Seven chance samples instead of five

The next disc has seven equally likely values, but the reference search only draws five samples at each point where the game deals one. This experiment changes the count to seven, so every possible next disc is covered exactly once, and asks whether the policy plays better.

The intuition

When the game deals the next disc, it picks uniformly from

1 2 3 4 5

6 7 — seven outcomes, each one chance in seven. The search cannot afford to expand all seven at every node of a four-move tree, so it draws a fixed handful of representative outcomes instead — the strata. The frozen reference draws five.

Five samples cannot represent seven equally likely things. Some disc values get drawn twice, and some do not get drawn at all. An independent audit of the reference search measured exactly how often: across the sampled nodes, five strata produced five distinct disc values on 62.65% of nodes, four on 33.86%, and three on 3.49% — a mean of 4.59, so on average 2.41 of the seven possible next discs are given probability zero at every node, while the ones that were drawn are inflated from one-in-seven to one-in-five or two-in-five (audit 02, H1).

That error does not wash out. The sampling is a deterministic function of the public position, so searching the same position again reproduces the same distortion. And it is not harmless: every column is judged against the same distorted disc distribution, so much of the distortion cancels in the ranking — but not all of it. A column that only dies if a 2 arrives looks perfectly safe at a node where 2 was never sampled.

Setting the count to seven makes each stratum land on exactly one disc value, which turns the next-disc average from a five-point sample into exact enumeration with the true one-in-seven weights.

How it works, step by step

1. Everything about the reference depth-4 search is held fixed: the leaf evaluator, the immediate-score semantics, the terminal value, the full-width action order, the position-derived sampling salt, iterative deepening, and the game rules.
2. The one changed constant is the number of chance samples per node: five becomes seven.
3. Because the branching factor goes from 7 columns × 5 samples = 35 to 7 × 7 = 49, the tree is much larger. The candidate therefore raises its work ceiling to 12,000,000 units per decision, against an enumerated worst case of 11,892,398, and keeps a 64 MiB memory ceiling.
4. Completion is proved from the work bound alone, not from cache retention: the cache is deliberately kept small (24,000 entries), so evicting an entry can repeat computation but can never make the search exceed the enumerated tree.
5. The output, as always, is one column.

What happened, in plain English

On the eight games used to fit it, the seven-sample search scored almost exactly the same as the five-sample reference — about 164 points less on average — while surviving 2.25 moves longer. It won three of the eight games. The rule the experiment had committed to in advance required both the score average and the survival average to improve, so a tiny score loss was enough to stop it, and the later cohorts were never opened. It also cost 3.79 times as much computation per move, and its clear and reveal rates per move both slipped slightly.

Two things about that outcome are worth knowing before quoting it.

First, the scoring mode. Those eight games were scored under the historical 7,000-point rule, not the corrected 17,000-point Hardcore rule this project uses. Because the level award is added after the search has chosen, rescoring does not change a single decision — the same games can be rescored exactly. Done that way, the −164-point gap becomes +4,336 points, and the experiment's own lower-tail numbers had already improved. On the corrected scoring, the gate this candidate failed would have passed (audit 03, C2).

Second, an independent re-test. A later exploratory experiment built a fresh parameterised search that reproduces the frozen reference exactly at its default settings, and compared five against seven samples on 64 paired games at corrected scoring. Seven samples scored +101,171 points (+34%) and +27.5 moves more, with a 95% lower bound of +47,457 and a 41–0–23 win record, on seeds that had never been read. The same work found that this is not "more samples is better" but an interaction: at five samples the fourth move of look-ahead is worth nothing measurable, and at seven it is worth about 86,000 points (finding 05).

That finding also offers a mechanism for why the original run might have looked flat. The reference search's frozen work ceiling of 3,200,000 units was sized to sit just above five-sample depth 4. A seven-sample search left at that ceiling silently runs out of budget and returns a completed depth-3 answer instead — so a careless comparison measures three moves with seven samples against four moves with five samples and reports it as a chance-sampling result. The recorded run declared a 12,000,000-unit ceiling, so it was not making that mistake; but the pattern it reported — a small score loss with a move gain — is also the signature of a shallower search, which is why the exploratory work flags the rejection as unsafe rather than wrong.

None of this is a re-run of the original configuration. It is arithmetic on the ledger's own numbers plus a separate experiment on different seeds.

The technical record

Source. approaches/fair-expectimax/chance-strata/fair-depth4-s7.cpp. Status in the experiment index: rejected — ledger-recorded, "score-neutral to worse, reduced flow, and cost about 3.8 times more work."

Fitting cohort (ledger, "Fair full-width D4 with seven chance samples", eight games on 0x3de10000...007). The section carries no scoring label; the audit below reconstructs it as historical 7,000-point Sequence scoring from the score identity, which is what makes the rescoring argument possible.

Arm	Mean score	Mean moves	Clears / move	Reveals / move	Lower-tail-25% score	Lower-tail-25% moves
Five samples (reference)	118,676	78.875	1.8875	1.0349	54,922	42.5
Seven samples	118,512.375	81.125	1.8598	0.9923	63,230	47.5

Paired change −163.625 points and +2.25 moves; three of eight score and move pairs won; paired 95% lower bound on score −38,398. No game censored at 1,000 moves. Work: 3,140,461,451 units total, 4,838,924 per move, against 806,741,660 and 1,278,513 for the reference — a 3.79× increase.

Heldout. The preregistered held-out cohort had already started when the fitting failure was computed and was stopped at a completed-game boundary. Four paired games are preserved in the artifact as explicitly exploratory and incomplete (−31,083 points, −17.5 moves, one win and three losses) and the ledger states they must not be used to compare the policy. Screen and confirmation ranges were never read. Timing and peak-memory fields were not retained after the interruption.

Verification. Optimised -Werror and ASan/UBSan builds passed the inherited engine and depth-4 tests, exact five-sample root/action/work/cache parity, complete seven-sample next-disc coverage, reveal-event stratification, determinism, reflection, metadata blindness, legality, fractional-tail arithmetic, and the cache-independent completion proof. Artifact /tmp/drop7-fair-depth4-s7.json, SHA-256 e67a28d445da4151c6d35603dfcc88bfc4c89ce2c9fe0cafd4dcf13ec2dc42ca.

The contesting records, both exploratory (development tier, not frozen protocol evidence):

• Audit 03 C2 — rescoring the ledger's own paired means as score + 2,000·moves − 9,250 turns −163.63 into +4,336. The audit checked every other paired comparison in the 7,000-point set for a sign change and found only this one and the leaf-tuning screen.
• Finding 05 — 64 paired games on 0xa51d1000–0xa51d103f, previously unread: five samples 297,327 points / 87.16 moves, seven samples 398,498 / 114.66; median, lower quartile and minimum all improve; measured work 1,296,034 versus 4,956,614 per move (3.82×). Its own limitations section is explicit: 64 games per arm, score standard deviation 38–64% of the mean, no fixed-time comparison, and seven samples makes only the next-disc expectation exact — covered-disc reveals are still sampled.

What this taught us, and what is still open

The recorded experiment rules out one exact configuration: seven chance samples, inside the frozen depth-4 search, judged by a gate requiring both the score and the survival mean to improve, on eight 7,000-point-scored fitting games. It does not rule out exact chance coverage as an idea, and the two exploratory documents above give concrete reasons to think this particular rejection was an artefact of the scoring mode and a small cohort.

What is still genuinely unknown:

• Whether the gain reproduces under a frozen, preregistered protocol at corrected scoring, on a cohort large enough for a heavy-tailed score distribution. Nothing here is that.
• Whether the reveal distribution is now the binding bias. Seven samples make the next disc exact, but a gray-disc reveal can expose a variable number of cells, so its expectation is still approximated. Fourteen samples — two per disc value — would separate "exactness" from "more samples", at roughly 7.8× the work of seven.
• What the right way to spend the extra computation is. Seven samples at depth 3 matched five samples at depth 4 in the exploratory work at one-eighth the work per move; if decision cost matters, that trade is unexplored.

Sources

• approaches/fair-expectimax/chance-strata/fair-depth4-s7.cpp — the candidate; the sample count and the completion-independent resource caps are its only differences from the reference.