Turning the reference search's constants into dials

The strongest policy in this repository has a handful of numbers baked into it that nobody had ever varied. This approach makes three of them adjustable without changing anything else, so they can be swept and measured instead of assumed.

Two sweeps have been run through it. The first — the one the approach is named after — is a valid negative: the death penalty is at its stop and is not a lever. The second is a replicated positive: the number of chance samples was too small to represent the game's own randomness, and fixing it was worth about a third more score.

The intuition

At the bottom of its look-ahead, the search puts a number on every board it can reach. A board where the game is over gets a fixed, very large negative number — the terminal utility, set to −1,000,000. Every other board gets a score from a hand-written leaf evaluator, and in practice those come out between roughly 200 and 11,000.

So the death penalty is about a hundred times larger than the entire spread of everything else. The decision rule is effectively: first avoid dying within four moves; only then prefer the nicer-looking board. Given that surviving is where 94% of the points come from (score decomposition), that constant is the single number pointed most directly at the objective — and it had never been swept. Perhaps the search was too timid, or not timid enough.

The second dial is subtler. At a chance node the search has to average over the next disc, which is equally likely to be any of seven values. The frozen reference averages over five representative samples. Five buckets cannot cover seven equally likely outcomes, so some disc values get no weight at all — the same ones, every time, at every node. That is not noise that cancels out; it is a fixed bias built into every estimate the search makes.

How it works, step by step

1. Input. Exactly what the reference reads: the visible board, the visible next disc, and how many drops remain before the next rise. Nothing else.
2. Reuse, not reimplementation. The leaf evaluator, the chance stratification, the board canonicalisation, the cache keys, the column ordering, the work accounting and the legal fallback are the unmodified frozen code, included as a library. Only the depth-limited driver is rewritten, so that three constants become command-line options.
3. A parity gate before any game is played. At its default settings the parameterised driver must pick the identical column as the frozen reference on every move. Recorded result: 50 moves compared, 0 mismatches.
4. Paired sweeps. Each setting plays the same ordered list of game seeds as the reference, so scores can be compared game by game rather than as two separate averages.
5. Output. A column, and per-game records through the family's shared harness, including the survival-flow rates that the whole family watches.

What happened — the death penalty

Nothing at all, in the direction that mattered. Making death fifty times more expensive produced byte-identical games: same scores, same lifetimes, 0 wins, 64 ties, 0 losses. The search is already as risk-averse as the constant can make it. Moving the other way, towards recklessness, only cost — and it cost in an instructive shape: at a penalty of −100,000 the policy cleared slightly more discs per move and still died sooner. Throughput is not the objective; throughput sustained without dying is.

That closes a cheap hypothesis. The gap between a 94-move average and the roughly 294 moves a million-point average would need cannot be bought by re-pricing death.

The technical record — terminal utility

Source: finding-04-terminal-utility-saturated, status exploratory, evidence tier development, recorded as a valid negative result. Six arms, 64 paired games each, seeds 0xa51d0000–0xa51d003f on the exploratory development lease SEEDLEASE-A51D, 2,000-move cap, common seeds across arms.

Terminal utility	Mean score	Mean moves	Clears/move	Paired delta vs reference	W–T–L	95% lower bound
−50,000,000	321,992	94.06	1.973	0	0–64–0	0
−10,000,000	321,992	94.06	1.973	0	0–64–0	0
−3,000,000	321,992	94.06	1.973	0	0–64–0	0
−1,000,000 (frozen)	321,992	94.06	1.973	0	—	—
−300,000	321,652	93.98	1.973	−340	1–62–1	−1,039
−100,000	320,161	92.92	1.987	−1,831	4–29–31	−4,563

The identical-game result at magnitudes at or beyond 1,000,000 is exact and needs no statistics. The −300,000 and −100,000 deltas are single-cohort estimates. Only the terminal utility was varied; the leaf weights are frozen, and a jointly re-tuned leaf could in principle move the saturation point — which is what the leaf reweight work went on to test, using this same parameterised search as its base.

What happened — the number of chance samples

Averaging over seven disc values instead of five was worth about a third more score, and it replicated on a second, previously unread set of games. On the confirmation cohort the seven-sample search scored 398,498 against 297,327, and lasted 114.66 moves against 87.16. Median, lower quartile and minimum all improved, so this is not one lucky long game; and the survival-flow rates moved in the direction the mechanism predicts, closing roughly a fifth of the gap to the clears and reveals per move that indefinite survival requires.

The more interesting part is that this is an interaction, not a "more samples is better" result. With the biased five-sample estimator, adding a fourth move of look-ahead bought nothing measurable. With the exact seven-sample estimator, the fourth move was worth about 86,000 points. Depth and estimator quality are complements. That offers a mechanism for one of the repository's most repeated conclusions — that deeper search is not automatically stronger — because every historical depth experiment sat on top of the same biased estimator.

It is also not free: the exact estimator costs 3.82 times the work per move at the same depth, and a variant one move shallower with the exact estimator ties the frozen reference at an eighth of the work.

The technical record — chance samples

Source: finding-05-chance-strata, status exploratory, evidence tier development, replicated across two independent cohorts for the headline arm. Positive result. Same parity gate (50 moves, 0 mismatches).

Comparison	Delta score	95% lower bound	Delta moves	W–T–L	Verdict
depth 4: 7 samples − 5 samples	+101,171	+47,457	+27.50	41–0–23	significant
depth 3: 7 samples − 5 samples	+7,276	−45,961	+2.42	34–0–30	not significant
7 samples: depth 4 − depth 3	+86,172	+26,468	+22.39	40–0–24	significant
5 samples: depth 4 − depth 3	+7,723	−42,743	+2.69	29–0–35	not significant

Cohorts: a first cohort on 0xa51d0000–0xa51d003f, whose seeds had already influenced other decisions and therefore cannot serve as confirmation (+71,138, lower bound +5,826); and a confirmation cohort on previously unread seeds 0xa51d1000–0xa51d103f (+101,171, lower bound +47,457). 64 paired games each.

Confirmation cohort detail: mean score 297,327 versus 398,498; mean moves 87.16 versus 114.66; clears per move 1.9489 versus 2.0571; reveals per move 1.0697 versus 1.1549; 0 censored games and 0 score-identity violations in both arms; games at or above one million: 0 versus 2.

Cost, at fixed depth 4: work per move 1,296,034 versus 4,956,614, a ratio of 3.824 against a predicted 3.79. Depth 3 with seven samples costs 156,834 per move and ties depth 4 with five samples (+14,999, 95% lower bound −31,029, 32–0–32).

Limitations as recorded. 64 paired games per arm; only the headline arm is replicated, and every other row of the interaction table rests on a single cohort. Seven samples makes the next-disc expectation exact, but gray-disc reveals are still sampled, so the reveal expectation is not exact — whether reveal sampling is now the binding bias was tested separately in the reveal sampling work. No fixed-time comparison is reported. And the headline mean of 398,498 is far below the 1,050,000 the frozen protocol requires before a candidate may even be frozen: no protected or final seed was opened or is justified by this result.

What this taught us, and what is still open

Two constants, two opposite outcomes, and the difference between them is the lesson. The risk constant was saturated: it had no reachable setting that changed a decision, so no amount of tuning could have helped. The sample count was biased: it was systematically wrong in a way that no amount of extra depth could average away. Sweeping a constant is cheap; assuming which way it will go is not.

The negative result also narrowed the search for the remaining headroom. The search already minimises death within its horizon as its first priority, and its horizon is four moves against a rise every five. Whatever is missing has to live in the long-horizon content of the leaf evaluator, not in the search's appetite for risk.

Still open, and cautionary: whether the depth gradient keeps climbing once the estimator is exact. A later, still in-progress test in the fast engine approach reports the fifth move of look-ahead reversing rather than continuing the gain, on interim partial cohorts. Read that record's own caveats before quoting it.