The Reflective Budget Governor: Equilibrium-Based Early Stopping for Iterative LLM Refinement

A three-signal stopping rule for iterative LLM refinement derived from the Möbius theory stack: an equilibrium-distance surrogate d̂_R (RZGM meaning-equilibrium layer), an update-free-recurrence and diversity-collapse detector pair (Reflective Homeostasis Layer), and a hard reflection-depth cap (MUSE bounded reflection). Pre-registered study: 500 English prose-refinement tasks across ten domains, two generator models (Gemma4 12B-IT-QAT and 26B-A4B-IT-QAT), 8,000 generations, 999 blind position-swapped pairwise judgments. The governor cuts refinement tokens by 36.6% / 33.1% (bootstrap 95% CIs 33.6-39.5% / 30.3-35.9%) versus an always-eight-iterations baseline — 2.2-3.4x a naive convergence stop — at a measured quality cost reported at equal prominence: 38.0% / 36.6% of judged comparisons lost overall, a majority among actual interventions (61.1% / 59.0%), with measured cost almost perfectly collinear with output length. A full-data ablation shows the saturation signal is simultaneously the savings engine and the dominant cost source, yielding a measured two-point dial (10.0-11.5% savings at 8.2-11.8% loss rate with saturation disabled). Pre-release adversarial verification (four independent refuter passes) confirmed all headline statistics against the raw data and caught an inverted flagship anecdote: the corpus's two catastrophic degenerations (~50 verbatim paragraph repetitions) were selected once and avoided once by the governor, undetected in both cases — set-based trigram signals are structurally blind to within-output verbatim repetition. All tasks, rollouts, verdicts, audit votes, and analysis code are released in the companion repository. AI co-observer: Claude Fable 5 (Anthropic) — working method only; blind pairwise judging by Claude subagents under a position-swapped two-vote protocol; four independent adversarial refuter passes preceded release; the registered author is the human author alone. Version 1.1 (2026-09-19): adds (i) a proof-backed clarification that at window 3 the saturation condition is exactly “three identical consecutive non-rewrite change types” (the 0.70 threshold is inert there; companion note DOI 10.5281/zenodo.22842506), (ii) a pre-registered evaluation of a repetition-ratio guard as a fourth, roll-back signal — fires on both known collapses and 200/200 synthetic inflations, on 0 of 7,998 remaining published generations, 0 of 60 template expansions, and 0 of 960 held-out generations (verdict GO; pre-registration commit a89f722 precedes all evaluation data), and (iii) an expanded AI-authorship disclosure. All v1.0 numbers are unchanged. Version 1.2 (2026-09-19): adds (i) a length-stratified reanalysis of the 621 judged governor pairs (loss rate 92% where the early text is under 60% of the final length, 16.1% at near-equal length; 31 losses vs 4 wins at matched length, sign test p < 10-5 — evidence of a content cost independent of length, not a decomposition of the headline cost), (ii) a held-out replication on 60 tasks authored after the fourth-signal pre-registration (token reduction 34.5% / 34.5%, loss rate 31.7% / 36.7%, main-study values inside the held-out intervals), and (iii) an online-intervention verification: an incremental governor reproduces all 1,000 published decisions (0 mismatches) and, run as a genuine online loop on the 60 held-out tasks with the 12B generator, generates 319 of 480 iterations with a stop distribution indistinguishable from the offline policy (permutation p = 0.46). All v1.0 and v1.1 numbers are unchanged. Version 1.3 (2026-09-19): adds a pre-registered length-matched re-judging (100 governor-loss pairs with early/final length ratio < 0.80; final text truncated at sentence boundaries to the early text's word count): the early text beats the length-matched final in 94/100 (loss 4%), and the full final beats its own truncation in 100/100. Pre-registered label: the losses on this stratum are attributable to the additional material, not to a per-length quality deficit; whether that material is value or judge-preferred bulk remains open, and truncation is a destructive proxy. All v1.0–v1.2 numbers are unchanged.