Projective Persistence of Full Histories: Compact Reconstruction, Max--Inf Margins, and Extension Recursions

We study the passage from compatible witnesses of every finite horizon to a single full history. For inverse systems of compact spaces we separate the existence of a compatible family from the stronger lifting of every prescribed prefix. An upper semicontinuous worst-margin functional attains the exact max–inf threshold on the space of full histories. The history spaces of finite horizon then form their own compact tower, so that local losses compose into a single limit cost. Layer mixing gives the exact minimax of the common margin and a sharp harmonic lower bound from the margins of the individual layers. Bellman recursions determine the optimal extension deficit, separate the existence of a good branch from the persistence of the whole tree, and yield a single measurable policy realizing all levels of the existential value. Lifting of terminal segments and an isometric step decomposition turn a contraction on the terminal quotient into an explicit lower bound on the residual, and a descent budget carries a common history without blurring and with controlled landing. Explicit families of countermodels separate compactness, compatibility, uniform margin, readout, survival mass, and local selection from the full Bellman profile. Cascades of full Gram matrices give a realization of these results with control of the complex entries and of the tail error. The relative cost detects a drop of the active rank through an infinite barrier and characterizes the persistence of a positive spectral gap, also for redundant columns and dependent choices of layers. The q³D realization with a nonzero regulator yields both persistence thresholds and a common limit of the full record in the physical sector of sixteen modes. The geometry of histories thereby determines the attainable margins.