The Projection–Admissibility Principle Descent, Recovery, and Structural Constraints on Effective Description

We replace the former noninvertibility obstruction by a typed theory of projection, descent, recovery, and admissibility. For an upper evolution, an initial reduction, and a final reduction, four different questions arise. A right section selects one compatible upper representative; a left decoder recovers the actual upper input; an autonomous reduced map exists exactly when upper evolution respects the initial equivalence classes; and an effective merger is noninjectivity of that reduced map. Noninjectivity of a cross-level projection does not forbid a right section. We prove the correct factor-through theorem, a valid no-right-section criterion for a genuinely contractive reduced self-map, a section-conditioning diagnostic, and locality descent for fiberwise coherent compression of a local operator net. Probability requires an upper measure and disintegration; entropy, irreversibility, and an arrow of time require additional structures. Applications to open systems, Wilsonian reduction, exterior gravity, and MTT are consequently reconstructions or conditional realizations rather than corollaries of noninjectivity alone.