Finding 01 — Hardcore score is survival time, and survival is a flow-balance problem

Status: exploratory, evidence tier development (64 paired whole games on a fresh exploratory lease). Reproduced in this checkout on 2026-08-20. Namespace: approaches/lifetime-objective/score-decomposition, run runs/RUN-A51D-d4/, seed lease SEEDLEASE-A51D. Nothing in docs/research/ was modified by this work.

Summary

Two facts, both measured rather than assumed:

1. Score is lifetime. Across 64 fair-D4 games the correlation between final score and moves survived is r = 0.9995. 94.3% of all points came from the flat 17,000-point row-rise bonus, 5.7% from chain waves, and 0.0% from board clears — there were none in 64 games. Points per move was 3,423, against a theoretical ceiling of 17,000 / 5 = 3,400 from rises alone.

2. Lifetime is a conservation law. Every five-move cycle inserts 5 placed discs + 7 risen covered discs = 12 discs onto a 49-cell board. A policy therefore survives indefinitely only if it sustains

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

   Fair D4 sustains 1.973 clears and 1.090 reveals per move. It is running a structural deficit of 18% and 22%. That deficit, not chain design, is what ends its games.

The million-point target is consequently a lifetime target: at 3,400 points per move a 1,000,000 mean requires a mean lifetime near 294 moves, versus the 94.06 measured here. That is a 3.1× survival problem.

Measured reference cohort

Fair D4, unmodified frozen source, 64 games, seeds 0xa51d0000–0xa51d003f, 2,000-move cap, 32 threads, 408 s wall.

This is consistent with, and independently reproduces, the ledger-recorded figure of 308,295.578 points at 90.031 moves over 64 games. It is a fresh-seed reproduction, not a replay of the historical cohort.

Score decomposition

The engine's score is exactly

score = 17,000 x rises + 70,000 x boardClears + Σ_waves popperCount x floor(7 x depth^2.5)

The runner asserts this identity on every game; it held for 64/64.

Chain-wave points are small by construction: a wave awards 7 points per disc at depth 1, 39 at depth 2, 109 at depth 3, 224 at depth 4, 391 at depth 5. A five-deep, seven-disc wave — a spectacular play — is worth 2,737 points, or 0.8 of one row rise. The 70,000-point board clear is worth 20.6 moves of survival but was never once achieved in 64 games of the strongest known policy.

Flow balance predicts lifetime

Sorting the 64 games by lifetime and grouping into quartiles:

Correlations over whole games: clears/move vs. moves r = 0.804, reveals/move vs. moves r = 0.770, score vs. moves r = 0.9995.

The single million-point game in the cohort (1,017,234 points, 280 moves, seed 0xa51d0033) reached 2.229 clears and 1.254 reveals per move — closer to the balance point than any other game, and still below it. Its 82,234 chain points are 8.1% of its score; the other 91.9% is survival.

Why this matters for the research program

It reframes the objective. The program's documents describe three competing jobs — stay alive, open the board, prepare chains — and much of the historical work went into chain structure and score prediction. The measurement says the third job is worth 5.7% of the score and only matters insofar as clearing discs is how you stay alive. There is one job with two names.

It explains a documented failure pattern. docs/research/status.md records that learned value models repeatedly fit visited states and then failed to rank unplayed siblings. Part of that difficulty is target conditioning: raw score is a 17,000-quantized, heavy-tailed variable with a standard deviation of 187,502 in this cohort — 58% of its own mean. Remaining lifetime is the same quantity divided by 3,400, with no heavy tail and no quantization. Per-move flow (clears and reveals) is denser still: it yields one labelled observation per move rather than one per game, with a bounded range.

It gives the 2.4 / 1.4 diagnostic a derivation. status.md describes "roughly 2.4 numbered clears and 1.4 reveals per move" as an empirical region "associated with stable long games" and warns they are "diagnostic targets from limited runs, not proven universal thresholds." They are neither empirical nor approximate: they are 12/5 and 7/5, the exact disc-conservation requirement of the five-move rise cadence on a 49-cell board. They are necessary conditions for unbounded survival, and this cohort confirms the engine obeys them.

It bounds what search can do. Rises occur every five moves; a depth-4 search sees at most one rise boundary and cannot observe the accumulation of a flow deficit over the eight-plus cycles that separate a 90-move game from a 294-move one. The deficit is a slow variable. This is a direct, quantitative argument for a learned long-horizon evaluator rather than deeper exact search, and it says what that evaluator should predict.

Trivial-policy control cohort

To separate "value of search" from "value of merely being alive", three deliberately weak policies were run on the same 64 seeds:

Fair D4 buys 3.2× the lifetime of uniformly random legal play. Reaching the target requires buying a further 3.1× on top of that.

Limitations

• 64 games on one exploratory lease. The score standard deviation is 58% of the mean, so the mean is known to roughly ±46,000 at one standard error.
• The flow-balance requirement is a necessary condition for indefinite survival, not a sufficient one; a policy can meet it locally and still die by stacking a single column into the top row.
• Board clears were absent here, so this cohort says nothing about whether a clear-seeking policy could change the economics. At 70,000 points a clear is worth 20.6 moves, so a policy achieving one clear per 20 moves would double the points-per-move rate. Nothing in this repository has measured whether that is reachable; that is a separate open question, not a refuted one.
• These figures describe the repository's simulator. See audit-01-engine-fidelity.md for two identified rise-boundary scoring discrepancies against the cited reference implementation, both of which act at rise boundaries and therefore act directly on the quantity measured here.

Reproduce

./approaches/lifetime-objective/score-decomposition/build.sh
./build/lifetime/decompose --policy fair-d4 --seed-start 0xa51d0000 \
    --games 64 --max-moves 2000 --threads 32 --output runs/RUN-A51D-d4/d4-standard64.json