Sequential, Never Layered: A Formal Theory of Governance Overlay Collapse — Why Stacked Overlays Interfere, When Stacking Turns Harmful, and the Capacity Law of the Overlay Portfolio

Abstract

Inference-time governance of language-model systems is built from *overlays*: system-prompt policies, metacognitive check organs, safety checklists, routing disciplines. Each overlay, alone, can improve output quality. The natural engineering move — stack every beneficial overlay into one context — fails in practice, and fails hard: in the July-2026 MOBIUS capacity×dose grid, the heaviest overlay dose (43,316 bytes) degraded judged quality on 26 of 27 model rungs, while light doses of single components helped many of the same models. The deployed MOBIUS architecture answered with a design law, "sequential, never layered" (Condition I): one overlay per context, stages composed one-way on committed outputs. That law was discovered empirically and stated as doctrine. This paper supplies its formal footing. We define an overlay economy in which each overlay carries a full-compliance benefit and an instruction-load dose, and all overlays share one bounded execution budget through a nonincreasing fidelity function φ of total dose. Four results follow.【proved】(i) Layered governance effects *never* add: for strictly decreasing φ, the layered effect of two overlays is strictly less than the sum of their individual effects — interference is universal and strictly negative (Proposition 1). (ii) Sequential composition dominates layering for every overlay set and every nonincreasing φ, strictly under mild conditions (Theorem 1) — the Condition I theorem. (iii) In the linear-fidelity instantiation with capacity parameter C, symmetric stacking of two overlays of dose d each is harmful exactly on d < C < 3d, and falls to the no-governance baseline at c = C — strictly below it whenever λ > 0 (Theorem 2); exact additivity survives only on a sub-saturation flat plateau, a property of general fidelity families rather than the clamped-linear instantiation (Proposition 2c); the same overlay pair flips from harmful-to-stack to beneficial-to-stack as capacity grows through 3d (Corollary 2.1). (iv) The optimal number of stacked overlays scales as m* ≈ C/(2d): the overlay portfolio is capacity-gated (Proposition 3). A frozen numerical protocol (deterministic boundary sweep plus 5 seeds × 1,000 randomized draws under exponential fidelity) verifies the subadditivity, dominance (two-overlay), and stacking-boundary claims with zero violations. An exploratory section reads the frozen July-2026 grid against the theory — the model was not fit to those data — and a falsification section states exactly how to break the result. Theory paper T2 of the MOBIUS 2026-08 theory series (six papers, T1–T6). Version 0.1, deposited as a preprint; journal submission of a revised version is planned, and the journal version may differ. AI co-observer: Claude Fable 5 (Anthropic), working method only; the registered author is the human author alone.