Why the Identity Debate Did Not Converge: Eight Theories of Persistence and the Structural Conditions They Could Not Derive

The philosophy of personal and material identity has been pursued for twenty-five centuries without convergence. Substance theories, bundle theories, psychological continuity theories, process ontologies, four-dimensionalism, sortal-relative identity: each developed with rigour, each encountered determinate structural limits, none resolved the debate. The standard explanation is that the problem is extraordinarily difficult, perhaps intractable. This paper argues that the standard explanation is wrong. The debate did not fail to converge because the problem is intractable. It failed to converge because no tradition succeeded in formally deriving the complete set of structural conditions that any non-trivial persistence theory must instantiate. Six such conditions are identified for the general persistence problem: the identity-bearing structure (F), the transformation-processing capacity (M), their coupling (K), the constitutive threshold (δ_F), the existence condition (Q1), and the identity condition (Q2). These are not theoretical choices. They are structural necessities generated by the persistence problem itself. For the personal identity subdomain — self-modeling, Σ-complete persistence subjects — the F·M·K architecture additionally generates Q3 (Recursive Constitutive Non-Externality), Q4 (Structural Self-Priority), and Q5 (Recursive F·M·K Integration). These three further conditions are not general persistence conditions but domain-specific consequences of the triadic architecture applied to self-modeling systems (P167). No tradition in the eight positions derives the full Q1–Q5 architecture for persons. The eight major traditions each independently instantiated some of these conditions and encountered determinate structural limits where further derivation was no longer possible from within their own resources. La Profilée (LP) is not a ninth theory in this debate. It is the formal derivation of the six conditions from three minimal assumptions — distinguishability (M1), real transformation (M2), and non-trivial persistence relation (M3) — and the formal proof that this derivation is exhaustive: no further structurally independent persistence condition exists within the admissibility class defined by M1–M3. The diagnosis this paper offers proceeds in three steps: identification of the six structural conditions; location of the eight traditions as partial instantiations in the resulting space; and derivation of four structural laws that explain why each tradition encounters the structural limit it encounters. Once the conditions are formally derived, the limits of the traditions become structurally locatable — not as failures, but as determinate structural limits generated by incomplete access to the same structural necessities.