Unanimity in Disguise: What an Observed-Support Entropy Threshold Actually Detects at Short Windows

Two deployed detectors in the Möbius stack — the saturation signal of the Reflective Budget Governor (RBG) and the saturation score of the Reflective Homeostasis Layer (RHL) — compute one minus the Shannon entropy of a short window of categorical observations, normalized by the logarithm of the number of categories actually observed, and fire at a threshold of 0.70. This note shows that at every window length up to 18 the condition is equivalent to unanimity. The result follows from a sharp envelope theorem: over all positive integer histograms with n observations and any number k ≥ 2 of observed categories, the normalized score 1 − H/log k is maximized, uniquely up to permutation, by the two-category histogram (n−1, 1). Hence for any threshold 0 < τ < 1 the smallest window at which a non-unanimous pattern can fire is n*(τ) = ⌈1/p_τ⌉ with h(p_τ) = (1−τ) log 2 — an exact threshold-to-window rule certified by integer inequalities (n* = 19 for 0.70, 78 for 0.90, 1163 for 0.99). Substituting the Boolean “all equal” for the numeric score reproduces every stop index and reason across the RBG study’s 1,000 refinement chains. A second family of results bounds why set-valued trigram similarity cannot see verbatim repetition: a word with at least 49 distinct trigrams keeps cosine ≥ 0.98 to any number of its own copies, and a nested-set identity matches the recorded low-distance signal of the study’s reported collapse case. The fixed-support entropy minimum is known since Beisel and Moreteau (1997); the cross-support comparison, the threshold-to-window inversion and the implementation consequences are the contribution here, with priority not claimed. Finite certificates are machine-checked in Lean 4; the universal proof is prose. Scripts, certificates, reviews and replay results are released in the companion repository. AI disclosure: the mathematical content (derivations, proofs, checks, Lean sources and prose) was generated by Codex (OpenAI) and Claude Fable 5.1 (Anthropic) under the author’s direction; the human registered author reviewed it and takes responsibility for it. Pre-release adversarial refutation passes were AI sessions of the same two model families, not human review. The registered author is the human author alone.