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Theorem · no-gain obstruction

FPRD-T142

Nested unconditional thresholds cannot improve the best envelope

Exact statement

If t1≥⋯≥tL≥0t_1\ge\cdots\ge t_L\ge0 are decreasing guard thresholds and Ai≥0A_i\ge0 are unconditional bounds for increasing disorder sublevels, then ML=tL+∑i=2Lmin⁡(ti−1−ti,Ai)+A1≥min⁡i(ti+Ai). M_L=t_L+\sum_{i=2}^L\min(t_{i-1}-t_i,A_i)+A_1 \ge\min_i(t_i+A_i).

StatusProved by an elementary induction and internally audited; no external review
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The theorem closes a tempting optimization route. Splitting the guard range into more bands cannot improve the best single-threshold estimate when every disorder count is only an unconditional marginal bound.

Definitions

  • The guard bands have widths Δi=ti−1−ti≥0\Delta_i=t_{i-1}-t_i\ge0.
  • AiA_i bounds the entire corresponding disorder sublevel without conditioning on the guard band.

Hypotheses and scope

  • t1≥⋯≥tL≥0t_1\ge\cdots\ge t_L\ge0.
  • Every AiA_i is nonnegative and unconditional; no joint distributional information is available.

Proof or evidence

Induct on LL. Put Δ=tL−1−tL\Delta=t_{L-1}-t_L. If Δ≤AL\Delta\le A_L, the final correction equals Δ\Delta, so ML=ML−1M_L=M_{L-1}. If Δ>AL\Delta>A_L, discard the earlier nonnegative corrections to get ML≥tL+ALM_L\ge t_L+A_L. Either way ML≥min⁡i(ti+Ai)M_L\ge\min_i(t_i+A_i).

Nested-threshold theorem and proof →

Verification notes

This run independently rederived the induction, checked the base case and both last-band cases, and confirmed that the claim does not assume independence. A new exhaustive falsification audit passed 532,700 small integer instances. The written induction remains the proof; no external review is recorded.

Limitations

  • The theorem applies only to envelopes built from unconditional marginal sublevel counts.
  • It does not rule out an improvement from a genuinely joint guard–disorder estimate.
  • It yields no Collatz or finiteness conclusion.

Open work

Do not add more unconditional Ferrers bands; seek a statistic that retains the joint guard–disorder correlation.

Notes

The proof is an elementary induction splitting on whether the final threshold gap is at most its disorder bound. This theorem is abstract and reusable beyond Collatz. It does not rule out gains from joint, conditional, or dynamically correlated guard--disorder information.