Phase 3: fixed-schedule comparison and a bounded biased-pair pilot
Scientific scope fixed on 2026-09-16 before these experiments. No manuscript or submission changes are included. All results, including zero gaps and resource failures, are retained. CPU only; no GPU, new training run, or outer-model search.
A. Fixed-schedule comparison
Reuse all nine phase-two rational witnesses without modifying their dimension, mixture weights, component probabilities, teacher, or budgets. Evaluate K=3,4. Compare the manuscript geometric cumulative sizes floor(n^(j/(K-1))), equally spaced cumulative sizes floor(j*n/(K-1)), and the best A-first deterministic fixed-batch schedule. The latter exhausts all ordered nonnegative integer batch sizes summing to n; empty batches allow fewer than K rounds. It is not the best among every deterministic adaptive policy.
For deterministic pair-label batches, compute full-output TV from the joint vector of per-batch mismatch counts, not just the final total count. As a separate, explicitly different policy, randomize pair labels uniformly and marginalize that private randomization; its total-count TV is then full TV. Do not attribute the difference between these restricted policies and the all-policy optimum to adaptation or private randomness alone.
Use exact rational arithmetic for finite sums and comparisons. Check all TVs against the existing certified all-policy lower bound and retain the complete schedule enumeration. Test independently against literal full-output laws at small n, the exact-target control, single-batch identities, and the contraction of TV under projecting the batch-count vector to its sum.
B. Fixed-bias pilot
Freeze the mismatch mixture of phase-two witness n8_B1_10_single_d1_4:
tau = 1/4
probabilities = [1/4, 1/2]
weights = [11681887/16777216, 5095329/16777216]
Use homogeneous representative biases a in {1/2, 2/5, 1/4}, changing the same bias in P and R. Required grid: n=4 pairs and K in {2,3,4}. A resource-gated extension tests n=6 and K in {3,4}. It is selected only by assembly resources, not by whether any observed error gap is positive. Gate: at most 120000 state-action variables, 3000000 transition entries, and 120 seconds for a representative n=6, a=2/5, K=4 exact-kernel graph profile. This gate excludes subsequent sparse-matrix assembly; the overall biased-stage limit covers that work as well. Any skip is recorded.
Changing n while preserving mixture parameters changes KL, so do not claim that these smaller models use the entire original 0.1 budget. At each fixed n, joint model KL and TV are identical across biases, since the representative-bit factor cancels. Report their actual values and verify feasibility under 0.1.
Crucially, bias can amplify reverse A|B head errors. Enumerate all partial-state heads and report the exact worst error; do not claim the fair Delta=1/4 head bound automatically persists. Do not exclude a bias because its head error is larger. This is a fixed-mismatch-model sensitivity study, not an equal-uniform- head-error comparison or a resolution of the fixed-bias sharp exponent.
The new controlled state records counts of nine pair types: empty, A0, A1, B0, B1, 00, 01, 10, 11. All legal actions include either singleton orientation, completion of each singleton type, and independent co-sampling of fresh pairs. Heads are frozen before the round. The component posterior must include the informative B-only observations. The terminal test and TV use all four completed pair-type counts, not the mismatch count alone.
An all-action Bellman lower certificate applies conditional on every literal history. A uniformly lifted histogram policy is pair-permutation invariant, so terminal histogram TV equals its full-output TV. Use exact or outward-rounded rational calculations to certify lower and feasible-policy upper endpoints. Cross-check small literal-coordinate kernels/LPs and compare the fair-bias case with the independent earlier fair-state solver. This is not Lean verification.
Resource and stopping rules
One numerical-library thread per process; 8 GiB address-space cap. Fixed-schedule stage: 600 seconds. Biased scientific job: 1200 seconds, each LP at most 90 seconds. Preserve timed-out or incomplete cases; never relabel them certified. No automatic expansion of n or parameter search to chase a favorable result.
Figures and interpretation
Three intended PDF/PNG figures: fixed-schedule comparison, all-nine-witness schedule overhead, and fixed-bias depth sensitivity. Palette and simple styling reference: https://github.com/ChenLiu-1996/figures4papers/ . Axes labels 22pt, tick/legend labels at least 18pt. PDF rendering is inspected before delivery. Certified endpoints are arithmetic bounds, not Monte Carlo confidence bands. If a gap is zero or negative, preserve it and explain the actual implication.