Entombed discs: the 3 that can never clear

Check an observation made while watching the depth-4 search play: it lets a 3 or a 4 land above its "safe zone": a column already taller than the disc's number, where it can never clear vertically, and a few rises later the game ends under it. The question is whether that disc is a cause the search cannot see, or a symptom of a board that is already lost.

The intuition

A column in Drop7 is packed by gravity, so the vertical run through every disc in it is simply the column's height. A 4 in a column of five can never clear downward until the column shrinks; if the row through it is also longer than four it cannot clear at all until a neighbour goes first. Call that disc entombed. The search's board scorer has terms for exactly this situation — but only for 1s and 2s (dead_low_numbers, low_number_height_risk). Its height penalties are blind to value: a 7 and a 3 at height five look the same. And the harm of entombing a disc arrives many moves later, outside a four-move look-ahead. So the hypothesis was that this is the missing foresight term, and a natural trigger for an "emergency mode".

How it was tested, step by step

1. Define the feature structurally from the public board: value n, column height > n, horizontal run > n; count them, weight them by height, count the gray discs trapped beneath them.
2. Dump what the scorer already knows. leafdump.cpp computes the frozen leaf's eighteen features for every one of the 5.26 million recorded positions (seven seconds).
3. Ask three questions of the corpus with analyze.py: how often the depth-1 to depth-4 behaviour policies have such a disc on the board, and at death; how many moves before death the first persistent one appears; and whether it predicts remaining lifetime beyond the eighteen leaf features, occupancy and the rise clock, on games the fit never saw.
4. Gate fixed in advance: present at ≥ 50% of depth-4 deaths, median lead time ≥ 10 moves, and a held-out partial correlation ≤ −0.05 with an incremental R² ≥ 0.005. All three were required.

What happened

The observation is real, and the explanation is not the one hoped for. Three quarters of the depth-4 policy's deaths (579 of 768 complete games) end with an entombed 3-or-higher on the board, and the first persistent one appears a median of 11 moves before the end — about two rises of warning. But the scorer already knows: adding the entombed features to the eighteen it has moves the held-out fit of remaining lifetime from R² 0.6952 to 0.6955, and at equal occupancy an entombed disc costs about two moves of life (35.0 versus 37.4 at 24–27 occupied cells). The disc is a marker of the last two rises, not their cause; the board is already at 31–40 occupied cells when it appears. The gate reads not supported as tested.

What this leaves open is the action question rather than the value question: when the search chose the move that entombed the disc, was there a non-entombing alternative that would have lived longer? That is a counterfactual replay from the entombing root, not a corpus statistic, and it is the experiment this one points at.

The record

Theory TH-20260822-entombed-disc-hazard-61faa529, experiment EX-20260822-entombed-disc-corpus-analysis-38be404e (frozen, CHECK, diagnostic, no seed opened), run RUN-20260822T051517Z-eecf1098, result in research/results/ (valid, fail, not-supported-as-tested, mechanics-only). Corpus runs/RUN-A51D-corpus/all.states, 4,863,627 non-explored positions, whole-origin split of afterstate-net/dataset.py. Depth-4: 19.1% of states carry an entombed ≥ 3 disc, 7.6% more than 40 moves before death, 69.8% within 5 moves of it; 75.4% of deaths; lead-time quartiles 5.5 / 11 / 15 moves, 57% ≥ 10, 8% ≥ 20. Held-out R² for log1p(moves to death): leaf scalar + occupancy 0.6593; 18 features + occupancy + rise clock 0.6952; plus entombed 0.6955; entombed + occupancy alone 0.5588. Held-out partial correlation of the entombed count −0.023. Depth-3 (4,096 games) agrees on every quantity. All numbers are in runs/RUN-20260822T051517Z-eecf1098/analysis.json.

Sources

• leafdump.cpp — frozen fast-leaf features for every corpus record
• analyze.py: the entombed features, prevalence, lead time, occupancy matching and the whole-origin incremental fit