ANP-05: Arbitrary-Depth Compatible C3 Paths, No-Terminal-Node, Adaptive Generation Renormalization, and the Singular-Horizon Compactness Gap

Arbitrary-Depth Compatible C3 Paths, No-Terminal-Node, Adaptive Generation Renormalization, and the Singular-Horizon Compactness Gap

ANP-04 completes universal causal-state entry (Type-I entry or non-Type-I adaptive entry); ANP-03 proves a weighted quasi-local source-parent edge (C3_W) for strongly selected source atoms. The remaining question: can these legal single-step edges actually be iterated to arbitrary finite depth? This round proves: yes. Every positive canonical weighted shell node has an earlier legal C3 parent node over any nonzero pre-singularity backward interval — either the same weighted shell state is inherited, or the exact shell-by-shell source ledger contains a positive dyadic source atom, giving a recursively legal weighted C3 parent node per ANP-03. This yields a "no-terminal-node" theorem; after adaptive generation-by-generation time renormalization, obtains an arbitrary finite-depth compatible C3 path starting from any late-time Type-I or non-Type-I entry seed, with each finite-depth path's transport product strictly positive. This round's inheritance criterion was later flagged as too strong by a correction note accompanying ANP-06 (see the correction below), but the conclusion itself — "arbitrary finite depth is realizable" — is unaffected.