| # 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. |
| |
