Partial Closure: Identity, Invariance, and Persistent Organization Through Relational Incompletion

The ordinary open/closed binary is inadequate for systems whose complete states remain variable while a deeper relational organization persists. We develop partial closure as persistent carrier retention under nonzero residual openness. A partial-closure system is represented by , where is a state space, an admissible evolution, an identity-bearing closure carrier, and a residual-openness functional. We distinguish state, invariant, relation, rule, domain, and lineage carriers; exact, approximate, and regenerative preservation; and multiple non-equivalent closure axes. A quotient-closure theorem shows that state-level nonclosure can coexist with exact closure in a carrier-defined quotient. An axis non-equivalence result establishes that no single scalar closure measure can generally classify multidimensional closure. Iterated-function-system fractals provide exact rule-closed but finite-form-open exemplars. For quantum mechanics, joint-state purity and local mixedness give an exact relation-closed/local-open profile, while a composition theorem derives the quadratic channel weight under explicit assumptions. For turbulence, we formulate a matched-amplitude test in which Fourier amplitudes are preserved while phase organization is altered through structured nulls. A translation-invariant triad-phase descriptor is required to predict future nonlinear transfer beyond energy, enstrophy, and dissipation; targeted phase disruption and matched repair provide intervention and rescue tests. The proposed cross-domain principle is restricted: quantum mechanics, fractals, and turbulence are not identified physically, but classified by the level at which identity is retained through incompletion. The paper concludes with falsification conditions, claim ceilings, and a staged computational program. Keywords: partial closure; relational identity; fractals; quantum foundations; turbulence; phase organization; closure carrier; regenerative dynamics; matched-amplitude nulls; Born rule.