Temporal Dynamics of Distinguishability Collapse: On the Breakdown of the Schwartau Inequality under Generative AI

The dynamic counterpart to distinguishability collapse: why Schwartau's twenty-five-year-old inequality Pt > Dt + Rt becomes structurally unsatisfiable when adversary tempos compress below the defender's structural observation floor.

Preprint — April 2026

We present the temporal counterpart of the Distinguishability Collapse Theorem [Rouxel, 2026]. Where the first paper established static information collapse — mutual information between observable and intent vanishing at each instant under polynomial-cost mimicry — this paper establishes its dynamic counterpart: under compression of the adversary adaptation tempo below the defender's structural observation floor, mutual information vanishes across the sampled trajectory.

The framework unifies four research traditions previously evolved in isolation: Schwartau's operational time-based security (1998), Boyd's decision-cycle theory (1995) and its cybernetic formalization by Brehmer (2005), the information-theoretic detection limits of Denning (1987) and Axelsson (2000), and the Nyquist–Shannon sampling theorem (1928, 1949). We anchor the argument in four convergent layers of empirical evidence: the Cybersecurity Dynamics program (Xu et al., 2013–2023), the time-to-exploit literature (Bilge & Dumitraș, 2012; Pauley et al., 2023), recent generative-AI offensive-capability evaluations (UK AISI 2025, 2026; Google DeepMind 2025), and direct industry breakout-time measurements (CrowdStrike 2021–2026, Mandiant 2025).

Schwartau's inequality holds operationally only when the sampled channel carries positive mutual information about intent. When the channel collapses, the inequality is not merely difficult to satisfy — it is structurally unsatisfiable.

Main contributions

Six formal results:

• Temporal Information Collapse Theorem (Theorem 4.X) — Under compression of the adversary tempo below twice the defender's structural observation floor, mutual information between sampled trajectory and intent vanishes on the confusable class. Architecture-invariant by the Data Processing Inequality.
• Observation-Tempo Floor Theorem (Theorem 4.1) — The defender's observation tempo has a strictly positive structural floor set by the maximum of physical, computational, statistical, and organizational constraints. None can be reduced to zero by investment.
• Irreversibility Theorem (Theorem 4.3) — Once the temporal collapse condition is met at any instant, it persists. The asymmetry has no symmetric defensive lever within the detection-prevention frame.
• Wienerian Statistical Floor Theorem (Theorem 5.0) — Observation tempo is bounded below by base-rate saturation of the signal-to-noise ratio: faster sampling buys no further mutual information about intent once this floor is reached.
• Window Closure Corollary (Corollary 5.2) — Under sustained tempo compression, no observation tempo is simultaneously operationally viable and Nyquist-compliant. The detectable window of adversary tempos closes from both sides.
• Hardening Lever Corollary (Corollary 4.3) — Within the Schwartau frame, the only operation that structurally reverses the collapse is raising the adversary tempo through architectural constraints uncompressible by generative AI.

We characterize a ternary partition of the adversary trajectory space — fast (aliasing collapse), detectable (the shrinking middle), patient (sub-resolution collapse) — that jointly exhausts the space of compressed adversaries. Three structurally distinct impossibility mechanisms cover the three regimes; their joint scope is decisive.

Implications

Acceleration-based defense — SOAR automation, XDR consolidation, MTTD/MTTR optimization — cannot satisfy Schwartau's inequality under temporal collapse. These investments operate on the defender side of the inequality and are bounded by the structural floor; they do not affect adversary tempo.

Three defensive paradigms match the three regimes: architectural hardening against fast adversaries, organizational-memory investment and deception against patient adversaries, and resilience–diversification–forensics against the residual mimicry-collapsed regime. A coherent defensive doctrine composes the three; no single paradigm covers all.

This formalizes — and scopes — the informal claim of Raghavan & Schneier (2025) that speed-based defense cannot prevail against AI-enabled adversaries. The architectural shift from detect and respond toward constrain and verify, identified in the first paper, extends here: within the Schwartau frame, only operations that raise the adversary's tempo escape the collapse.

Read the full preprint on Zenodo →

License: CC BY-NC 4.0