Blame for server/phase3/PROTOCOL.md

# 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.