The Crossover Theorem: Existence, Uniqueness, and the Impossibility of Axis-Blind Governance Gates — Benefit–Burden Monotonicity, Single-Crossing as a Falsifiable Signature, and a Corrected Folk Claim About Dose Ordering

Abstract

The July 2026 MOBIUS measurement campaign reported an operational law for inference-time governance: the same governance prompt that damages a small language model improves a large one, with the sign of the effect flipping at roughly 4B parameters on the content axis — and flipping the *other* way on the route-discipline axis. This paper supplies the mathematical layer beneath that law. In a two-term benefit–burden response model — benefit nondecreasing and burden nonincreasing in executor capacity (assumption B1) — we prove three results.【proved】(i) The net governance effect is monotone in capacity; with opposite signs at the capacity endpoints (B2), a crossover capacity c∗ exists, the zero set is a (possibly degenerate) closed interval, and strictness of either monotonicity forces uniqueness (Theorem 1). The load-bearing consequence is *single-crossing*: under B1 the effect can flip sign at most once in capacity, a falsifiable prediction — a second sign flip at higher capacity refutes B1, it does not merely add noise. (ii) Under increasing differences in (dose, capacity), the optimal-dose correspondence is monotone nondecreasing in capacity (Theorem 2, by monotone comparative statics【derived】). We then record a deliberate *non-theorem*: increasing differences do NOT imply that per-dose crossover capacities are ordered — a two-line counterexample corrects the folk claim that "heavier doses cross later" follows from supermodularity; crossover ordering is an independent empirical property, to be checked, not assumed. (iii) When two task axes have distinct optimal doses at a given capacity, every axis-blind gate — any policy mapping capacity to a single dose applied on both axes — suffers worst-axis regret at least the disagreement gap Δ(c) > 0, while the two-argument gate (capacity, axis) achieves zero regret (Theorem 3, Corollary 3.1). A deterministic script validates single-crossing on a 256-cell parametric grid (0 violations) and audits the frozen 27-rung × 3-dose empirical matrix — and the audit's headline must be stated plainly: strict sign-level single-crossing *fails* for both light doses (5 and 7 family-smoothed sign changes; only one flip per dose sits at the ≈4B crossover). The failure is confined to shallow wobble above the crossover: every post-crossover negative light-dose cell has magnitude at most 2.97 judge points and lies within ±2 standard errors of zero (per-cell SEs recovered from the primary per-question scores, n = 30 per cell), whereas below-crossover harms reach −37.3 and exceed 2 SE at 11 of 14 light-dose cells. We read this magnitude asymmetry as a degenerate-to-flat crossover interval seen through measurement noise — an interpretation supported by the per-cell error analysis but graded exploratory, not established fact, with a pre-stated reproducibility condition — a second sign flip whose cells depart from zero beyond sampling error under replication, at *any* magnitude — under which the wobble would instead count as a refutation of B1. The 42KB overlay never crosses within the observed range, and the plug-in disagreement gap is strictly positive at 5 of 8 shared rungs — maximally at the bottom rung (0.8B) and the 8B reasoning rung, exactly where the mirror-image data pattern lives.【exploratory-empirical】The reasoning-family exception is treated as a violation candidate for B1 under the parameter-count capacity proxy, with a falsifiable effective-capacity refinement. We close with break conditions and a pre-stated rejection condition. Theory paper T4 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.