Exact search

Three attempts to fix a search by giving it more — more independent chance estimates, more depth, more voters — and the discovery that none of them repairs a biased judgement of the board.

The intuition

A search that looks ahead has two sources of error, and they need different cures. It can be noisy — its handful of sampled chance outcomes happen to flatter one column — or it can be biased — the evaluator at the bottom of the tree is simply wrong about which boards are good. Noise averages out with more samples; bias does not average out with anything.

These three programs are, in effect, a controlled experiment on that distinction. Each one attacks noise, in a different way, while leaving the evaluator alone:

• Independent root quadratures. Run the same complete search three times with three independent sampling salts and average the results. Pure variance reduction, no other change.
• A fully completed fourth ply. No shortlisting, no learned pruning: every legal action expanded at every depth, with the same chance sampler.
• A three-member policy ensemble. Let three complete searches vote, and when they split — no majority — keep the reference member's move rather than trusting a divided committee.

How it works, step by step

1. Read the public position — board, next disc, rise clock.
2. Expand every legal column at every choice node. "Exact" here means full-width decision nodes; the gray-disc reveals are still sampled, and the ledger says so explicitly.
3. Vary only the chance sampling between members: each gets a different fixed, deterministic salt derived from the public state.
4. Combine. Average the root values across members, or count votes, or simply search one ply deeper.
5. Fall back when the committee splits. A split vote is treated as containing no robust direction, so the reference member's action is retained.

What happened, in plain English

The first two were rejected, and together they make a single argument that shaped everything after them.

Averaging three independent chance estimates passed a four-game screen and then lost its eight-game confirmation decisively, losing seven of the eight paired games while spending about three times the work per move. The lesson recorded in the ledger is sharper than "it did not help": reducing sampling variance can remove accidental regularisation that one fixed quadrature was providing. A noisy estimate of a wrong value was, by luck, better behaved than a precise estimate of the same wrong value.

Completing a full extra ply lost its screen too, by 25,449 points and 18.75 moves, at about 24 times the cost. The ledger's conclusion is the one this family is remembered for: "a fully completed extra ply is therefore not the missing ingredient: the phase heuristic at the horizon is the central source of bias."

The voting ensemble is a looser end. A small replicated gain was reported against its own comparator, but it was never tested against the strongest reference search, so it certifies nothing.

The technical record

Independent root-quadrature ensemble — rejected, ledger-recorded. Three members, each expanding every legal action through three plies with five stratified chance samples, differing only in a fixed public-state sampling salt; root-Q arrays averaged before selection. A four-game screen passed (128,858.75 points and 88.75 moves versus 96,304 and 67.5), and the gated eight-game confirmation reversed decisively: 117,066.5 points and 80.625 moves versus 162,492.125 and 109.375, losing seven of eight paired games, at 157,047 work units per move versus 53,778. Optimised -Werror and ASan/UBSan self-tests passed determinism, reflection, public-state isolation, legality, resource bounds, scoring and single-member action parity.

Fully completed depth 4 (older leaf) — rejected, ledger-recorded. Every legal decision action expanded with the same five-stratum public chance sampler at depths 3 and 4, with compile-time worst-case bounds proving the search fits in 3,134,950 work units and 45,430 cached nodes, and a hard assertion that the requested iteration completed. Depth 3 averaged 149,621.25 points and 103.75 moves; depth 4 averaged 124,172.25 and 85 — a paired difference of −25,449 points and −18.75 moves with a 2–2 record, at 1,303,982 work units per move (about 24.1× depth 3) and 0.943 moves per second. It failed the screen, so confirmation seeds were not read.

Both of those are historical 7,000-point Sequence-scored runs, using the 7,000-point level award of the 30-drop rules instead of the 17,000-point award of the mode studied here. Their absolute scores are not evidence about the million-point target; the paired comparisons on the same games, the work measurements and the win records remain valid (mode-specific scoring correction). Four and eight games are small cohorts.

Three-member exact policy ensemble — completed, task-record only. The experiment index records "a small replicated gain was reported against its comparator, but it was not qualified against the strongest D3/D4 reference." Task-record only means the comparison lives in a research conversation and was never promoted into the ledger; no score, cohort or margin for it is retained in this repository. Repository-verified from the source: three or five members drawn from a fixed table of five policy seeds, modes vote, qmean and screen, a strict majority requirement ("A split vote contains no robust direction; retain the reference member"), defaults of 4 games at a 1,000-move cap from 0x3d707000.

What this taught us, and what is still open

• Precision is not accuracy. Three independent estimates of a biased value gave a confidently wrong answer where one noisy estimate had been accidentally hedged. This is the cleanest demonstration in the repository that variance reduction is not a free improvement.
• Depth is conditional on the leaf. The identical extra ply was harmful with the older evaluator and helpful with the fair one. "Search deeper" is not a strategy; it is a multiplier on whatever judgement sits at the bottom.
• A committee cannot invent information its members do not have. Voting and majority fallbacks are safety mechanisms, in the same category as the conservative overrides the strategy catalog describes — useful for preventing damage, not for producing insight.
• What is still open is a properly powered comparison for the ensemble against the current reference search. As it stands it is an unqualified win over an unqualified comparator, which is not evidence of anything.