BU50 Appearance-Side Minimal Admissible Structure under B_U First Lawful Nonzero Retention, Finite Projection Audit, and the Reclassification of Formal Divergence / Finite Energy

Abstract This paper develops BU50 as a structural account of the appearance-side minimal admissible structure under Single-Universe Boundary Consistency, B_U. Its central claim is that the first lawful “nonzero” item on the appearance side is not a minimal object, particle, or epistemic observation limit, but the smallest admissible retained structure that survives microscopic zeroing, finite projection, coherence audit, and settlement. Total reality cannot enter the phenomenal ledger as an unfiltered ontic intensity. It must pass through B_U, the fixed boundary B_0, the dynamically complete interface D(M), and the fast-slow decomposition in which microscopic fast variables are projected into a zero-mean subspace while retainable slow coherent components become candidates for standing. BU50 formalizes this transition through an admissible set \mathcal A_{L_n} and a scalar evaluation functional J_{L_n}. The appearance-side minimal structure S_{\min,L_n} is defined as the lowest-cost, nonzero, settleable item within the current resolution window, satisfying microscopic zero-debt, fast-slow orthogonality, coherence, and settlement predicates. This prevents the minimum from collapsing back into background zeroing while preserving its status as a lawful finite readout. The paper further reclassifies formal energy divergence. A \delta-type local projection produces arbitrarily high-frequency spectral components and may yield formal infinite energy in backward-mirror expressions. Physical finitude arises through finite resolution, finite cutoff, and bounded projection, by which ontic energy becomes readable, auditable, amplifiable, and settleable on the appearance side. BU50 concludes that the appearance-side minimal admissible structure is the first lawful nonzero standing after the cleared background. It is the seed of reality that consciousness and civilization can recognize, amplify, combine, and write back into the total-reality network as higher-order standing.