Constraint Inheritance and Entropy-Capacity Contraction in OUXSPACE

This preprint establishes a proven finite-state and local-geometric core for the OUXSPACE program of hierarchical self-organization. P1 proves the exact capacity-contraction identity ΔS_cap=-k_BΔC under a fixed master partition. P2 proves the exact Gibbs-Shannon/Kullback-Leibler decomposition S_GS=S_cap-k_BD_KL(p||u)+S_intra. P3 proves the regular local rank-surplus identity F_{n+1}-F_n=g-q, and P4 gives its gauge-aware extension. A transparent non-OUXSPACE toy model demonstrates an actual q>g case and the corresponding finite-state capacity contraction. The OUXSPACE nuclear source sequence is recast here as Operative Manifolds OM-D1 through OM-D11. For alpha closure, this revision goes beyond an unspecified f_comp by declaring a periodic compensation potential V_comp=κ_α[1-cos(ξ̃12-ξ*)] and the associated regular residual contribution f_comp=κ_α sin(ξ̃12-ξ*), κ_α≠0. Consequently, once a physical non-gauge OM-D1 product tangent with dξ12≠0 exists, alpha transversality is mathematically proven for this declared minimal operator and yields q=1, g_cross=0 and ΔR=+1. The existence of that physical OM-D1 tangent and the physical adequacy of the declared operator remain prospective, so alpha is conditionally proven within the operative model but not yet physically confirmed. The paper additionally connects the formalism to OM-D4-OM-D11 nuclear architecture, shell-bridge closure, published μ/τ no-fit audits (formalized here as verified numerical propositions P8-P9), and a biological constraint-intersection corollary. These extensions are status-separated so that numerical support is not confused with theorem-level proof or universal thermodynamic claims. Scope. The proof layer consists of P1-P7 and their stated assumptions. Operative Manifolds OM-D1-OM-D11, shell-bridge, μ/τ, and biological closure are included as explicit applications, operative extensions, or support tests rather than being silently used to prove the finite-state identities. The stronger claim that total thermodynamic entropy universally decreases with organization remains outside the proven core. P8-P9 are verified numerical propositions, not members of the theorem-level proof layer.