Stage D0: the fair value of oracle-visited boards

Measures whether the boards a clairvoyant oracle chooses to stand on are still good boards when the future is replaced by ordinary public randomness, by re-valuing them and matched fair-search boards under the same 32 independent public futures.

The intuition

The oracle sees the whole future: every hidden gray-disc value and every disc it will be dealt. Its games last far longer than any fair policy's. The open question of the oracle-curriculum family (stated as "the single missing measurement" in docs/exploratory/audit-05-optimistic-curriculum.md §5.4) is why: does the oracle survive because it builds genuinely better boards, or because it knows the tape and threads a needle no public player could see?

The two explanations dictate opposite research programs. If the boards are genuinely better, a student can be trained on them with honest labels — the H-pool idea. If the advantage is the tape, those boards teach nothing transferable and the program should close.

Stage D0 separates the two by a controlled measurement, with no model and no training:

1. Take boards the oracle actually stood on (pool O).
2. Take boards a fair depth-4 search stood on, matched to pool O on rise phase, occupancy and maximum height (pool F), so the comparison is between boards of the same coarse shape.
3. Strip both to public information only, then re-value every board the same way: play 25 further moves with a fixed public depth-1 fair policy under 32 independent futures derived (through domain-separated hash streams) from the board itself — never from the game's seed. The mean number of moves survived is the board's fair value, R_fair.

If oracle boards are better boards, R_fair(O) should exceed R_fair(F) by a sizable fraction of the realised-lifetime gap the oracle actually enjoyed. That fraction is the transferable fraction

τ = (R_fair(O) − R_fair(F)) / (R_tape(O) − R_real(F)),

where R_tape/R_real are the remaining moves each board actually had on its own trajectory, both capped at the same 25-move horizon.

How it works

The implementation reuses the repository's frozen machinery rather than rewriting it; d0.cpp builds into two binaries so that the privilege boundary is enforced by the linker, not by discipline.

1. d0-generate (privileged, teacher-side) plays 64 oracle games (depth 4, beam 128, 500-move cap — the constants of approaches/oracle-curriculum/state-curriculum/oracle-curriculum.cpp) on the leased training seeds 0xa52e0000–0xa52e003f, and samples up to 31 public states per game from move 50 onward — one from each of 31 equal stretches of the game, with a seed-keyed offset inside each stretch so the sample cannot alias with the five-move rise cadence. Each state is exported as the public tuple (board, next disc, moves until rise) plus labels: the oracle's column, fair D4's column at the same root (a public computation), and the realised remaining moves.
2. The same binary then plays fair-D4 reference games (approaches/fair-expectimax/reference/fair-only-depth4.cpp: depth 4, five strata) on 0xa52e0100–0xa52e01ff, lazily in fixed batches of 16 seeds, and matches their post-move-50 states 1:1 to pool O on the exact bucket (rise phase, occupancy ÷ 4, maximum height ÷ 2) — the binning of approaches/oracle-curriculum/topology/oracle-topology-audit.cpp — at most two per bucket per game, stopping when every O state is matched or the 256 seeds run out. Unmatched O states are dropped and counted.
3. d0-relabel (public-only; its symbol table is checked to contain no oracle planner and no headless-tape accessor) reads back only the public tuple of each record and computes R_fair: 32 independent restarts at horizon 25 under fair depth-1 continuation, with reveal and visible-disc streams drawn from the domain-separated "CRRV"/"CRVS" streams of oracle-curriculum.cpp, seeded from the canonical public-state hash. It also values every legal first column of every O root under the same 32 futures (common random numbers), which gives the fair-top-1 comparison: how often the oracle's column is fair-best at its own roots versus how often fair D4's column is.
4. analyze.py computes τ, the four R numbers, origin-game halves, the kBlocked flow-band fraction, the top-1 rates, and a 10,000-resample cluster bootstrap over origin games (seed 0xb0071eaf).

Two deliberate differences from the original curriculum relabeller, both visible in d0.cpp's header comment: K = 32 scenarios instead of 7, and the restart is played in the canonical mirror orientation at every step, which makes R_fair of a board and of its mirror image equal exactly rather than approximately.

The CHECK gates (gate.sh) run before any leased seed is read, on the already-opened probe range 0xa5278000–0xa52784ff only: the symbol-table privilege check, byte-identical generation at 4 and 16 threads, byte-identical relabelling across two runs and thread counts, exact mirror invariance, metadata independence (score, level and move counter cannot change any restart), domain separation of the streams, and label blindness (editing every label field in the input changes no relabel output byte).

What happened

The measurement came back against the idea, and the H-pool program closed. On the same 32 public futures, the boards the oracle stood on were only about a quarter of a move better than matched fair-search boards over a 25-move lookahead — and the oracle's own choice of column at those boards was fair-best less often than the fair search's choice at the very same boards. Boards the oracle reaches are fine boards; its moves from them are tape-reading, not transferable judgement.

All numbers below are from the result record RS-20260823T205143Z-ead14c9d (run RUN-20260823T195750Z-1cb2f9de).

• Pool O: 1,984 states from 64 oracle games (63 of 64 reached the 500-move cap). Pool F: 1,271 matched states from all 256 fair-D4 games (match rate 0.641; 713 O states from buckets fair D4 rarely visits were dropped, as preregistered).
• R_fair(O) = 24.783 vs R_fair(F) = 24.520: a fair-value advantage of +0.263 moves (cluster bootstrap 95% [+0.156, +0.367]), the same positive sign in both origin-game halves — real, but small, and compressed by the 25-move ceiling (92–96% of futures survive the horizon).
• R_tape(O) = 24.338 vs R_real(F) = 24.611: with almost every oracle game censored at the 500-move cap, 94–95% of both realised remainders sat at the 25-move cap, so the realised-lifetime gap the preregistered τ divides by came out at −0.274 moves instead of the large positive gap the definition assumed. τ = −0.959 (95% [−3.069, −0.390]) — a fail against the preregistered τ ≥ 0.25 gate, though a fail driven by a degenerate denominator rather than by a negative fair-value difference; the result record states this limitation explicitly.
• The action-quality criterion failed on its own, unaffected by the ceiling: the oracle's column was fair-top-1 at its own roots 76.6% of the time against fair D4's 81.4% at the same roots (difference −0.049, 95% [−0.069, −0.029]; on the 194 roots with a unique best column, 26.8% vs 38.7%).
• Under fair randomness almost no oracle board is a trap: 0.5% of matched O states fall in the blocked flow band (F: 2.6%).

Run record

Preregistration EX-20260823-hpool-stage-d0-e0ad1c65 (CHECK tier, diagnostic), theory TH-20260823-hpool-optimistic-states-fair-labels-bb707bc9, lease SL-20260823T200000Z-a52e0000 (training role, 0xa52e0000–0xa52e01ff, opened once). Gates passed before the lease was opened (runs/RUN-20260823T191900Z-b9f8f80d/d0/gates.log); main run 3,132 wall- seconds on 16 threads, peak RSS 152 MB, artifacts under runs/RUN-20260823T191900Z-b9f8f80d/d0/. Outcome: run valid, scientific outcome fail, theory assessed not-supported-as-tested at pilot tier. Two of the three preregistered refusal conditions triggered (τ < 0.25; oracle column fair-top-1 less often than D4's); the sign-agreement condition passed.

Preregistered gate

Stage D0 refuses the H-pool theory (TH-20260823-hpool-optimistic-states-fair-labels-bb707bc9) if any of:

• τ < 0.25 pooled;
• the two origin-game halves of pool O disagree in the sign of R_fair(O) − R_fair(F);
• the oracle's column is fair-top-1 at O roots less often than fair D4's column at the same roots.

A refusal closes the optimistic-teacher program; a pass licenses exactly one follow-on (the fair / optimistic / mixed pool-comparison experiment), and is itself a diagnostic — never a policy claim.

What this taught us / what is still open

The measurement audit-05 called "the single missing measurement" now exists, and it says the oracle's advantage lives overwhelmingly in what it does with the tape, not in the boards it builds: the fair value of its boards is only marginally above matched fair boards, and its actions are worse than fair D4's under honest randomness even on its own boards. Per the preregistration, no student is trained on the optimistic state pool and the E1 ordering ablation is not run.

Still open, and not answered by this experiment: whether the small but real +0.26-move board-quality edge would grow at a longer relabelling horizon or a deeper continuation policy (the 25-move horizon capped both sides of the comparison), and whether the 713 unmatched O states — boards whose shapes fair D4 never reaches — are better or worse than the matched ones. Both questions would need a new preregistration; neither revives the H-pool teacher on this evidence.