---
title: "Entombed discs: the 3 that can never clear"
family: lifetime-objective
summary: Measures how often low numbered discs become impossible to clear, when death follows, and whether the evaluator notices.
status: completed
evidence: repository-verified
reads: teacher
---

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.

<EvidenceLabel status="completed" evidence="repository-verified" reads="teacher" />

## 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.

<TechnicalDetails title="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`.

</TechnicalDetails>

## 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