Structural Collapse of Admissible Second Variations in High Dimension

This paper develops a finite-rank theory of admissible second variations in high-dimensional response systems. The main idea is that a system may contain many primitive or readable directions, but only a uniformly bounded finite core can remain simultaneously admissible when admissibility is controlled by a finite-dimensional margin structure. The paper distinguishes raw variation, readable variation, and admissible variation, and shows that these notions should not be identified in high-dimensional settings. The primitive object is an ordered second-order response form, whose symmetric part determines the readable quotient and whose alternating part contributes through compatibility burdens rather than scalar quadratic curvature. Admissibility is formulated through finite margin spaces, burden assignments, and compatibility classes. Under independent burden growth, every genuinely new admissible core direction consumes a new independent margin degree of freedom. Since the margin rank is finite, the admissible core is uniformly finite-dimensional. The paper also discusses stabilization of the admissible core, finite-window codimension growth, and a primitive-to-margin projection mechanism explaining how high-dimensional response data can give rise to finite admissibility structures. The result is a structural collapse theorem: high-dimensional readable abundance does not imply high-dimensional simultaneous admissibility. Instead, admissibility is controlled by a finite projected margin rank.