---
title: Seven chance samples instead of five
family: fair-expectimax
summary: The next disc has seven equally likely values, but the reference samples only five. This tests all seven.
status: rejected
evidence: ledger-recorded
reads: public
---

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.

<EvidenceLabel status="rejected" evidence="ledger-recorded" reads="public" />

<Callout title="The record is contested" tone="warn">
The ledger rejected seven samples. Two later exploratory documents in this
repository argue that rejection does not hold: one shows the losing margin
reverses sign when the games are rescored under corrected Hardcore rules, and
one re-ran the comparison from scratch and measured a large gain. Both are read
in full at the bottom of this page. Nothing has been re-run under a frozen
protocol, so the question is open, not settled in either direction.
</Callout>

## The intuition

When the game deals the next disc, it picks uniformly from
<Disc n={1} /> <Disc n={2} /> <Disc n={3} /> <Disc n={4} /> <Disc n={5} />
<Disc n={6} /> <Disc n={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](/learn/glossary). 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](/docs/exploratory/audit-02-fair-d4), 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 <Disc n={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

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](/docs/exploratory/audit-03-claim-arithmetic), 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](/docs/exploratory/finding-05-chance-strata)).

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.

<TechnicalDetails title="The technical record">

**Source.** `approaches/fair-expectimax/chance-strata/fair-depth4-s7.cpp`.
Status in the [experiment index](/docs/research/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](/docs/exploratory/audit-03-claim-arithmetic) 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](/docs/exploratory/finding-05-chance-strata) — 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.

</TechnicalDetails>

## 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.