Looking five moves ahead

The reference search looks four of its own moves ahead. These three experiments each try to buy a fifth — for the most promising columns only, only when a row rise is about to happen, or everywhere at once by taking fewer chance samples to pay for it.

Two were rejected on gameplay and one was stopped by its own runtime budget before it could produce a result. None displaced the four-move search.

The intuition, and the arithmetic that makes it hard

Every extra move of look-ahead multiplies the tree by the number of columns times the number of chance outcomes considered — seven times five, or 35, in the reference configuration. Four moves is already about 1.35 million unit operations per decision. A fifth, done naively, is 35 times that.

So each of these experiments buys the fifth move by giving something up:

• Search it only where it might matter. Rank the columns cheaply, then search deeply only the best two or three. Everything the shallow ranking got wrong is now invisible to the deep search.
• Search it only at the moment it might matter. A row rises every five drops; a four-move search standing one move before a rise cannot see past it. Deepen only there.
• Search it everywhere, but look at fewer futures. Drop from five chance samples to three and spend the savings on depth. That makes the averaging over chance cruder — the thing the whole family is built on.

There is also a subtler trap, which the first experiment guarded against correctly: if a deeper search runs out of its work budget it must fall back to a shallower completed answer, so a "depth 5" policy can quietly be a depth-4 policy on most decisions. The recorded runs report their fallback rates explicitly for that reason.

How each one works, step by step

Selective extension. (1) Every legal column is searched at the root, full width. (2) At internal decision points, an exact one-move fair evaluation using the same five chance samples deterministically orders the legal columns. (3) Only the leading two or three receive the deeper search. (4) A work cap and cache bound every attempt, and an incomplete attempt returns the separately completed four-move answer. The menu of configurations — five moves with the top two columns, five with the top three, six with the top two — was frozen before any data was read.

Rise-boundary extension. (1) The policy reads how many drops remain before the next rise. (2) If exactly five remain, it uses the proved-complete five-move, top-two-columns search. (3) Otherwise it uses the unchanged four-move reference. Nothing is retuned. A fallback plan covering the last two drops of the cycle existed but was only eligible if the primary configuration failed its fit.

Full-width fifth move with three samples. (1) Three policies are compared on the same games: the reference at five samples, the reference at three samples as a sampling control, and the full-width five-move search at three samples. (2) Every legal column is searched at every decision point. (3) The five-move search carries a compile-time proof that it completes within 8,791,020 unit operations per move without depending on cache retention. (4) A mandatory pilot projects the total runtime from the first paired games before a second seed may be read.

What happened, in plain English

Selective extension fitted well and then failed on fresh games. On the four games used to choose the configuration, the five-move top-three-columns search gained about 55,000 points and 32 moves. On the eight-game held-out pass it kept a smaller gain. On the eight-game fresh screen that followed it lost — about 7,200 points and 3.5 moves — winning only three of eight games. The gain in the earlier rounds had come mostly from single long games: remove the biggest winning game from the screen and the average moves to −45,549 points. It cost 2.77 times the reference's work per move and fell back to the four-move answer on 69% of decisions.

The rise-boundary version never got a statistical result. Its runner is required to play one pair of games, project the cost of the full protocol from that measurement, and stop if the projection exceeds 40 minutes. It did: the projection came out at 44.8 minutes. On that single pilot pair the four-move reference scored 231,290 in 150 moves and the candidate 107,668 in 75 — a very large loss, with both flow rates moving the wrong way as well. One pair is not a rejection, but it gave no reason to override the time guard.

The full-width fifth move with three samples stopped the same way, and even earlier: its mandatory pilot projected 46.7 minutes against a 45-minute limit, so it paused after one paired triple. On that one game the reference scored 269,141 in 170 moves, the three-sample control 151,153 in 95, and the five-move three-sample search 182,622 in 120. The fifth move beat the sampling control it was paying for, but stayed well behind the plain reference — which is the shape you would expect if chance-sample quality matters more than depth.

The technical record

Selective extension — fair-selective-depth.cpp; rejected — ledger-recorded ("the screen lost score, survival, and flow"). None of the three sections carries a scoring label; audit 03 classifies every figure on this page as historical 7,000-point Sequence scoring from the score identity. Rescoring does not change any of these outcomes: score and survival moved together in all three.

Stage	Cohort	Reference	Candidate	Paired change
Fit (menu of 3)	4 games 0x3dd00000...003, 500-move cap	—	five moves / top three columns selected	+55,436 pts, +32.5 moves, robust utility +0.1848
Training heldout	8 games 0x3dd10000...007	120,580.75 / 82.875	133,977.125 / 89.75	+13,396.375 pts, +6.875 moves
Fresh screen	8 games 0x3ea70000...007, 1,000-move cap	182,041.125 / 115.875	174,808.375 / 112.375	−7,232.75 pts, −3.5 moves

Both held-out 95% lower bounds were negative (4/8 score wins, 3/8 move wins with one tie), and removing one +201,765-point pair made the held-out means negative. On the screen, removing the largest +260,980-point win moved the means to −45,548.86 / −25.43, and only two of eight leave-one-pair-out means were positive on both endpoints. Clears fell 2.0647 → 2.0378 per move, reveals 1.1672 → 1.1324. The candidate switched 76 of 899 same-position four-move actions (8.45%) and fell back on 620 decisions (68.97%); 60.5% of switches occurred in the last two drops of the rise cycle, which held 40.0% of decisions — diagnostic, not causal. Work 3.839M per move versus 1.387M (2.77×); the cache hit its 45,000-entry cap; 6,825.3 seconds for the gated run; no censoring. Confirmation 0x3ea80000...00f never read. Artifact SHA-256 730caf83ad6dfb9b7f473a0a058bd10a4e29ed9306f9a56daec7b710125a2ab7.

Rise-boundary extension — fair-cycle-boundary-depth5.cpp; rejected — ledger-recorded ("the pilot was adverse and also missed its runtime gate"). Pilot pair 0x3de70000: reference 231,290 / 150 moves, candidate 107,668 / 75, change −123,622 points and −75 moves; clears 2.1200 → 1.9600, reveals 1.1933 → 1.1067; every deep decision completed without fallback. The pair took 256.071 seconds and projected 2,688.743 seconds against a fixed 2,400-second ceiling, so the process exited paused before the second fitting seed, the fallback ablation, the held-out, screen and confirmation ranges. The artifact marks this pilot-only, with no formal inference. Source SHA-256 b2f2f2987c63fe878e46a6d375f27c61c6c4fafc823909554b91145291d5e680.

Full-width fifth move, three samples — fair-depth5-s3.cpp; runtime-paused — ledger-recorded. Fitting gate (never reached) required the candidate to improve both means against both controls while retaining 98% of each control's clear and reveal throughput. First paired triple on 0x3de40000: reference (four moves, five samples) 269,141 / 170; three-sample control 151,153 / 95; five moves, three samples 182,622 / 120 — behind the reference by 86,519 points and 50 moves. Throughput 2.0750/1.1417 per move versus 2.2000/1.2824 and 1.9895/1.0842. Work per move 1.521M / 0.226M / 3.904M. The triple took 623.250 seconds and projected 2,804.627 seconds against a 45-minute limit, so the run paused before 0x3de40001; no gate conclusion may be drawn from its single game. Screen and confirmation ranges never read. Artifact SHA-256 c94a28dffe9d10c3a4a2026f2acdfe14b7e5f71bd4c6c0ef0289b7612848f448.

All three passed their optimised -Werror and ASan/UBSan self-tests, including exact parity with the reference search where required, completion proofs, determinism, reflection safety, and metadata and game-seed blindness.

What this taught us, and what is still open

The three runs rule out three specific ways of buying a fifth move inside the frozen five-sample search: shallow-ranked selective extension, rise-boundary extension, and paying for depth with coarser chance sampling. They also established a practical habit that later work reused — projecting the cost of a protocol from a measured pilot and stopping before consuming seeds you cannot afford to finish.

They do not establish that a fifth move is worthless. Two open threads:

• The chance estimator may be the reason. A later exploratory experiment found that the fourth move is worth almost nothing at five chance samples and about 86,000 points at seven (finding 05). Every depth experiment in the ledger was run on top of the five-sample estimator, so "depth does not pay" and "depth does not pay here" are not distinguished by this evidence.
• The cost projections that cancelled these runs were too pessimistic. An in-progress exploratory measurement found that worst-case work bounds overestimate real work by roughly ten times at depth 5, because repeated positions collapse most of the tree, and that at depth 5 with seven chance samples the frozen 60,000-entry cache costs 2.19× more work than a 200,000-entry one for identical play — a budget decision, not a scientific one. Its own interim five-move, seven-sample arm is nevertheless far behind the four-move comparator on the games finished so far — on a partial sample its author explicitly warns is biased against it (finding 15).

Sources

• fair-selective-depth.cpp — bounded, public-state-only selective extension; every root column stays full width.
• fair-cycle-boundary-depth5.cpp — the deeper search only when the visible phase is exactly one full five-drop cycle from the next rise.
• fair-depth5-s3.cpp — the reference, a three-sample control, and the full-width fifth move at three samples, at a cycle boundary.