Finding 06 — Flow balance is achievable: Drop7 Hardcore is a control problem, not a losing battle

Status: exploratory, evidence tier development/pilot. Built and measured in this checkout on 2026-08-20. Namespace: approaches/lifetime-objective/flow-ceiling/, run runs/RUN-FLOW-044da902f8e8/, seed lease SEEDLEASE-A52-FLOW = 0xa5230000–0xa5233fff. Nothing in docs/research/, artifacts/, research/, or any existing approach source was modified by this work. The only files created are under approaches/lifetime-objective/flow-ceiling/, build/flow-ceiling/, runs/RUN-FLOW-*/ and docs/exploratory/.

CORRECTION NOTICE (added after publication). One sentence in §3 of this document — "The gap between the two is a control gap" — is not supported by the evidence here and is superseded by finding-07. The planner measured below is privileged twice over: it plans exactly over a window and it reads the hidden board. This document never separated those two privileges. finding-07 does. It finds the sentence directionally right but unproven: a legal receding-horizon planner with the same objective, the same horizon and the same master tapes closes 58.8% of the clairvoyant-minus-D4 gap, so at most 41.2% of it is hidden information — an upper bound that was still falling when compute ran out. Knowing the future disc tape is worth nothing. What is genuinely weakened is a different claim: that the ceiling below is a reachable target. No legal policy tested reached flow balance — all 87 fair-planner games ended and every fair arm's board filled monotonically, although the best legal arm reached 2.2309 clears per move (93.0% of the requirement) with an occupancy slope still falling with more sampling. The original text is left in place unaltered below. Everything else in this document — the conservation arithmetic, the clairvoyant flow ceiling, the occupancy-equilibrium mechanism, the score composition and the wave-depth results — stands as measured. What is weakened is the reachability of that ceiling by a legal policy, which this document did not measure.

SECOND CORRECTION NOTICE (added after publication). The eight master tapes used below are unrepresentative — they favour long games — and every absolute figure measured on them is inflated in proportion to how long the policy survives. finding-12 §2 measures the size: fair depth 4 drops from 117.75 to 93.56 mean moves on 128 fresh tapes, lowest column drops only 3.7%.

The clairvoyant results in this document are not affected. Re-run on 64 fresh tapes, the clairvoyant planner at H = 7 measures 2.3663 clears and 1.3832 reveals per move against 2.3601 and 1.3785 here — a shift of +0.006, in the opposite direction to fair D4's −0.060 — with 59 of 64 fresh games reaching the move cap alive against 6 of 8. A policy that is censored at the cap on both cohorts has no room to be inflated by an easy tape draw, which is why the ceiling is robust and the comparator is not.

The one thing to re-read: every comparison against fair depth 4 below uses a D4 baseline that is 26% too generous on lifetime. "396.88 moves against fair depth 4's 117.75" should be read as "against 93.56", which makes the contrast larger, not smaller. No clairvoyant figure needs revision.

The question

finding-01 establishes two measured facts. Hardcore score is 94.29% flat row-rise bonus and correlates with lifetime at r = 0.9995, and lifetime obeys an exact conservation law: five placed discs plus seven risen covered discs enter a 49-cell board every five moves, so surviving indefinitely requires sustaining

• ≥ 2.400 numbered clears per move (12 / 5), and
• ≥ 1.400 covered reveals per move (7 / 5).

Fair depth 4 at five chance strata sustains 1.9489 and 1.0697. finding-05's best configuration, depth 4 at seven strata, sustains 2.0571 and 1.1549. Both are below the requirement and both die.

That leaves the research program facing two incompatible worlds:

• World A — strictly losing. If no line of play can sustain 2.400, every game necessarily ends, mean lifetime is bounded, and a 1,000,000-point mean cannot come from survival. It would have to come from chain scoring, which is convex in wave depth (popperCount x floor(7 x d^2.5): 7 points per disc at depth 1, 391 at depth 5, 2,213 at depth 10, 12,521 at depth 20).
• World B — a control problem. If flow balance is sustainable, survival remains the lever and the whole problem is finding a public policy that holds the balance.

The evidence in this document supports World B, and does so with a large margin.

Summary

Claim	Evidence
Flow balance is achievable	A clairvoyant receding-horizon planner holds 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 — above the 2.400 requirement. Board occupancy is flat at ~19–20 of 49 cells for 200 consecutive five-move cycles
Games stop ending	All 6 games reached the 1,000-move cap alive, mean score 3,865,157. On a separate 8-game cohort capped at 400 moves, 7 of 8 survived. Fair depth 4 on identical futures died at a mean of 117.75 moves
The mechanism is a stable equilibrium	The achievable clear rate rises with occupancy and crosses 2.400 near 20 occupied cells, so ~20 cells is an attracting fixed point. Fair D4's clear rate peaks at 2.59 near 27 cells and then falls as the board crowds, so its equilibrium is unstable and it dies
Survival dominates chains, even for a chain-seeking optimum	Maximizing score over the same window earns 1.75x the points per move (6,807 vs 3,891) and reaches wave depth 22, but cuts lifetime by 58% and loses on the mean (165.0 vs 396.9 moves; 1,123,130 vs 1,544,461 points)
Deep cascades are real but not the lever	Clairvoyant score-seeking play puts 21.5% of its waves at depth ≥ 11 and 7.8% at depth ≥ 15, against 0.22% and 0.00% for fair D4. The engine pays far more than fair play collects — and collecting it still loses
The 70,000 board clear stayed unreachable	Zero board clears in 19,610 measured moves across every policy on a realistic board, including two clairvoyant ones. audit-01's H2 fifth-drop bonus bias therefore inflates nothing in this document — though on deliberately sparse boards a clairvoyant solver schedules 9 of its 14 clears onto a fifth drop, exactly as H2 predicted

The headline number must be read carefully: the receding-horizon planner is privileged. It reads the hidden board. It is a teacher and a ceiling measurement, never a deployable policy. What it establishes is that the ceiling is above the requirement — that the target is a control problem with a solution, not an arithmetic impossibility.

Method

Why this cannot be measured in the base engine

audit-01 finding M2 records that this engine draws a covered disc's number from the reveal RNG at the moment of reveal, so there is no persistent hidden board and no well-posed clairvoyant optimum. All work here therefore runs inside the scenario engine of finding-02, whose LatentRevealSource fixes every hidden value in advance and whose StreamRevealSource is proven trajectory-identical to drop7::playHeadlessMove over 8,192 game-plays and 218,470 moves. Scores measured here are not bit-comparable with base-engine ledger figures; flow rates are comparable in expectation because the reveal marginal is identical.

The master tape: one fixed future per game

A MasterTape fixes a whole game's randomness once: the start position (the repository's real game start — an empty board with a solid covered bottom row, five moves before the first rise), one numbered disc per absolute move index, and one hidden row per absolute rise index. Rises land on fixed move indices whatever columns are chosen, so a master tape defines the same future for every policy. Nothing is redrawn when a policy deviates, and every policy in the tables below plays the same eight futures.

Receding-horizon clairvoyant play, and exactly what it bounds

A single nine-move optimum cannot answer the sustainability question. Nine moves is under two rise cycles, so a nine-move line can build height it never has to pay for; its flow rate is an over-estimate of anything sustainable. §1 below measures that over-estimate explicitly so the size of the effect is on the record.

The measurement that answers the question plays whole games: at every move, cut an exactly-solvable horizon-H window out of the fixed future, solve it exactly, play the first move of the optimal line, and re-solve from the new position.

• This is a lower bound on sustainable optimal flow. A globally optimal player is at least as good as one that re-plans every move with an H-move exact lookahead.
• It is not an upper bound and not globally optimal. A rate measured here that falls short of 2.400 would not prove 2.400 unreachable.
• Because the bound is a lower bound, a rate that reaches balance is conclusive in the direction that matters. That is what happened.
• --horizon is swept over 5, 7 and 9 so the reader can see whether the measured rate is still climbing with lookahead or has flattened.

Two objectives, because score is not flow

scenario/solver.hpp maximizes points. The conservation law is about discs. flow-solver.hpp therefore provides a pluggable per-move objective:

• rh-points maximizes the window's score delta — exactly the frozen solver's objective;
• rh-clears maximizes numbered discs cleared in the window, the quantity the conservation law is about, and therefore the more direct probe of the ceiling.

Both read the hidden board and are diagnostics only. fair-d4 (the frozen comparator, unmodified, through chooseDepth4Action) and lowest-column read only public state and are the controls.

Validation before any number was taken

Gate	Scale	Result
flow-run --self-test: master tape determinism	2 tapes	ok
a cut window reproduces the long game move for move (board, latent, score delta, wave list)	24 windows x 6 moves	ok
per-move score identity delta = 17,000 x rise + 70,000 x clears + sum wave points	a whole game, and every move of every game below	0 violations in 19,610 moves
each objective's optimum dominates on its own quantity	8 windows	ok
flow-run --cross-check: flow-solver.hpp with the points objective vs the frozen exact solver	128 scenarios of suite-h9-v1, H = 9	0 mismatches, 0 incomplete
every solved window's principal variation replayed through the engine must earn the reported value	9,175 windows across the two primary cohorts	0 mismatches
lowest-column whole-game control vs the base-engine figure of finding-01	8 games	33.12 moves / 97,110 points here vs 33.28 / 100,050 there
fair-d4 whole-game control vs finding-01	8 games	1.9808 clears and 1.0896 reveals per move here vs 1.973 / 1.090 there

The last two matter: they show the latent randomness model and this long-game driver reproduce known public-policy behaviour, so the clairvoyant numbers are being measured on the same game.

1. The nine-move optimum, and how much it over-states flow

pv-replay replays the stored principal variation of every scenario in approaches/lifetime-objective/scenario/data/suite-h9-v1.jsonl (128 positions, H = 9, all solved exactly) and measures it move by move. Fair depth 4 is played over the identical positions, tape and latent board.

Over 128 scenarios, H = 9	clairvoyant optimum	fair depth 4
moves played	1,115	1,103
lines that died inside the horizon	9	15
numbered clears / move	2.7444	2.0762
covered reveals / move	1.4565	1.0879
per-line mean clears / move	2.6951	2.0444
occupied cells, mean start → end	24.23 → 20.80	24.23 → 26.34
covered cells, mean start → end	15.61 → 14.68	15.61 → 17.61
score share: rise / board clear / chain	53.77% / 0% / 46.23%	90.86% / 0% / 9.14%
waves, deepest	2,249, depth 20	1,615, depth 14
board clears	0	0
replayed score equal to the stored optimum	128 / 128	—

Two things are visible immediately. Over nine moves the optimal line empties the board (24.23 → 20.80 occupied) while fair D4 fills it (24.23 → 26.34), which is the flow deficit made concrete on a single instrument. And the optimum clears 2.7444 per move, comfortably above 2.400.

That 2.7444 is not a sustainable rate. It is a nine-move number and the horizon is doing part of the work: a line ending at move 9 is free to spend structure it would otherwise need later. The evidence shows that it is an upper bracket. The receding-horizon measurement below is the lower bracket, and the truth lies between them.

2. Whole games under receding-horizon clairvoyant control

Eight master tapes, seeds 0xa5230000–0xa5230007, identical for every policy. 400-move cap. Commit one move per solve.

A note on the cohorts before the tables. A master tape draws its disc sequence first and its risen rows afterwards, so its length — and therefore the hidden rows — depends on the move cap. The 400-move cohort below and the 1,000-move cohort in §2.1 share seeds but are independent futures, not a continuation. That makes §2.1 an independent replication rather than a longer look at the same games, which is the stronger of the two readings.

policy	mean moves	censored at the cap	mean score	clears / move	reveals / move	occupancy slope
lowest column	33.12	0 / 8	97,110	1.1774	0.5019	+4.47
fair depth 4	117.75	0 / 8	409,985	2.0467	1.1423	+0.99
rh-points H = 9	165.00	0 / 8	1,123,130	2.2000	1.2561	+0.44
rh-clears H = 5	345.00	5 / 8	1,296,404	2.3496	1.3641	+0.053
rh-clears H = 7	366.25	6 / 8	1,422,001	2.3601	1.3785	+0.014
rh-clears H = 9	396.88	7 / 8	1,544,461	2.3663	1.3839	−0.009
requirement	∞			2.400	1.400	0.000

Clears per move is pooled over all moves of all games. The per-game means are 1.1537 / 1.9808 / 2.0812 / 2.3423 / 2.3553 / 2.3662 in the same order, so the pooled figures are not a survivorship artifact. Occupancy slope is the mean over games of the least-squares slope of occupied cells against cycle index from cycle 2 onward, in cells per five-move cycle.

The slope column is the cleanest single diagnostic in this document, and it is not an independent measurement — it is the conservation law integrated. Over any stretch of play, twelve discs enter per five-move cycle and five times the clear rate leave, so occupancy drifts by 12 − 5 x clears-per-move cells per cycle. A policy whose occupancy rises monotonically is losing the battle regardless of what its instantaneous clear rate looks like in any one position.

That slope, drawn per policy alongside the fair-planner arms of finding-07 §3 and the fresh-tape corrections of finding-12 §2:

Source data

Slope is the mean over games of the least-squares slope of occupied cells against cycle index from cycle 2 onward (finding-06 section 2). The eight-tape series (findings 06/07, 8 games per arm, paired master tapes 0xa5230000-0xa5230007) is now known to be unrepresentative: finding-12 measured that those tapes favour long games, inflating fair D4's mean lifetime by 26%, so the fresh-tape series (finding-12 section 2; 128 tapes for fair D4 and planner H5 K256, 32 tapes for planner H7 K256) is the representative one and the two series are never pooled. The finding-12 cohort table prints fair D4's fresh slope as +1.481; its paired H=5 table prints the same arm as +1.4806 — both recorded, the cohort-table value is charted. Recorded but not charted (the plan's selection): clairvoyant rh-clears fresh-tape slopes +0.075 at H=7 (64 tapes) and +0.099 at H=9 (32 tapes) (finding-12 section 6), and rh-clears H9 cap-1000 slope -0.0015 over cycles 2-200 (finding-06 section 2.1). Every fair arm's slope is strictly positive and every fair game ended: 0 of 87 eight-tape fair-planner games and 0 of 160 fresh-tape fair games reached the cap. Evidence tier: development/pilot; small paired cohorts throughout. The eight-tape clairvoyant rows are privileged diagnostics (they read the hidden board), never deployable policies.

Series	policy (planner = fair arm B of finding-07; rh- = clairvoyant)	occupancy slope	Bounds	n	Source
eight master tapes (findings 06/07; cohort-inflated)	lowest column public control; dead in 7 cycles; mean 33.12 moves	4.47 cells per five-move cycle	—	8	docs/exploratory/finding-06-flow-ceiling.md section 2 whole-game table, occupancy slope column
eight master tapes (findings 06/07; cohort-inflated)	fair D4 0/8 survive, mean 117.75 moves on these tapes (93.56 fresh)	0.99 cells per five-move cycle	—	8	docs/exploratory/finding-06-flow-ceiling.md section 2 whole-game table, occupancy slope column
eight master tapes (findings 06/07; cohort-inflated)	planner H5 K256 fair arm B; 0/8 survive, mean 128.75 moves	0.58 cells per five-move cycle	—	8	docs/exploratory/finding-07-fair-planning-ceiling.md section 3 occupancy-trend table
eight master tapes (findings 06/07; cohort-inflated)	planner H7 K16 fair arm B; 0/8 survive, mean 76.25 moves	1.27 cells per five-move cycle	—	8	docs/exploratory/finding-07-fair-planning-ceiling.md section 3 occupancy-trend table
eight master tapes (findings 06/07; cohort-inflated)	planner H7 K64 fair arm B; 0/8 survive, mean 155.00 moves	0.83 cells per five-move cycle	—	8	docs/exploratory/finding-07-fair-planning-ceiling.md section 3 occupancy-trend table
eight master tapes (findings 06/07; cohort-inflated)	planner H7 K256 fair arm B; best fair slope; 0/8 survive, mean 200.88 moves	0.31 cells per five-move cycle	—	8	docs/exploratory/finding-07-fair-planning-ceiling.md section 3 occupancy-trend table
eight master tapes (findings 06/07; cohort-inflated)	rh-points H9 clairvoyant score-seeker; 0/8 at the cap, mean 165.00 moves	0.44 cells per five-move cycle	—	8	docs/exploratory/finding-06-flow-ceiling.md section 2 whole-game table, occupancy slope column
eight master tapes (findings 06/07; cohort-inflated)	rh-clears H5 clairvoyant clear-seeker; 5/8 reach the 400-move cap alive	0.05 cells per five-move cycle	—	8	docs/exploratory/finding-06-flow-ceiling.md section 2 whole-game table, occupancy slope column
eight master tapes (findings 06/07; cohort-inflated)	rh-clears H7 clairvoyant clear-seeker; 6/8 reach the cap alive	0.01 cells per five-move cycle	—	8	docs/exploratory/finding-06-flow-ceiling.md section 2 whole-game table, occupancy slope column
eight master tapes (findings 06/07; cohort-inflated)	rh-clears H9 clairvoyant clear-seeker; 7/8 reach the cap alive; the only flat board	-0.01 cells per five-move cycle	—	8	docs/exploratory/finding-06-flow-ceiling.md section 2 whole-game table, occupancy slope column
fresh master tapes (finding-12 re-baseline)	fair D4 128 fresh tapes; mean 93.56 moves; paired H=5 table prints +1.4806	1.48 cells per five-move cycle	—	128	docs/exploratory/finding-12-fair-planner-ceiling-extended.md section 2 re-baselining cohort table
fresh master tapes (finding-12 re-baseline)	planner H5 K256 128 fresh tapes; slope delta vs D4 -0.1054 straddles zero (LB -0.3069)	1.38 cells per five-move cycle	—	128	docs/exploratory/finding-12-fair-planner-ceiling-extended.md section 2 paired re-baselined table (H=5)
fresh master tapes (finding-12 re-baseline)	planner H7 K256 32 fresh tapes; slope delta vs D4 -0.4338, upper bound -0.0509: advantage established	1.31 cells per five-move cycle	—	32	docs/exploratory/finding-12-fair-planner-ceiling-extended.md section 2 paired re-baselined table (H=7)

Spec: web/content/figures/occupancy-slope-by-policy.json · 3 source records

The whole-game clear rate above is dragged down by the opening, where the board is too sparse to clear anything (§3). Excluding the first five cycles — measuring only moves 26 and later, once the board has reached its operating point — gives the steady-state rate, which is what the conservation law is really about:

policy	clears / move, moves 26+	implied drift, cells / cycle
fair depth 4	2.1213	+1.394
rh-points H = 9	2.2643	+0.679
rh-clears H = 5	2.3855	+0.072
rh-clears H = 7	2.3963	+0.018
rh-clears H = 9	2.3966	+0.017
requirement	2.400	0.000

In steady state the clairvoyant clear-seeker sustains 99.86% of the exact requirement. The residual drift of 0.017 cells per five-move cycle would take roughly 1,700 cycles — 8,500 moves — to fill a board from its 20-cell operating point, which is well past the point where the measurement itself is the limit rather than the game.

• Lowest column loses 4.47 cells per cycle and is dead in seven cycles.
• Fair depth 4 loses 0.99 cells per cycle. It is not collapsing; it is bleeding, and 49 cells of board buys it about 24 cycles.
• rh-clears at H = 9 has a slope of −0.009 cells per cycle, which is flat. Mean occupancy at cycle 2 is 21.0 and at cycle 80 is 19.7, over eighty cycles. The board is in a steady state.

The occupancy trend, cycle by cycle (400-move cohort)

Mean occupied cells (of 49) after each five-move cycle; the count of surviving games at that cycle is in parentheses.

policy	c1	c2	c5	c10	c20	c30	c40	c60	c80
lowest column	16.5 (8)	24.8 (8)	37.9 (8)	—	—	—	—	—	—
fair depth 4	15.8 (8)	19.4 (8)	22.8 (8)	25.9 (8)	30.0 (4)	25.0 (1)	28.0 (1)	—	—
rh-points H = 9	16.8 (8)	24.6 (8)	21.0 (8)	26.0 (7)	32.8 (4)	23.0 (3)	24.0 (3)	33.0 (2)	—
rh-clears H = 9	16.0 (8)	21.0 (8)	19.1 (8)	17.8 (8)	18.2 (8)	20.1 (8)	19.8 (8)	21.9 (8)	19.7 (7)

Read the surviving-game counts as well as the means: fair D4 and rh-points appear to stabilise only because their crowded games have already died. All eight rh-clears games are still in the average at cycle 60.

2.1 The same policy on six fresh futures, run four times as long

The 400-move cap was itself the binding constraint on seven of eight games, so the primary cohort was re-run on six fresh futures with a 1,000-move cap.

rh-clears H = 9, 6 games, cap 1,000	value
games reaching the cap alive	6 / 6
mean moves	1,000.00 (censored)
mean score	3,865,157
clears / move, whole game	2.3875 (99.5% of 2.400)
reveals / move, whole game	1.3963 (99.7% of 1.400)
clears / move, moves 101+	2.3993
clears / move, moves 501–1000	2.4023 — above the requirement
implied occupancy drift, moves 501–1000	−0.012 cells per cycle
occupancy slope, cycles 2–200	−0.0015 cells per cycle
mean occupied cells at cycle 2 / 100 / 200	20.67 / 20.67 / 19.50
final occupied cells, per game	19, 23, 25, 18, 19, 13
board clears	0
score-identity violations / PV mismatches	0 / 0
windows that exceeded their solve budget	1 of 6,000

Over the second half of these games the clairvoyant clear-seeker sustains 2.4023 numbered clears per move, which is the conservation requirement, and its board drifts downward by 0.012 cells per cycle. Mean occupancy at cycle 200 is 19.50 against 20.67 at cycle 2. This is not a policy that is slowly losing; it is a policy in equilibrium, measured over 200 consecutive rise cycles and 6,000 moves with no death.

The residual whole-game deficit (2.3875 rather than 2.400) is entirely the opening: the first hundred moves are played on a board too sparse to clear (§3), and that debt is never repaid because it does not need to be — the board stabilises at 20 cells, not at 7.

3. The mechanism — a stable equilibrium at ~20 cells

Disc conservation is exact, so the number of discs a move clears can be reconstructed from the recorded occupancy trace without further instrumentation: cleared_i = occupied_before + 1 + 7 x rise_i − occupied_after. Every game below passed the check that this reconstruction reproduces the engine's own clear total.

Clears per move, conditioned on how full the board was before the move:

occupied cells before the move	rh-clears cap 1,000	rh-clears cap 400	fair depth 4	rh-points H = 9
0–9	1.08 (n=274)	0.92 (n=154)	0.87 (n=38)	0.27 (n=33)
10–14	1.48 (n=1296)	1.44 (n=667)	1.38 (n=63)	0.60 (n=73)
15–19	2.16 (n=2279)	2.08 (n=1183)	1.91 (n=173)	0.66 (n=205)
20–24	2.97 (n=1568)	2.89 (n=862)	2.10 (n=266)	0.97 (n=288)
25–29	4.21 (n=533)	4.10 (n=269)	2.59 (n=239)	1.99 (n=314)
30–34	5.70 (n=50)	8.58 (n=38)	2.26 (n=78)	4.66 (n=258)
35–39	—	13.00 (n=2)	1.58 (n=45)	4.20 (n=123)
40–44	—	—	1.40 (n=30)	3.58 (n=26)
45–49	—	—	1.00 (n=10)	—
required	2.400	2.400	2.400	2.400

n is the number of moves observed in that band. Each policy generates its own distribution over occupancies, so the bands are not matched samples and the sparsely populated cells (rh-clears above 30 cells, which it visits 50 times in 6,000 moves) should not be over-read. The two rh-clears cohorts are independent futures and agree band for band, which is the strongest internal check available on this table.

This is the whole finding in one table.

Source data

From finding-06 section 3's occupancy-conditional table. Bands are 0-9, 10-14, 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49 occupied cells; x positions are band midpoints chosen for drawing (the bands are the recorded unit). Bands are not matched samples across policies; each point carries its recorded move count n, and sparse cells should not be over-read. rh-clears cap-1000 bands above 34 cells and the rh-points 45-49 band are em-dashes in the source table (not recorded) — those points are omitted. The flat series is the exact disc-conservation requirement 2.400 = 12/5 (finding-01); the generator has no reference-line primitive, so it is drawn as a two-point polyline. rh-clears is a clairvoyant (privileged) diagnostic, not a legal policy. Evidence tier: exploratory mechanism study.

Series	occupied cells before the move (band midpoint)	numbered clears per move	Bounds	n	Source
rh-clears H=9, cap 1000 (clairvoyant)	4.5 band 0-9	1.08 clears/move	—	274	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-clears H=9, cap 1000 (clairvoyant)	12 band 10-14	1.48 clears/move	—	1,296	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-clears H=9, cap 1000 (clairvoyant)	17 band 15-19	2.16 clears/move	—	2,279	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-clears H=9, cap 1000 (clairvoyant)	22 band 20-24	2.97 clears/move	—	1,568	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-clears H=9, cap 1000 (clairvoyant)	27 band 25-29	4.21 clears/move	—	533	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-clears H=9, cap 1000 (clairvoyant)	32 band 30-34	5.7 clears/move	—	50	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	4.5 band 0-9	0.87 clears/move	—	38	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	12 band 10-14	1.38 clears/move	—	63	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	17 band 15-19	1.91 clears/move	—	173	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	22 band 20-24	2.1 clears/move	—	266	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	27 band 25-29	2.59 clears/move	—	239	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	32 band 30-34	2.26 clears/move	—	78	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	37 band 35-39	1.58 clears/move	—	45	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	42 band 40-44	1.4 clears/move	—	30	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
fair depth 4	47 band 45-49	1 clears/move	—	10	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	4.5 band 0-9	0.27 clears/move	—	33	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	12 band 10-14	0.6 clears/move	—	73	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	17 band 15-19	0.66 clears/move	—	205	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	22 band 20-24	0.97 clears/move	—	288	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	27 band 25-29	1.99 clears/move	—	314	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	32 band 30-34	4.66 clears/move	—	258	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	37 band 35-39	4.2 clears/move	—	123	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
rh-points H=9 (clairvoyant)	42 band 40-44	3.58 clears/move	—	26	docs/exploratory/finding-06-flow-ceiling.md section 3 occupancy-conditional table
survival requirement 2.400 = 12/5	4.5 exact disc-conservation rate	2.4 clears/move	—	—	docs/exploratory/finding-01-score-is-survival.md steady-state requirement (12/5)
survival requirement 2.400 = 12/5	47 exact disc-conservation rate	2.4 clears/move	—	—	docs/exploratory/finding-01-score-is-survival.md steady-state requirement (12/5)

Spec: web/content/figures/occupancy-clear-rate-equilibrium.json · 2 source records

Source

diagram-occupancy-fixed-point.svg — source and reading guide

Mechanism diagram D4 of runs/RUN-20260823T191900Z-b9f8f80d/kimi-k3-figure-plan.md. Hand-written, self-contained SVG. This is a schematic phase diagram: the two curves are smoothed shapes that follow the recorded occupancy-band means of finding-06 §3, but no data points are drawn and the caption says so. The only numbers on the canvas are the axis scales, the recorded constant 2.400 = 12/5, and the two documented crossing locations (≈20 and 25–29 cells).

What it explains

The mechanism behind docs/exploratory/finding-06-flow-ceiling.md §3 (and the measured companion figure occupancy-clear-rate-equilibrium, P7 of the plan): why ~20 occupied cells is an attracting equilibrium for the clairvoyant clear-seeker, and why fair depth 4 death-spirals.

Element-by-element

• Axes: x = occupied cells before the move (0–49 of the 7×7 board); y = numbered clears per move (0–6).
• Dashed accent line at 2.400: the exact disc-conservation requirement — 5 placed discs + 7 risen covered discs enter per five-move cycle, so survival needs 12/5 clears per move (finding-01; the companion 7/5 = 1.400 reveals per move is not shown).
• Accent S-curve (clairvoyant rh-clears, schematic): the achievable clear rate rises with occupancy and crosses 2.400 at ≈20 cells. Shape follows the recorded band means 1.08 / 1.48 / 2.16 / 2.97 / 4.21 / 5.70 (finding-06 §3, cap-1000 cohort).
• Restoring arrows around the crossing: the negative feedback — below the crossing the board fills and the clear rate rises; above it the board empties and the rate falls. Measured counterpart: occupancy stationary at 19–20 of 49 cells for 200 consecutive cycles, slope −0.0015 cells/cycle, and 2.4023 clears/move over moves 501–1000 (finding-06 §2.1).
• Dark hump (fair depth 4, schematic): peaks at 2.59 in the 25–29-cell band — barely above the requirement — then falls (2.26, 1.58, 1.40, 1.00 across the higher bands, finding-06 §3). It has no stable fixed point.
• Danger dashed arrow past the hump: "deficit compounds — the death spiral". Once the board is pushed past ~30 cells the rate drops under the requirement and the shortfall feeds itself; occupancy slope +0.99 cells per cycle on the eight tapes (+1.481 on 128 fresh tapes, finding-12 §2). This is finding-06's explanation of heavy-tailed lifetimes.
• Caption strip: the required disclaimer that this figure is schematic and the measured curves live in occupancy-clear-rate-equilibrium.
• Popovers: the requirement's derivation; the measured fixed point; the feedback mechanism; D4's hump with its recorded band values; the death spiral with its recorded slopes.

Simplifications (stated explicitly)

1. Schematic, not measured. The curves are smooth interpolations through the shapes of the finding-06 §3 band table; the plan mandates this ("a schematic phase diagram (not measured data)") because the measured version is the companion chart P7. Crossing locations are marked with ≈.
2. Bands are not matched samples across policies (finding-06 limitation 9); the schematic hides that caveat, and the popovers point back to the table.
3. The reveals-per-move requirement (1.400) is omitted to keep one axis; it is named in the requirement popover.
4. The clairvoyant curve is drawn from the cap-1000 cohort's band means; the two rh-clears cohorts agree band for band (finding-06 §3).

Sources

• docs/exploratory/finding-06-flow-ceiling.md — §2 whole-game table (slopes), §2.1 (the 1,000-move cohort), §3 (the occupancy-conditional table and the fixed-point / death-spiral mechanism).
• docs/exploratory/finding-01-score-is-survival.md — the 12/5 and 7/5 conservation constants.
• docs/exploratory/finding-12-fair-planner-ceiling-extended.md §2 — the fresh-tape D4 occupancy slope (+1.481).
• Figure spec: runs/RUN-20260823T191900Z-b9f8f80d/kimi-k3-figure-plan.md, D4.

Conventions

Same as the other diagrams in this directory: theme-aware CSS variables with light fallbacks, fig-pt/fig-pop pure-SVG hover/focus popovers matching web/content/figures/score-vs-depth.svg and the .research-fig block of web/app/globals.css.

Source: web/content/figures/diagrams/diagram-occupancy-fixed-point.source.md

A near-empty board cannot clear. Drop7 clears a disc when its number equals the length of the row or column run it lands in, so a sparse board offers almost nothing to match against: at 0–9 occupied cells even a clairvoyant clear-seeker manages 0.92 per move. The clear rate is an increasing function of occupancy.

rh-clears therefore sits on a fixed point. Below about 20 cells it clears less than 2.400 and the board fills; above 20 it clears more and the board empties. Negative feedback, an attracting equilibrium at roughly 19–20 occupied cells, and a measured steady state that held for 200 cycles. Flow balance is not merely reachable in principle — it is the stable operating point of a policy that aims at it.

For the record, and closing a cheap hypothesis: fair D4 is not dying from excessive height aversion. Its mean occupancy is 23.91 cells over all moves and 25.63 after the opening, against the clairvoyant clear-seeker's 17.93 and 17.98. D4 runs a board about six cells fuller than the clairvoyant equilibrium and still extracts less at every band below. It is not conservative; it is ineffective at the height it already plays at. See finding-07 §5.

Fair depth 4 has no such fixed point. Its clear rate peaks at 2.59 in the 25–29 band, only barely above the requirement, and then falls — 2.26 at 30–34, 1.58 at 35–39, 1.40 at 40–44, 1.00 at 45–49. Once its board is pushed past about 30 cells the rate drops below the requirement and the deficit compounds. That is a death spiral, and it explains why lifetime is heavy-tailed rather than concentrated: a fair-D4 game lives as long as it happens to stay under the threshold and then goes quickly.

The gap between the two is a control gap. At 20–24 occupied cells the clairvoyant clear-seeker extracts 2.89 clears per move and fair D4 extracts 2.10 from the same regime.

CORRECTED. The sentence above is left as originally written and is wrong as stated. This document did not separate the planner's two privileges — exact windowed planning, and reading the hidden board — so it could not attribute the gap to either. finding-07 removes them one at a time and measures the split: a legal receding-horizon planner with the same objective, the same horizon and the same master tapes closes 58.8% of this gap at H = 7 (40.0% at H = 5, where both planners are saturated), knowing the future disc tape closes none of it, and the remaining at most 41.2% is hidden board plus the known suboptimality of hindsight optimization. So the gap is majority planning — but the legal planner still died in every game.

4. Score composition, and what chains are actually worth

policy	rise bonus	board clear	chain waves	points / move
finding-01 fair D4, base engine, 64 games	94.29%	0.00%	5.71%	3,423
lowest column	98.47%	0.00%	1.53%	2,932
fair depth 4	93.81%	0.00%	6.19%	3,482
rh-clears H = 5	89.99%	0.00%	10.01%	3,757
rh-clears H = 7	87.27%	0.00%	12.73%	3,882
rh-clears H = 9	87.23%	0.00%	12.77%	3,891
rh-clears H = 9, cap 1,000	87.97%	0.00%	12.03%	3,865
rh-points H = 9	48.44%	0.00%	51.56%	6,807
clairvoyant optimum, H = 9 windows	53.77%	0.00%	46.23%	6,097

The rh-points row is the direct test of the World A hypothesis. Given the same futures and the same exact nine-move lookahead, a planner told to maximize score finds a completely different game: more than half its points come from chain waves, and it earns 6,807 points per move against rh-clears's 3,891 — a 1.75x better rate. It still loses, badly, on the only statistic that matters:

• rh-points: 165.00 mean moves, 1,123,130 mean score, 0 / 8 alive at the cap.
• rh-clears: 396.88 mean moves, 1,544,461 mean score, 7 / 8 alive at the cap.

Chain construction is real, large, and currently unexploited — and it is not the lever. Maximizing it over a nine-move window buys 1.75x the scoring rate and pays 2.4x the lifetime for it.

Wave-depth distribution

Share of all waves at each depth. finding-01 measured 44.0% at depth 1 and a deepest-ever depth of 11 across 8,444 waves in 64 fair-D4 games.

	fair D4 (this run)	rh-clears H = 9	rh-clears cap 1,000	rh-points H = 9	H = 9 optimum
waves observed	1,374	5,018	9,501	2,179	2,249
depth 1	42.36%	36.09%	36.50%	24.78%	24.68%
depth 2	24.38%	21.32%	21.49%	13.17%	13.38%
depth 3	15.87%	14.63%	14.64%	8.08%	9.96%
depth 4	8.59%	9.92%	10.20%	6.10%	8.05%
depth 5–10	8.66%	17.32%	16.50%	26.34%	31.58%
depth ≥ 11	0.22%	0.72%	0.67%	21.53%	12.36%
depth ≥ 15	0.00%	0.00%	0.06%	7.76%	2.67%
deepest	12	14	19	22	20

The same distribution as a chart, where the rh-points tail stands apart from the near-identical survival profiles:

Source data

finding-06 section 4 wave-depth table, copied verbatim; share of all waves at each depth per policy. Cohorts: fair D4 (this run), rh-clears H9 and rh-points H9 are 8-game cohorts on the eight master tapes (400-move cap); rh-clears cap-1000 is 6 games on six fresh futures at a 1,000-move cap; the H9 exact optimum column is the 2,249 waves of the 128 suite-h9-v1 optimal lines, a window sample rather than whole games. Waves observed per policy: 1,374 / 5,018 / 9,501 / 2,179 / 2,249 (carried per point). CAVEAT on the bands: 'depth >=15' is a subset of 'depth >=11', not a partition — both rows are charted exactly as recorded. The rh-points tail (21.53% at depth >=11, 7.76% at >=15, deepest 22) is the chain-construction regime that earns 1.75x the points per move and still loses on the mean (1,123,130 vs 1,544,461) because it cuts lifetime by 58%; the survival policies' profiles are near-identical (0.22-0.72% past depth 10). finding-01's independent base-engine measurement (44.0% at depth 1, deepest-ever 11 over 8,444 waves in 64 fair-D4 games) is a different cohort and engine — noted here, not charted. The eight-tape cohort is now known to favour long games (finding-12); this is a descriptive share table, not a lifetime figure. Evidence tier: development/pilot.

Series	cascade wave depth band	share of all waves	Bounds	n	Source
fair D4 (this run)	depth 1 1,374 waves observed	42.36 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
fair D4 (this run)	depth 2	24.38 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
fair D4 (this run)	depth 3	15.87 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
fair D4 (this run)	depth 4	8.59 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
fair D4 (this run)	depth 5–10	8.66 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
fair D4 (this run)	depth ≥11 deepest wave 12	0.22 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
fair D4 (this run)	depth ≥15 subset of the >=11 band	0 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth 1 5,018 waves	36.09 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth 2	21.32 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth 3	14.63 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth 4	9.92 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth 5–10	17.32 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth ≥11 deepest 14	0.72 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9	depth ≥15	0 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth 1 9,501 waves; six fresh futures, 1,000-move cap	36.5 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth 2	21.49 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth 3	14.64 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth 4	10.2 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth 5–10	16.5 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth ≥11 deepest 19	0.67 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-clears H9 cap-1000	depth ≥15	0.06 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth 1 2,179 waves	24.78 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth 2	13.17 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth 3	8.08 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth 4	6.1 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth 5–10	26.34 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth ≥11 deepest 22 — the chain-seeking tail	21.53 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
rh-points H9	depth ≥15	7.76 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth 1 2,249 waves in the 128 suite-h9-v1 optimal lines	24.68 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth 2	13.38 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth 3	9.96 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth 4	8.05 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth 5–10	31.58 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth ≥11 deepest 20	12.36 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table
H9 exact optimum (windows)	depth ≥15	2.67 %	—	—	docs/exploratory/finding-06-flow-ceiling.md section 4 wave-depth table

Spec: web/content/figures/wave-depth-distribution.json · 1 source record

So the answer to "does clairvoyant play routinely reach depth 15+?" is: only when it is told to maximize points. A clairvoyant survival player lives at depth ≤ 14 and puts 0.72% of its waves past depth 10 — barely more than fair D4. Deep cascades are not lottery tickets; they are reliably constructible with perfect knowledge, at a price in lifetime that exceeds what they pay.

5. Board clears and the known upward bias

audit-01 finding H2 (cited as "M1" in finding-02 §6 and in the brief for this work; the audit's own label is H2) records that this engine tests for an empty board before the row rise as well as after it, so a board clear that lands on a fifth drop is paid 70,000 points that the cited reference does not award at all: 87,007 here against 17,007 there. The audit's warning is specific — "a search-based candidate that can see the rise coming has a direct incentive to schedule clears there" — and a clairvoyant solver is exactly such a candidate.

Every cohort in this document is therefore reported with its board-clear accounting split out. clear awards counts 70,000 bonuses paid; fifth-drop counts awards that landed on a rise move and are therefore H2-affected; double award counts the literal both-checks-fired-in-one-move case.

cohort	moves	board-clear moves	70,000 awards	fifth-drop (H2-affected)	double awards
suite-h9-v1 clairvoyant optimal lines	1,115	0	0	0	0
suite-h9-v1 fair depth 4	1,103	0	0	0	0
lowest column, 8 whole games	265	0	0	0	0
fair depth 4, 8 whole games	942	0	0	0	0
rh-points H = 9, 8 whole games	1,320	0	0	0	0
rh-clears H = 5 / 7 / 9, 24 whole games (cap 400)	8,865	0	0	0	0
rh-clears H = 9, 6 whole games (cap 1,000)	6,000	0	0	0	0
sparse-probe-h9-v1 clairvoyant optimal lines	2,304	14	14	9	0

The H2 bias inflates nothing in the flow measurement. No policy measured on a realistic board — including two privileged clairvoyant ones with exact nine-move lookahead — ever emptied the board, in 19,610 measured moves. The verdict of this document does not touch the bonus.

The one cohort where clears do occur is sparse-probe-h9-v1, 256 deliberately sparse positions of 3–14 occupied cells, re-solved here (256/256 exact, 12 scenarios whose optimum clears, two of them twice — reproducing finding-02 §6 exactly). There the audit's prediction is confirmed quantitatively:

• 9 of the 14 clears (64%) landed on a fifth drop. H2 estimates "one board clear in five falls on a fifth drop by position alone"; a clairvoyant planner puts nearly two thirds of them there.
• Those 9 awards are 630,000 points of the cohort's 980,000 board-clear points and 4.9% of its total score of 12,872,636 — points the cited reference would not pay.
• The literal double award never fired, consistent with H2's own note that the post-rise check "is almost unreachable, since it requires all seven newly risen grays to be revealed and popped within the same move".

Anyone using board clears as a scoring lever must subtract that. The sparse cohort's flow is also worth recording, because it is the counter-example that makes §3's mechanism concrete: with a board of 3–14 cells, even the exact optimum manages only 2.2739 clears and 1.1037 reveals per move, and mean occupancy rises over the nine moves (8.44 → 9.28). An empty board is not a safe board; it is a board that cannot clear.

6. Horizon sensitivity

The receding-horizon estimate is a lower bound whose tightness depends on the lookahead. Sweeping it:

lookahead	clears / move	reveals / move	mean moves	alive at 400	occupancy slope
H = 5	2.3496	1.3641	345.00	5 / 8	+0.053
H = 7	2.3601	1.3785	366.25	6 / 8	+0.014
H = 9	2.3663	1.3839	396.88	7 / 8	−0.009
requirement	2.400	1.400	∞	8 / 8	0.000

The rate is monotone in the lookahead and nearly flat: four extra plies of exact clairvoyant search buy 0.0167 clears per move, 0.7% of the requirement. Even H = 5 — a lookahead barely longer than one rise cycle — already reaches 99.4% of the steady-state requirement and keeps five of eight games alive to 400 moves.

The horizon axis is close to exhausted: finding-02 measured H = 9 as the largest horizon solvable exactly in seconds, H = 11 as the practical limit, and H = 12 as out of reach. Whole games at H = 11 were not affordable under this work's resource budget and were not attempted, so the trend above stops at nine. The direction of the missing points is not in doubt — the sequence is monotonically increasing and a longer lookahead can only weakly improve an exact receding-horizon controller — but the exact limit is not measured.

For the strategic question this does not matter. §2.1 measures the steady-state rate at H = 9 directly and finds 2.4023 over 3,000 moves, so the requirement is already met at the horizon that is affordable. A larger horizon would raise the margin, not decide the verdict.

7. Verdict

Flow balance is achievable. The evidence supports World B: Drop7 Hardcore is a control problem, not a strictly losing battle.

Stated precisely, and with the bound direction made explicit:

• A lower bound construction — receding-horizon clairvoyant play with a nine-move exact lookahead, committing one move at a time against a fixed future — sustains 2.3875 numbered clears and 1.3963 covered reveals per move over 6,000 moves, and 2.4023 clears per move over the second half of those games, which is the conservation requirement rather than an approach to it. The whole-game shortfall is the sparse opening, and the opening is not part of any steady state.
• Under it, board occupancy is stationary at ~19–20 of 49 cells for 200 consecutive five-move cycles, with a fitted trend of −0.0015 cells per cycle and a mean of 20.67 cells at cycle 2 against 19.50 at cycle 200. Occupancy does not rise monotonically. The battle is not being lost.
• Every game reached the move cap alive: 6 / 6 at 1,000 moves and 7 / 8 at 400 moves on a separate cohort. Mean lifetime is a censored lower bound of 1,000 moves, against fair depth 4's 117.75 on identical futures. finding-01 put the lifetime a 1,000,000-point mean needs at ~294 moves; the ceiling is at least three times that.
• The mechanism is an attracting equilibrium: the achievable clear rate is increasing in occupancy and crosses 2.400 near 20 cells, so the operating point is self-correcting. Fair D4's rate crosses 2.400 only in a narrow band around 27 cells and falls beyond it, so its operating point is not — which is why its games end and why its lifetime is heavy-tailed.
• The alternative world is positively excluded, not merely unsupported. Chain-maximizing clairvoyant play on the same futures produced 1.75x the points per move, 21.5% of waves past depth 10 and cascades to depth 22 — and a mean score 27% lower, because it lost 2.4x the lifetime.

What this does not establish:

• It does not produce a policy. The planner reads the hidden board. Every measurement here is a ceiling and a teacher signal, not a candidate.
• It does not prove balance holds forever. It holds over 6,000 measured moves and 200 cycles with a slightly negative drift; nothing here excludes a slow failure on a scale ten times larger. §6.
• It does not make a qualification claim. The censored mean of 3,865,157 points is a privileged policy's score in the latent randomness model on six development seeds. It is not comparable with a ledger figure and it is not evidence about any public policy.

What this changes for the research program

The program's target is a mean above 1,000,000 in corrected five-move Hardcore. finding-01 reduced that to a lifetime target of roughly 294 moves at 3,400 points per move. This document shows a lifetime of at least 1,000 moves is achievable in the game and identifies precisely what a policy must do to get it: hold board occupancy near 20 cells, where the achievable clear rate exceeds 2.400.

That is a far more specific and far more learnable objective than "play better". It also supplies the target variable. The gap between fair D4 and the ceiling is concentrated in one measurable place — the 20–29 occupied-cell band, where the clairvoyant planner extracts 2.89 and 4.10 clears per move and fair D4 extracts 2.10 and 2.59 — and a receding-horizon clairvoyant player generates as many labelled (position, best-column) pairs as anyone wants, all of them on the trajectory a strong policy would actually visit rather than on positions a weak policy stumbles into.

QUALIFIED by finding-07. Up to 41% of this teacher's margin comes from information the student will never have, so a student fitted to clairvoyant labels is partly being asked to learn a function of hidden state. finding-07 §6 argues the better-posed target is to distil the fair planner instead — legal by construction, already 0.18 clears per move ahead of D4, and expensive enough (256 exact window solves per move) that amortising it is exactly what a learned evaluator is for.

Conversely, the chain-construction line of hope should be de-prioritised as a primary lever. It is real: the engine pays it, and chain-seeking clairvoyant play collects 3,510 chain points per move against fair D4's 216, a factor of 16. It is also, measured directly and on paired futures, a net loss when pursued.

Limitations

1. Six to eight games per policy, on one exploratory lease. Every whole-game figure here is a small cohort. The paired design (identical master tapes for every policy within a cohort) removes most of the between-game variance from the comparisons, but the absolute means carry wide intervals. The flow rates are far better determined than the scores: they average thousands of moves rather than a handful of games.
2. Every primary game is censored at the move cap — 6/6 at 1,000 moves and 7/8 at 400. Mean lifetime and mean score are lower bounds, and the occupancy trend is measured over 200 cycles, not forever. Nothing here excludes a slow failure on a much longer scale.
3. The planner is privileged. rh-clears and rh-points read the latent board. Nothing here is a deployable policy or a step toward one on its own.
4. The latent randomness model is not the base engine's model (audit-01 M2, finding-02 §2). Scores are not bit-comparable with any ledger figure. The two public controls reproduce their base-engine flow rates and lifetimes closely, which is the evidence offered that the games are the same game. Absolute scores also inherit audit-01 H1 (the level bonus is forfeited on the terminating rise, a −17,000 bias per death), which is negligible here because almost every game is censored rather than terminated.
5. A master tape is a function of the seed and the move cap, because its risen rows are drawn after its disc tape. The 400-move and 1,000-move cohorts therefore share seeds but not futures. This makes them independent samples rather than nested ones, which is the more useful arrangement here, but the two cohorts must not be described as the same games run for longer.
6. Receding-horizon play is not globally optimal. It is a lower bound. A globally optimal player would do at least as well, so the direction of the conclusion is safe, but the exact ceiling is unknown.
7. The horizon is nearly exhausted. H = 9 is the largest horizon finding-02 solves exactly within seconds; 37 of 3,175 windows (1.2%) in the 400-move run (and 1 of 6,000 in the 1,000-move run) exceeded their 12-second budget at H = 9 and were re-solved exactly at H = 7, which if anything biases the measured rate down.
8. rh-clears is not the flow-optimal policy. Maximizing clears over a nine-move window is a heuristic proxy for maximizing sustained flow; it is myopic about where it leaves the board. The true clairvoyant ceiling is at least what it achieved.
9. The occupancy-conditional table is descriptive, not causal. Each policy generates its own distribution over occupancies, so the bands are not matched samples; a policy that rarely visits 30–34 cells is measured there on few moves. The counts are reported for that reason.
10. The opening position is this engine's, not the reference's. audit-01 finding H3 records that the repository opens Hardcore above a solid gray row while the cited reference's Blitz opens with 11–21 numbered discs and no grays. Every whole game here starts from the repository's opening, so the first cycle's very low clear rate (§3) is partly an artifact of that choice. It affects the opening, not the steady state, which is why §2 reports both.
11. The timings are not timing-grade. Every run shared a 16-core / 32-thread machine with other jobs at load averages of roughly 36–40, using at most 10 threads. docs/benchmarks.md requires exclusive resources for a performance claim; none is made here. Node counts are exact and machine-independent.
12. A model contribution record under research/contributions/ is owed and was not written, because this work was scoped to create files only under approaches/lifetime-objective/flow-ceiling/ and docs/exploratory/. The coordinator should add one (level L3 for the flow-ceiling theory, the window solver, the receding-horizon measurement and this result) before it is promoted. The same debt is open for finding-02.
13. docs/research/status.md still describes 2.4 / 1.4 as "diagnostic targets from limited runs, not proven universal thresholds." finding-01 derived them exactly; this document measures a line of play that meets them. Only the coordinator edits that table.

Reproduce

# build (clang++ explicitly; the Makefile's CXX ?= clang++ loses to make's
# builtin CXX=g++, which trips a false -Werror=array-bounds in
# src/core/native/public-behavior.hpp)
./approaches/lifetime-objective/scenario/build.sh
./approaches/lifetime-objective/flow-ceiling/build.sh

# gates
./build/flow-ceiling/flow-run --self-test
./build/flow-ceiling/flow-run \
    --cross-check approaches/lifetime-objective/scenario/data/suite-h9-v1.jsonl \
    --threads 10

# 1. the nine-move optimum's flow, with a paired fair-D4 control
RID=RUN-FLOW-044da902f8e8
./build/scenario/solve \
    --input approaches/lifetime-objective/scenario/data/suite-h9-v1.jsonl \
    --threads 10 --jsonl runs/$RID/suite-h9-solved.jsonl
./build/flow-ceiling/pv-replay --input runs/$RID/suite-h9-solved.jsonl \
    --also-fair --jsonl runs/$RID/suite-h9-pv-replay.jsonl

# 2. whole games: the clairvoyant flow ceiling and its controls
for H in 5 7 9; do
  ./build/flow-ceiling/flow-run --policy rh-clears --games 8 \
      --seed-start 0xa5230000 --horizon $H --commit 1 --max-moves 400 \
      --threads 8 --time-limit 12 --jsonl runs/$RID/rh-clears-h$H-c1.jsonl
done
./build/flow-ceiling/flow-run --policy rh-clears --games 6 \
    --seed-start 0xa5230000 --horizon 9 --commit 1 --max-moves 1000 \
    --threads 6 --time-limit 12 --jsonl runs/$RID/rh-clears-h9-c1-long.jsonl
./build/flow-ceiling/flow-run --policy rh-points --games 8 \
    --seed-start 0xa5230000 --horizon 9 --commit 1 --max-moves 400 \
    --threads 8 --time-limit 12 --jsonl runs/$RID/rh-points-h9-c1.jsonl
./build/flow-ceiling/flow-run --policy fair-d4 --games 8 \
    --seed-start 0xa5230000 --max-moves 400 --threads 2 \
    --jsonl runs/$RID/fair-d4.jsonl
./build/flow-ceiling/flow-run --policy lowest-column --games 8 \
    --seed-start 0xa5230000 --max-moves 400 --threads 1 \
    --jsonl runs/$RID/lowest-column.jsonl

# 3. board clears and the audit-01 H2 bias, on the sparse probe
./build/scenario/solve \
    --input approaches/lifetime-objective/scenario/data/sparse-probe-h9-v1.jsonl \
    --threads 4 --time-limit 120 --jsonl runs/$RID/sparse-h9-solved.jsonl
./build/flow-ceiling/pv-replay --input runs/$RID/sparse-h9-solved.jsonl \
    --jsonl runs/$RID/sparse-h9-pv-replay.jsonl

# 4. the tables
python3 approaches/lifetime-objective/flow-ceiling/analyze.py runs/$RID/*.jsonl

Seed lease

SEEDLEASE-A52-FLOW = 0xa5230000–0xa5233fff, assigned to this work by the coordinator, a sub-range of the SEEDLEASE-A52 reserve recorded in lease-map.md. Role: exploratory development diagnostic; once read these seeds are development data permanently. Only 0xa5230000– 0xa5230007 were opened. flow-run refuses to start outside the lease. suite-h9-v1 and sparse-probe-h9-v1 were consumed as already-minted data from SEEDLEASE-A51D-SCEN; no new seed from that lease was drawn.

Environment

AMD clang 23.0.0git, -O3 -std=c++20 -pthread -Wall -Wextra. 16 physical cores / 32 logical, 125 GiB RAM, shared with other jobs throughout at load averages of roughly 36–40; at most 10 threads were used. Frozen sources were consumed unmodified and their hashes are recorded at build time in build/flow-ceiling/sources.sha256. The frozen fair depth-4 comparator is compiled from a generated build-tree copy whose only changed line is the entry point, and the build refuses to proceed if more than that one line differs.