Phase 2: finite all-policy execution-gap witnesses
Frozen before scientific runs: configs/phase2.json. This is an exploratory
finite candidate search followed by independent deterministic certification.
It is not a global minimax optimizer, an asymptotic exponent experiment, or a
formal verification of the manuscript.
Target quantity
For a fixed positive fair-pair oracle R, let d_K(R)=inf_policy TV(P,Q_R^policy), over every legal full-history, privately randomized K-round policy from the paper. Seek a certified bracket L<=d_K(R)<=U and a positive execution gap G=L-TV(P,R). Sequential original-head sampling can deliver R, so G>0 is unavoidable additional error at depth K relative to that feasible serial law. G is not a claim that the infinite-depth optimum equals TV(P,R).
Each candidate obeys actual KL(P||R)<=B and uniform conditional head TV<=1/4. Within the rational Bernoulli-mixture family used here, all component rates lie in [1/4,1/2], so the latter also follows from posterior convexity; the certificate independently enumerates all completed contexts to check it.
Candidate search and controls
n=8 searches eight specified families at budgets .02,.05,.1: three single alternative shifts, the manuscript two-level ladder (lambda=.2,w<=1/8), and four relaxed two-level spacings with equal alternative weights and w<=1/2. The relaxed family remains in the paper's general exchangeable oracle class, but is not the particular geometric ladder of the finite lower theorem. n=12,16 evaluate the three singles, manuscript ladder, and the n=8 winning relaxed spacing recalibrated at the same budget. Selection maximizes floating d_3(R)-TV(P,R); every candidate and failure is retained.
The common actual-KL budget target is B*(1-1e-6). The calibrated total alternative weight is rounded downward onto a 2^-24 grid. Mixture parameters are then exact rationals; no result treats serialized binary floats as the intended oracle. If a family's weight cap prevents reaching the target, label it under-budget; do not describe it as a matched-KL comparison. In particular the theorem ladder may not reach the requested budget in these small dimensions.
One finalist per (n,B) is frozen for K=2,3,4, using the exact same R at all K. Finite candidate selection is not an optimization certificate for the outer supremum over R. Independent certification of a feasible witness nevertheless gives a valid lower bound on that supremum at its verified budget.
Independent certification
The floating LP proposes a bounded terminal count test f in [0,1] and a policy. An independent standard-library-only verifier uses exact rational oracle heads and transition probabilities, not the LP transition matrices. All legal actions are included in the backward Bellman minimum. Values are rounded downward on a fixed integer grid. L=V_0-E_P[f] is therefore a valid lower bound even if the proposed test is suboptimal or the floating dual is imperfect.
The recovered policy is rationalized and normalized to define an actual legal policy; zero-mass unspecified states receive a specified legal fallback. Forward propagation rounds down masses. If q_low is pointwise below its true terminal count law and e=1-sum(q_low), then U=(sum_m abs(P_m-q_low_m)+e)/2 is a rigorous upper bound. Clip final U at one.
Model TV uses exact rational count masses. Actual KL is enclosed using rational logarithm series with explicit remainder bounds; all conditional heads are checked. Certified claims require positivity, KL and head checks to pass. Record exact rational endpoints, not just decimal display values.
The connection to all literal policies uses the fair-model controlled-state aggregation and symmetric-lifting proof, documented in phase 1 and rechecked here. This finite implementation is not Lean/formal-kernel certification.
Resources and outputs
CPU only; numerical-library and HiGHS threads limited to one. Whole run timeout one hour, 8 GiB address-space cap, per-LP solve at most 120 s, at most 250,000 state-action variables. Increasing n beyond16 requires a separate profiling decision; no dense high-dimensional search is silently launched. No optional large-n Monte Carlo or bias study is included in this first phase2 run. Preserve source/config hashes, raw candidates, selected rational witnesses, policy files, exact certificates, validation output and all failures.
Figures
Continue the user-requested figures4papers-inspired palette (#0F4D92, #B64342, #42949E, #9A4D8E), white background, redundant line/marker encodings, frameless legends, large text. Reference: https://github.com/ChenLiu-1996/figures4papers/ Target axes labels22pt, ticks/legends18pt, no more than two main panels per figure. Certified intervals are arithmetic enclosures, not Monte Carlo confidence bands. Show the model-TV baseline, exact actual-KL values and whether a budget is inactive. Do not imply any family is globally hardest or that a positive gap must be large. Export PDF/PNG and inspect rendered PDFs before delivery.