Local Incidence Profiles of Segment Arrangements in a Quadrilateral: Finite Realization Order, Exact Strict Classification through N=4, and Endpoint-Meeting Sectors at N=3

Version v2.5r1 This work introduces a typed local-incidence profile for finite arrangements of straight-line segments inside a convex quadrilateral with labelled corners. The profile is a polynomial obtained from the planarization vertices of the original segment family. Its variables distinguish corner incidences, non-corner boundary endpoints, transversal crossings of full segments, and interior endpoint-meeting vertices. The profile is deliberately coarser than the complete embedded arrangement. The main inverse problem is therefore to determine how many corner-labelled topological types realize the same profile. This number is called the realization order, or equivalently the profile multiplicity across topological types. The paper proves that every realizable profile has finite realization order and establishes an endpoint-incidence identity that recovers the number of original segments from the profile. It also derives a defect identity for strict triangulations and uses it to obtain exact classifications for small segment families. The principal results include: exact strict classification for $N = 1, 2, 3, 4$, with respectively $$2,\ 13,\ 60,\ 245$$ corner-labelled topological types; exact coarse-profile realization orders throughout the strict $N \le 4$ classification; rigidity of the corner-and-side refined profile throughout the strict $N \le 4$ range; exact classification of the maximal-corner one-leaf sector at $N = 3$, including $$\mu(4x+y+z+u) = 12,$$ $$\mu(4x+y+2z+u) = 24,$$ $$\mu(4x+y+3z+u) = 24;$$ complete classification of the degree-two interior endpoint-meeting sector at $N = 3$, consisting of $896$ boundary skeletons, $1792$ corner-labelled topological types, and $28$ coarse profiles; complete classification of the trivalent-star sector at $N = 3$, consisting of $88$ topological types and four coarse profiles; an explicit polynomial lower bound showing that realization order is unbounded; a bounded order-two U-statistic formulation for the crossing coefficient in a random endpoint model. The proof architecture separates geometric normal-form reductions from finite exact enumeration. The accompanying reproducibility package contains the LaTeX source, three deterministic Python verifiers, machine-readable JSON verification reports, complete CSV type and profile ledgers, licenses, a manifest, and SHA-256 checksums. The verification scripts use only the Python standard library, reject optimized python -O execution, and do not rely on floating-point arithmetic, random sampling, or proof-critical assert statements. The profile is defined directly from the original segment family and its planarization vertices. It is independent of auxiliary constrained edges used only to triangulate residual faces. The present version does not claim a complete classification of all non-strict $N = 3$ arrangements; it closes the strict $N \le 4$ range and the stated endpoint-meeting sectors at $N = 3$. Author: Lee Byoungwoo(이병우) leeclinic@protonmail.com , (Daejeon Math Forum, South Korea)