ANP-06: Singular-Horizon Infinite Ancestry Extraction, Dual-Propagator Correction, Horizon Persistence, and Chain-Necessity Closure Audit

Singular-Horizon Infinite Ancestry Extraction, Dual-Propagator Correction, Horizon Persistence, and Chain-Necessity Closure Audit

ANP-05 claims that for any N, a compatible C3 ancestry path of depth N exists, and correctly notes this does not imply an infinite chain exists. But before attempting infinite extraction, one inheritance step needs correction: the condition e_k^χ(a)>0 only proves an earlier state exists, not that this earlier state makes a forward propagation contribution to a later-time child observable — the state may already have been dissipated/canceled, with the later state force-reconstructed. This round first repairs this causal ledger: replacing bare earlier-weighted-shell-energy positivity with an exact dual-propagator Duhamel contribution, proving a genuine forward-causal-inheritance-or-source dichotomy, and a finite-depth continuation theorem for the corrected dual/footprint causal-node class. Abstractly proves: arbitrary finite-depth realizability does not imply the existence of an infinite branch with no finite branching, compactness, or horizon persistence. Separates the trivial full-solution causal spine from the strongly-marked/source ancestry chain that Chain Necessity actually seeks. Defines the horizon-persistent node, proves a sufficient recursive-persistence criterion for an actual singular-horizon chain. Identifies the missing theorem as preserving/extracting a horizon-persistent marked C3 branch, not local source-parent existence.