Intrinsic vs Extrinsic Topology — Outcome Classification Across Constraint Regimes

This Related Signal Note (Taxonomy) establishes a minimal, outcome-based classification of topological structure across the current Tuckwell corpus. Systems are partitioned by what the governing constraint does to the topology: whether it forces the topology intrinsically, permits it under external boundary conditions, or reconstructs it analytically without physical enforcement. Three mutually exclusive classes are defined: Class 1 — Forced (Intrinsic Constraint)Topology necessarily arises from internal constraints. This class is subdivided by tier:T1 (algebraic/spectral) — topology follows from mathematical conditions on the manifold (e.g., commuting S¹ actions in the D3 framework).T2 (physical/material) — topology arises from internal physical coupling in a real system (e.g., lattice mode coupling in bulk perovskites). Class 2 — Permitted (Extrinsic Stabilisation)Topology is stabilised by external boundary conditions (interfaces, strain, fields, or processing). The topology is possible but not necessary and may not persist if the boundary is removed. Class 3 — Reconstructed (Analytic/Representational)Topological structure appears in a mathematical representation (e.g., wavefunction or action) but is not enforced by the underlying physical manifold. The topology is reproduced in the model, not forced by the system. The taxonomy explicitly separates outcome from mechanism and enforces a strict T1/T2 tier distinction. Cross-tier reference is permitted; cross-tier equivalence claims are excluded. The classification is designed as a minimal mapping layer for the corpus, not a survey of mechanisms or literature. The current corpus anchors include: -D3 main paper (T1 forced; algebraic necessity condition) -Xu et al. RSN (T2 forced; intrinsic topological order in bulk material) -Lohmiller–Slotine RSN (reconstructed; analytic constraint without physical topology) -Prior art cluster (permitted; extrinsic stabilisation in thin films and interfaces) This document introduces no new empirical results. It formalises existing corpus entries into a coherent classification system to prevent fragmentation and to provide a consistent vocabulary for comparing constraint regimes. This note is not peer reviewed and is filed as a corpus classification layer. Keywords topology classification, intrinsic topology, extrinsic topology, constraint-driven systems, forced topology, permitted topology, reconstructed topology, boundary conditions, algebraic topology, spectral methods, perovskites, ferroelectricity, octahedral rotations, phonon coupling, topological materials, TRIDENT framework, TTR-1, D3 topology, constraint regimes, system classification, condensed matter physics