ANP-04: Non-Type-I Ancestry Entry, Adaptive Weak-L³ Seeds, UV Square-Tail Extraction, and Universal Causal-State Initialization

Non-Type-I Ancestry Entry, Adaptive Weak-L3 Seeds, UV Square-Tail Extraction, and Universal Causal-State Initialization

ANP-00 through 03 build a recursively legal weighted C3 causal edge on the Type-I branch, conditioned on a strongly selected source atom. The remaining universal-entry problem: on the branch sup_(t<T*) ‖u(t)‖_(L^(3,∞))=∞, no single finite constant can define a uniform Type-I core geometry. This round proves this does not block causal-state initialization. The correct substitute is adaptive: adaptive amplitude, adaptive frequency, adaptive footprint, adaptive causal time step. From weak-L³ super-level-set geometry and the global finite-energy bound, extracts instants t_n→T*, amplitude levels λ_n→∞, a contracting effective volume scale r_n, and a dyadic threshold J_n→∞ such that low frequencies cannot account for the super-level-set amplitude, obtaining a quantitative global UV vorticity square-tail lower bound of order M_n⁴/E_2². Although no globally uniform weak-L³ bound exists, every pre-singularity seed admits an adaptive backward causal window Δ_n~R_n²/(1+M̄_n) over which ANP-02's footprint aperture remains uniformly controlled. Combined with the Type-I core entry, obtains a universal two-branch causal-state entry theorem.