Finding 05 — Search depth pays only when the chance estimator is unbiased

Status: exploratory. Evidence tier development, replicated across two independent cohorts for the headline arm. Positive result. Namespace: approaches/lifetime-objective/risk-calibration, runs runs/RUN-A51D-risk/ and runs/RUN-A51D-s7confirm/. No existing file was modified. The frozen reference is untouched; this is a new parameterized candidate that reproduces it exactly at default settings (CHECK gate: 50 moves compared, 0 mismatches).

Summary

The frozen fair-D4 reference draws five stratified samples at each chance node. The next visible disc is uniform on seven values. Five strata cannot represent seven atoms: an independent audit measured that on average 2.41 of the 7 disc values receive zero weight at every node, deterministically, so the error never averages out (audit-02-fair-d4.md, H1).

Setting the sample count to seven makes the next-disc expectation exact. Doing so is worth +101,171 mean points (+34%) and +27.5 moves (+32%) over the frozen reference on previously unread seeds.

But the effect is not "more samples is better." It is an interaction:

With a biased chance estimator, the fourth ply buys nothing. With an exact one, it is worth 86,000 points. Depth and estimator quality are complements, not independent knobs.

This offers a mechanism for one of the repository's most-repeated conclusions — that "deeper search is not automatically stronger" (strategies.md). Every depth experiment in the historical record was run on top of the same five-stratum estimator. If the extra ply mainly propagates and amplifies a systematic chance-node bias, that is exactly what one would observe.

Two cohorts

The confirmation cohort is the evidential one; the first cohort cannot be confirmation because those seeds had already influenced other decisions.

Confirmation cohort detail

Median, Q25 and minimum all improve, so the gain is not one lucky tail game. The flow rates move in the predicted direction: the gap to the 2.400 clears and 1.400 reveals per move required for indefinite survival (finding-01) narrows by about 19% and 22%. That is the mechanism that has to be true for the effect to be real.

The work bound is part of the result

Worst-case depth-4 work is Σ_{d≤4} [Σ_{l≤d} b^l + b^d] for branching b = 7 columns × strata:

The frozen bound was sized to sit just above five-stratum depth 4. A seven- stratum run left at that bound silently hits the work limit and falls back to a completed depth 3 — so a naive comparison measures depth-3-with-7-strata against depth-4-with-5-strata and reports it as a chance-sampling result. Every arm here declares its bound explicitly; the seven-stratum arms ran at 16,000,000.

Measured work per move confirms the projection: 1,296,034 at five strata versus 4,956,614 at seven, a ratio of 3.82× against a predicted 3.79×.

This is a candidate for what happened historically. The ledger's seven- stratum experiment was rejected on a small score loss with a move gain — which is also the signature of a shallower search. Combined with audit-03's finding that the same rejection reverses sign under corrected scoring (−164 → +4,336), there are now two independent reasons that rejection is unsafe. This is a hypothesis about the historical run, not a claim about it: the original configuration was not re-run.

Cost, stated plainly

This buys strength with compute. At fixed depth 4 it costs 3.82× the work per move. Under benchmarks.md that makes it a fixed-depth comparison owing a fixed-work account:

Two things follow, and the second is uncomfortable for the frozen reference:

1. Depth 3 with 7 strata matches depth 4 with 5 strata at one-eighth the work (+14,999, 95% bound −31,029, 32–0–32 — a tie, not a win). If decision cost matters, that configuration is strictly better value.
2. On this cohort the frozen depth-4 reference is statistically indistinguishable from depth 3 at five strata (+7,723 for depth 3, 95% bound −42,743, 29–0–35). The ledger's D4-over-D3 result rests on an eight-game cohort. This does not refute it — different seeds, and 64 games is still small — but it is a reason to re-examine it.

Limitations

• 64 paired games per arm. The headline d4s7-vs-d4s5 arm is replicated on two cohorts; every other row in the interaction table rests on one cohort and should be treated as a single observation.
• Score standard deviation is 38–64% of the mean, so 64-game means carry roughly ±19,000 to ±32,000 at one standard error.
• Seven strata makes the next-disc expectation exact. Covered-disc reveals are still sampled, and a reveal wave can expose a variable number of cells, so the reveal expectation is not exact at seven strata. Whether reveal sampling is now the binding bias is untested; 14 strata (two samples per disc value) would distinguish "exactness" from "more samples", at roughly 7.8× the work of seven.
• No fixed-time comparison is reported; arms ran at different thread counts on a contended machine, so only work-per-move is comparable.
• These are this repository's simulator semantics, including the two rise-boundary scoring discrepancies in audit-01.
• Mean 398,498 is far below the 1,050,000 the frozen protocol requires before a candidate may be frozen. No protected or final seed was opened or is justified by this result.

Reproduce

./approaches/lifetime-objective/risk-calibration/build.sh
./build/lifetime/risk-calibration --parity --seed-start 0xa51d0100 --parity-games 2 --parity-moves 25
B=./build/lifetime/risk-calibration
$B --depth 4 --chance-samples 5 --max-work  3200000 --seed-start 0xa51d1000 --games 64 --threads 30 --output fresh-s5.json
$B --depth 4 --chance-samples 7 --max-work 16000000 --seed-start 0xa51d1000 --games 64 --threads 30 --output fresh-s7.json
$B --depth 3 --chance-samples 7 --max-work 16000000 --seed-start 0xa51d1000 --games 64 --threads 30 --output s7d3.json
$B --depth 3 --chance-samples 5 --max-work  3200000 --seed-start 0xa51d1000 --games 64 --threads 30 --output s5d3.json