| # 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 |
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|
| 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. |
| |
