.ZLT-P77 — Atomization Failure: Structural Impossibility of Zero Leap Convergence Under Buffer Depletion

A system can know exactly what to do, have the structural plasticity to do it, and still be unable to act — because the buffer that makes reorganization admissible has run out. This is Atomization Failure: the third and final failure mode of the Zero Leap Convergence triple.

Unlike Idolatry Failure (P75), where the system is deceived by a corrupted proxy, and Pharisee Failure (P76), where the system is frozen by structural rigidity, Atomization Failure is fully symptomatic. The system cannot sustain normal function under load. Every perturbation propagates into damage. The buffer gap is directly measurable. The paradox is that perfect coherence (Pi > 0) and intact plasticity (K > 0) are not enough: without kappa ≥ kappa_min, no V_minus_2 reorganization pathway is admissible, and convergence is structurally impossible.

This paper proves three theorems. The Atomization Failure Theorem (P77.T1): kappa < kappa_min implies liminf R(t)/I(t) ≥ ε > 0 under NDb and NOT-P3 — within the regime Pi > 0, K > 0 — regardless of their specific values. The Depletion Monotonicity Theorem (P77.T2): under NDb with kappa < kappa_min, if no external kappa injection occurs, kappa is non-increasing — the system cannot self-restore buffer via internal mechanisms alone. The Irreversibility Threshold Theorem (P77.T3): there exists a threshold kappa_irr < kappa_min below which even external kappa injection becomes inadmissible under NOT-P3, closing the last recovery pathway.

Three new diagnostic quantities are introduced: the Atomization Rate (λ), measuring buffer depletion velocity; the Half-Depletion Time (T_half), measuring time to kappa = 0; and the Exhaustion Index (EI), measuring depth past kappa_min. A fourth quantity, the Structural Fragility Index (Gamma), unifies BRC and EI into a single monotone signal valid across the kappa range.

Cross-domain validation covers 12 domains including organ failure under ischemia, sovereign debt crisis, species extinction, post-conflict institutional collapse, and AI context exhaustion. The paper closes the Structural Failure Mode Triad (P75–P76–P77): Idolatry, Pharisee, and Atomization are the only three structural failure modes consistent with the ZLC axiom set. This closure is formally established in P78.