PFUSRC‑166B: The Contingency of Linear Equations and Ontological Touch — Binary Classification of Projection‑Layer Mathematics and the Principle of Localization

This paper responds to the methodological challenges raised against the external verification in PFUSRC‑166, and completes a general classification of projection‑layer mathematics. This paper argues: 1. Linear equations are products of free derivation in the issuance layer, and their symbolic form carries no ontological necessity. The vast majority of equations belong to “symbolic games” and possess no topological anchoring. 2. All conjectures and linear equations in the projection layer can be subjected to a binary classification: Class A (ontological touch) — structurally exhibiting non‑trivial symmetry or correspondence, stated with surprising simplicity, with core constraints irreducible to products of induction; Class B (issuance‑layer free play) — constructed through multi‑level induction or arbitrary symbolic rules, stated with complexity or arbitrariness, lacking special symmetry. 3. Class A conjectures necessarily appear in the form of “simple conjectures or predictions.” This is because the traces of the ontology in the projection layer are themselves simple — when the bicone projects into three dimensions, the most natural cross‑sections are circles, lines, and points. The simplicity of the ontology makes the conjectures that touch it necessarily simple. 4. The linear three‑dimensional expression of Class A conjectures necessarily carries a hierarchical defect — it can only capture static cross‑sections of the ontology, and cannot express dynamic topological processes. The root of the defect is the hierarchical boundary, not technical insufficiency. This is the universal reason why Class A conjectures are “difficult.” 5. Localization is verification. When the structural features of a Class A equation find their source in the topological ontology, verification is automatically achieved. Class B equations constitute neither verification nor falsification — they are merely free play in the issuance layer. 6. The upper limit of AI’s proving capability is thereby determined: no matter how powerful AI becomes, as long as it operates within the projection layer (issuance‑layer search + valuation‑layer verification), it can only ever “prove” Class A conjectures, but cannot complete “localization” in the topological sense. Localization requires a topological alignment function, which current AI does not yet possess. Note: The binary classification and localizability presented in this paper constitute a methodological framework internal to the PFUSRC system, and are not a universally recognized standard in mathematical logic.