The Independence of Existence and Identity: Why Persistence Theory Requires Q1/Q2 Separation

Every theory of persistence must answer two questions: does this system continue to exist? And: does the continuing system remain the same system? These are not the same question. La Profilée formalizes the first as the persistence condition IR ≤ 1 and the second as the Frame Continuity Condition (FCC). The Q1/Q2 Separation Theorem establishes that Q2 presupposes Q1, while Q1 does not imply Q2. This paper presents the formal proof of structural non-equivalence, the four-regime architecture that follows (Persistence, Transmutation, Collapse, and Dissolution, with one structurally impossible combination that closes the space), and the Asymmetry Theorem: Q2 (identity) implies Q1 (existence), but Q1 does not imply Q2. Identity without existence is structurally impossible; existence without identity (Transmutation) is structurally possible. The fourth regime — ¬Q1 ∧ Q2 — is not a missing case but a structural impossibility. The paper demonstrates Q1/Q2 separation across six domains: gradual cognitive change, material replacement, organisational identity, personal development, political continuity, and mind-upload scenarios. In each domain, the four-regime architecture generates determinate structural verdicts where single-condition theories produce indeterminate or contradictory results. The structural limits identified in P164 are therefore not contingent features of particular philosophical frameworks. They are the forced consequences of attempting to answer two structurally distinct but asymmetrically related questions with a single condition. The formal proof makes this visible. This paper provides the formal proof. Theorems 1–5 establish that Q1 and Q2 are structurally non-equivalent and independently derived, with asymmetric dependence: Q2 presupposes Q1, while Q1 does not imply Q2. The Asymmetry Theorem establishes that the independence is not symmetric: Q2 presupposes Q1 (identity presupposes existence), but Q1 does not imply Q2 (existence does not guarantee identity). Together, the two theorems generate a complete four-regime architecture and explain why every tradition that derived one condition without the other encountered indeterminate verdicts in precisely the cases where the two conditions diverge. The structural diagnosis of the identity debate established in the companion paper (P164) identifies a recurring pattern across eight major traditions: Parfit glimpses the Q1/Q2 separation without formalising it; Sider derives Q1 with exceptional precision while importing Q2; Wiggins correctly identifies F-selection without deriving the admissibility conditions that Q2 requires. In each case, the structural limit is not accidental. It follows from the fact that Q1 and Q2 are structurally distinct but asymmetrically related conditions that cannot be jointly derived from any single-entry-point approach to the persistence problem.