Why Reflection Should Be Shallow: Contractive Gains, Finite Optimal Depth, and the Rumination Regime — A Cost–Benefit Theory of Iterated Self-Revision, with a Capacity-Gated Crossover

Abstract

How many times should a system re-read and revise its own answer? Empirical work in the MOBIUS measurement campaign found a consistent answer across thirteen open-weight model capacities: once or twice — never zero above a capacity threshold, never many, and *less than zero* below the threshold, where every reflection pass makes output worse [1]. This paper supplies the missing mathematical layer: a minimal cost–benefit model in which that entire phenomenology is a theorem. We model reflection as a sequence of stochastic revision passes with marginal quality gains bounded by a geometric envelope (Assumption A1) and a strictly positive per-pass cost. The geometric envelope is *motivated* — not proved — by an information-theoretic idealization: if each pass processed only the previous output, the chain would be Markov and the data-processing inequality with its strong (contraction-coefficient) refinements would force geometric decay of extractable information; the measured protocol re-presents the input at every pass, so the chain is not Markov, and the surviving motivation is finiteness of the re-readable information in a fixed input — A1 is an explicit modeling premise, stated as such. Under this premise we prove four results.【proved】(i) The net-value-optimal depth d∗ is finite, with the explicit bound d∗ ≤ 1 + ⌈log(κ/g₁)/log ρ⌉, net value strictly decreasing beyond the bound, and a depth-independent value ceiling g₁/(1−ρ) − κ (Theorem 1). (ii) When the first-pass gain does not cover the per-pass cost, d∗ = 0, and under distortion-injecting passes reflection is monotonically harmful at every depth — the rumination regime (Theorem 2). (iii) Parameterizing gains by capacity yields a reflection crossover capacity c∗ whenever the gain path crosses the cost (otherwise rumination holds at every capacity and c∗ = +∞): below c∗ depth zero is optimal, above it positive depth becomes admissible, and the depth budget is nested increasing in capacity (Theorem 3). (iv) When gains meet the envelope with equality, d∗ has a closed form — the number of passes whose marginal gain strictly clears the cost — and the shallow optima {1, 2} occupy the wide parameter region κ/g₁ ∈ [ρ², 1) (Proposition 4). A deterministic 1,080-cell parameter sweep verifies every claimed formula, in generic position (boundary ties are excluded by construction and settled by proof), against direct argmax with zero violations. We then juxtapose — without fitting — the frozen July-2026 depth curves: the 0.8B rung is all-negative at every depth (rumination); measured optimal depth is 1–2, or within 1.2 judge points of a shallow depth — a reading criterion derived from the source study's noise remark — on all capable rungs, with one rung (4B) monotone increasing to the depth-3 measurement boundary under decaying increments; and the measured crossover ≈2–4B instantiates c∗.【exploratory-empirical】A falsification section states how to break each assumption and a pre-registered rejection condition that fires on observable violation of the gain-decay premise. Theory paper T3 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.