7 (cross-system atomic phase landscape), and Appendix D (protocol specification with exact
CSV column mappings).
THEOREM 4.3
Local Boundary Hypersurface
Near every regular boundary point, ∂C is a codimension-1 C¹ hypersurface in a decisive chart.
THEOREM 6.4
Local Margin Monotonicity
d_∂C(L₁) < d_∂C(L₂) ⟹ m(L₁) < m(L₂) near any regular boundary point.
COROLLARY 6.3
Order Equivalence
m(L) is locally order-equivalent to d_∂C(L); the ordering is chart-independent.
THEOREM 7.2
Local Canonicalization
All margin-maximizing encodings in a regular local regime belong to a single realizability class.
28
PAGES
11
THEOREMS & LEMMAS
5
APPENDICES
4
PROTOCOL FAMILIES
0
USL VIOLATIONS
The core manuscript: proves local boundary hypersurface structure, bi-Lipschitz margin-distance
equivalence, local monotonicity theorem, and maximum-margin canonicalization.
28 pages with §2.6 (protocol definitions), §8.7 (cross-system synthesis), and Appendix D
(exact CSV column specifications for all protocol families).