08 / C3-F: Phase-Space Quasi-Locality, the Ancestry Cone, and the Finite-Branching Reversal

C3-E compressed the local heterochiral survivor down to three simultaneously necessary conditions: rapid viscous renewal, phase/amplitude efficiency, and a provenance-preserving genealogy. This round, for the first time, brings physical space directly into the proof route. The core is the Off-Diagonal Critical Interaction Lemma (Theorem 3.1): the kernel of the annular Leray nonlinearity, at frequency scale $\lambda_q$, has Schwartz-class off-diagonal decay $(1+\lambda_qd)^{-N}$ — the farther apart the parents' spatial supports, the faster the direct interaction contribution decays; no turbulence-scaling assumption is used, this is pure Fourier geometry. Cutting space into admissible dyadic packets of side length $O(\lambda_q^{-1})$, it proves each output packet has only finitely many core parent tuples (Proposition 7.1), with the tail contribution controlled by Theorem 8.1's packet tail bound. Locality–Coherence Tradeoff (Theorem 10.1): if the phase efficiency $\eta_q$ is not too small, one can fix a radius $R_\ast$ independent of $q$ such that at least half of the actual positive production is attributable to the spatial parent core within this radius; as $\eta_q\to0$ the required radius grows, but as long as it does not collapse at superpolynomial speed, the radius remains far smaller than the macroscopic scale. Combining this with the viscous-window renewal from C3-E, it proves Ancestry Center Convergence and the Parabolic Ancestry Cone Theorem (Theorem 19.1): the coherent genealogy's spatial center converges to a single point $x_\ast$ and its time converges to $T_\ast$, with errors compressed to $O(\lambda_n^{-1})$ and $O((\nu\lambda_n^2)^{-1})$ respectively — geometrically compatible with Barker–Prange's Type-I concentration result, though the document explicitly states the two are independent theorems and must not be substituted for one another. The paper's most important course correction is in §22: the intuition that "each parent has only finitely many children" sounds like it should rule out an infinite cascade, but this is wrong — by Kőnig's infinity lemma, a rooted tree with a finite root set, finitely many children per node, and arbitrarily large depth must contain an infinite ray. So finite branching does not annihilate infinite paths; it instead compresses a vague, aggregate cascade into a concrete, extractable genealogy. The document splits C1c into three sub-problems: C1c-a (static packet source selection, for which this round gives a finite-core selection lemma), C1c-b (infinite-path extraction, a purely combinatorial theorem), and C1c-c (causal direction, still missing) — the genuine remaining gap is that an instantaneous interaction graph at a single time can contain cycles (A→B and B→A simultaneously), which is not the same as strictly earlier causal parenthood; genuine causality can only be established using Duhamel time ordering.