09 / C3-G: The First-Crossing Causal Frontier, Critical Shell Ancestry, and the Monotone Depletion No-Go

The largest gap left by C3-F is that an instantaneous interaction is not the same as a strictly earlier causal parent. This round instead uses the dimensionless critical shell amplitude $a_q^\sigma=\|u_q^\sigma\|_\infty/(\nu\lambda_q)$ (scale-invariant, and directly connected to the dissipation-wavenumber framework), defining the first crossing time $\tau_{q,\sigma}$ for each shell-sign node. The core result is the Critical First-Crossing Parent Lemma (Theorem 9.1): under an "eventual local-source dominance" assumption (earlier rounds established suppression of strongly nonlocal production, but this assumption has not yet been proved unconditionally for every possible blow-up, so this round's result is explicitly flagged as conditional), there is a fixed small threshold $\beta_\ast$ such that when a child first crosses the threshold, some local parent must already have crossed the same threshold earlier — a proof by contradiction using only the normalized Duhamel formula and the finiteness of the parent-type count. This gives the strict time ordering $\tau_p<\tau_c$, and the resulting Critical Activation DAG cannot contain cycles, filling the gap left by C3-F. It further proves the First Frontier Crossing Lemma (Theorem 15.1): high-frequency activity cannot "teleport" to an arbitrarily high frontier; it must first pass through a shell boundary of bounded width — combining this with the unboundedness of the dissipation wavenumber (from $\Lambda\notin L^{5/2}$ under blow-up), a Kőnig-type argument extracts a Conditional Ancestry Ray (Theorem 18.1): a causal path exists that is arbitrarily long, strictly time-ordered, and unbounded in frequency. But this round also tests another intuition — that "once a parent has passed energy to a child, it should permanently lose the corresponding resource" — and proves this cannot be derived from the conservation laws: energy and helicity conservation only constrain the transfer vector to a fixed one-dimensional signed direction $\dot{\mathbf e}=\Theta_\tau(t)\mathbf v_\tau$, with no constraint whatsoever on the sign of $\Theta_\tau(t)$. Proposition 26.1 gives an explicit counterexample: taking $\Theta(t)=\sin t$, this abstract transfer ledger satisfies triadwise energy and helicity conservation exactly at every instant, yet the energy repeatedly swaps back and forth — the conservation algebra alone is not enough to prove monotone depletion. Only viscosity provides a genuinely irrecoverable loss, but C2 already showed its cost shrinks to $O(\lambda^{-1})$, which remains summable along a geometric scale sequence. The research frontier shifts to C3-H: applying a critical rescaling to the causal genealogy already obtained, to see whether it forces out a renormalized limit object that collides head-on with existing backward-uniqueness/rigidity theorems.