---
title: Is the required clearing rate reachable at all?
family: lifetime-objective
summary: Lets a privileged planner test whether any policy can remove discs as fast as the game adds them.
status: completed
evidence: development-tier exploratory findings
reads: oracle
---
Let a planner cheat — read the hidden numbers under the gray discs and plan
against a future it already knows — to find out whether **any** way of playing
can remove discs as fast as the game adds them.
<EvidenceLabel status="completed" evidence="development-tier exploratory findings" reads="oracle" />
<Callout title="Everything headline on this page is a cheat, and that is the point" tone="warn">
The planner measured here reads the hidden value under every gray disc and the
whole future sequence of discs. It is an **oracle**: a measuring instrument and
a possible source of training labels, never something that could play the game
for real. Its numbers are a *ceiling*. They say what the game permits, not what
a legal player achieved. This page also measures a **legal** planner with the
same design and the cheating removed; every number below says which one it is.
</Callout>
## Why the question is worth asking
The [family page](/approaches/lifetime-objective) has the arithmetic: twelve
discs enter a 49-cell board every five moves, so surviving forever needs
**2.400 numbered clears and 1.400 gray-disc reveals per move**. The reference
policy manages 1.973 and 1.090. Two completely different research programmes
follow from what that means:
- **If no line of play can sustain 2.400**, every game necessarily ends, average
lifetime is capped, and a million-point average would have to come from
spectacular chain reactions instead.
- **If flow balance is sustainable**, survival is the lever and the whole
problem is finding a legal policy that holds the balance.
Nobody had checked which world this is. That is what this approach is for.
## The intuition: an empty board is not a safe board
The obvious guess is that the reference policy dies because it lets the board
get too full, and that a tidier player would live longer. The engine says the
opposite about the sparse end. A disc clears when its number equals the length
of the run it lands in, so a nearly empty board offers almost nothing to match
against.
<BoardCompare
before="0000000000000000000000000000000000047000008888888"
after="0000000000000000000060005598074988849888888888888"
beforeLabel="9 discs · 14 of 49 drops clear"
afterLabel="25 discs · 39 of 49 drops clear"
caption="Two positions from one game played through the repository's rules engine (move 2 and move 16). For each board, every one of the seven disc values was dropped into every legal column and the engine asked whether anything popped. On the sparse board only 14 of the 49 possible drops clear anything; on the fuller board, 39."
/>
That is the mechanism behind everything below. **The rate at which a board can
clear discs rises with how full it is** — up to a point. So there is some
occupancy at which the achievable clear rate crosses 2.400, and a policy that
can hold the board there is in equilibrium: below it the board fills, above it
the board empties. The question is whether that crossing point exists on a real
board and whether anything can sit on it.
## How the measurement works
1. **Fix a whole future in advance.** A *master tape* pins the starting
position, one disc per move index, and the hidden numbers of every row that
will ever rise. Every policy in a comparison plays the same tapes, so nothing
is redrawn when one of them deviates. This needs a simulator with a
persistent hidden board, which is why the work runs inside the
[scenario engine](/approaches/lifetime-objective/scenario) rather than the
base one.
2. **Plan exactly over a short window.** At every move, cut a window of `H`
moves out of that fixed future, solve it exactly: every legal move, every
consequence, no [leaf evaluator](/learn/glossary) and no guessing — play the
first move of the best line, and re-solve from the new position. Windows of
5, 7 and 9 moves were used; 9 is the largest that solves exactly in seconds.
3. **Choose what to maximise.** One objective maximises the window's **points**;
the other maximises **numbered discs cleared**, which is the quantity the
conservation law is about. Both read the hidden board. The reference depth-4
search and a trivial "always drop in the lowest column" player are the
controls, and both read only public information.
4. **Then take the cheating away, one privilege at a time.** The same planner,
given no hidden values, draws `K` complete guesses of what is under the
covers, solves the window exactly against each guess, and plays the column
with the best average. A third arm is given the future disc sequence but not
the hidden board. A mechanical gate proves the legal arm is legal: over
20-plus consecutive real decisions, replacing *every* hidden value and
substituting a completely different future left its chosen column identical.
## What happened
**The ceiling is above the requirement.** Told to maximise clears, the
clairvoyant planner with a nine-move window sustained **2.3875 clears and 1.3963
reveals per move over 6,000 moves**, and **2.4023 clears per move over the
second half of those games — at the requirement rather than approaching it**.
Board occupancy sat flat at 19–20 of 49 cells for 200 consecutive five-move
cycles, and all six games reached the 1,000-move cap still alive. The reference
policy on identical futures died. So flow balance is achievable *in the game*:
this is a control problem with a solution, not an arithmetic impossibility.
**And the equilibrium is self-correcting.** Conditioning on how full the board
was before each move, the clairvoyant clear-seeker extracts 2.16 clears per move
at 15–19 occupied cells, 2.97 at 20–24 and 4.21 at 25–29, so below about 20
cells it clears less than 2.400 and the board fills, above it clears more and
the board empties. The reference policy has no such fixed point: its rate peaks
at 2.59 near 27 cells and then *falls* — 2.26, 1.58, 1.40, 1.00 as the board
crowds, which is a death spiral, and explains why its lifetimes are so
unevenly distributed.
**Chain building is real, and it is not the lever.** Given the same futures and
the same exact look-ahead, a planner told to maximise *points* plays a
completely different game: more than half its score comes from chain reactions,
it reaches 22-deep cascades, and it earns 1.75 times the points per move. It
also lost 2.4 times the lifetime and finished with a **lower** mean score. Deep
cascades are constructible with perfect knowledge; collecting them still loses.
**No legal policy got near the ceiling.** Take the hidden board away and the
same planner, averaging over 256 guesses, closes only part of the gap — and
much less than was first reported. On 32 fresh futures played by all four arms it
sustains **2.0260 clears per move**, against **1.9004** for the reference policy
and **2.3637** for the clairvoyant planner on those same futures: about **27% of
the gap** at the longer of the two affordable windows, and 22% at the shorter
one. Across every legal arm ever
run in this approach — 87 games in the first study, then 128 and 32 more on
fresh futures — **not one game survived to the move cap**, and every one of
those boards filled up monotonically.
**Knowing the future is worth nothing; knowing what is under the covers is
worth everything.** The arm handed the exact future disc sequence was no better
than the fully legal arm. But the shortfall against the clairvoyant planner
grows sharply with how much is hidden: **+0.05 clears per move on boards with
10–14 discs, +3.54 on boards with 30–34**, where about twenty covered cells are
unknown. More guessing closes the sparse end and does essentially nothing at the
crowded end, which is exactly the regime a policy running a flow deficit gets
pushed into.
**More guessing eventually makes it worse.** The obvious lever — raise the number
of guessed futures — turns over. Going from 256 guesses to 1,024 was worse on
five of six paired futures and cut mean lifetime from 182 to 100 moves. That is
the expected behaviour of this kind of planner: inside every guess it plans as
if the guess were true — [strategy fusion](/learn/glossary), so more compute
buys a more faithful optimisation of a biased objective. There is no setting of this dial at which the planner
approaches 2.400.
<Callout title="One of the numbers on this page corrects an earlier one" tone="info">
The first study reported the legal planner at **2.2309 clears per move**, closing
**58.8%** of the gap. Both figures were measured on eight futures that turned out
to favour long games: the reference policy scores 117.75 mean moves on those
eight and **93.56** on 128 fresh ones: a 26% inflation, confirmed by two
independent measurements of the same policy at 94.06 and 93.78 moves. The
clairvoyant planner, which was hitting the move cap either way, was **not**
affected (2.3601 → 2.3663). So the ceiling stands and the legal planner's share
of the gap was roughly halved. Both the original claim and its correction are
retained in the record rather than edited away.
</Callout>
<TechnicalDetails title="The record: cohorts, configurations and gates">
Three retained exploratory findings, all at the repository's `development` /
`pilot` evidence tier, all measured in this checkout on 2026-08-20, none with a
row in the [experiment index](/docs/research/experiment-index) or an entry in the
[ledger](/docs/research/history):
- [`finding-06-flow-ceiling`](/docs/exploratory/finding-06-flow-ceiling) — the
clairvoyant ceiling and the equilibrium mechanism.
- [`finding-07-fair-planning-ceiling`](/docs/exploratory/finding-07-fair-planning-ceiling) —
removing the two privileges one at a time.
- [`finding-12-fair-planner-ceiling-extended`](/docs/exploratory/finding-12-fair-planner-ceiling-extended) —
the re-baselining on fresh futures, and the correction above.
**Clairvoyant clear-seeker, nine-move window, one move committed per solve**
(finding-06 §2.1): 6 games, 1,000-move cap, 6/6 censored alive, mean score
3,865,157 (not comparable with any ledger figure — different randomness model),
2.3875 clears / 1.3963 reveals per move whole-game, 2.4023 clears per move over
moves 501–1000, occupancy slope −0.0015 cells per cycle over cycles 2–200, mean
occupancy 20.67 at cycle 2 and 19.50 at cycle 200, 0 board clears.
**Paired controls on the same eight master tapes** (finding-06 §2): lowest
column 33.12 mean moves and slope +4.47; reference depth-4 search 117.75 moves
and slope +0.99; points-maximising clairvoyant planner 165.00 moves, 1,123,130
mean score; clear-maximising clairvoyant planner at the nine-move window 396.88
moves, 1,544,461 mean score, 7 of 8 censored alive.
**Occupancy bands, clears per move** (finding-06 §3): clairvoyant clear-seeker
1.08 / 1.48 / 2.16 / 2.97 / 4.21 / 5.70 across the 0–9, 10–14, 15–19, 20–24,
25–29 and 30–34 bands; reference depth-4 search 0.87 / 1.38 / 1.91 / 2.10 / 2.59
/ 2.26, then 1.58, 1.40 and 1.00 in the three fullest bands. The bands are
descriptive, not matched samples: each policy generates its own occupancy
distribution and the counts are published for that reason.
**The legal planner, re-baselined** (finding-12 §2, §9): at the seven-move
window with 256 guesses on 32 fresh master tapes, 2.0260 clears and 1.1256
reveals per move against the reference search's 1.9004 and 1.0339 on the same
tapes; +18.81 mean moves with a one-sided 95% bootstrap lower bound of +0.56;
mean score +66,365 with a lower bound of −505, i.e. straddling zero. At the
five-move window with 256 guesses on 128 fresh tapes, only the flow rates have
lower bounds above zero (+0.0580 clears, lower bound +0.0157) — lifetime and
score straddle zero and the planner wins the lifetime race on 58 of 128 tapes.
Corrected gap shares on the matched 32-tape cohort: 22.2% at the five-move
window, 27.1% at the seven-move window, against finding-07's 40.0% and 58.8%.
**Gates, all before any number was taken** (finding-06, finding-07, finding-12):
master-tape determinism; a cut window reproduces the long game move for move;
the per-move score identity `17,000 × rises + 70,000 × board clears + wave
points` held with **0 violations in 19,610 moves**; the window solver reproduced
the frozen exact solver on **128 scenarios with 0 mismatches**; every solved
window's best line replayed through the engine for its reported value, 9,175
windows, 0 mismatches; and the information-boundary gate on the legal arm
described above. All seven `--self-test` gates still pass after finding-12's
changes.
**Limitations the findings state about themselves:** eight to 128 games per arm;
every clairvoyant game is censored at a move cap, so its lifetime is a lower
bound; the planner is a determinization (hindsight-optimisation) controller and
not the optimal legal planner, so its rate is a floor and the "information
share" is an upper bound; a nine-move fair arm was never affordable and is not
reported; the K = 1024 arm is 6 of 8 games, stopped after four hours; the
scenario engine's randomness model is not the base engine's, so scores here are
not comparable with ledger figures; and no timing on this page is timing-grade —
the machine carried load averages of 36–60 throughout.
</TechnicalDetails>
## What this taught us, and what is still open
- **The programme has a target, and it is not a score.** Hold roughly 20 of 49
cells occupied, where the achievable clear rate crosses 2.400. That is a
learnable objective in a way that "play better" is not.
- **A negative result with a direction.** Determinized planning — imagine many
complete boards, plan in each, average — has now been pushed to its limit on
both of its dials, more guesses and a longer window, and neither gets there.
The family is measured out, and that rules something out rather than merely
failing to confirm it.
- **The residual has a shape.** It is concentrated on crowded boards, where the
hidden state is largest, and it is not something a better [leaf](/learn/glossary)
can absorb. Whatever comes next has to *reason about* what is under the covers
instead of averaging over guesses of it: a belief-state search, or an explicit
reason to open a cover in order to learn from it. Nothing in this repository
does that yet.
- **One tempting repair is already ruled out.** "Make the reference search keep
a lower board" is the wrong variable: it already runs a board about six cells
*fuller* than the clairvoyant equilibrium and still extracts less at every
occupancy band. It is not too conservative; it is ineffective at the height it
already plays at.
- **Still open, and stated as such by the record:** whether flow balance holds
beyond the 6,000 measured moves; where the true legal ceiling is, since the
planner measured here is not the best possible legal planner; and what the
nine-move legal planner would do, which was never affordable.
Related: the [distillation attempt](/approaches/lifetime-objective/planner-distill)
that tried to compress this page's legal planner into a network, and the
[benchmark validation](/approaches/lifetime-objective/suite-validation) that used
the same exact-window machinery to grade positions.