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Entombed numbered discs are an early, unpriced death predictor for the fair depth-4 search

not-supported-as-testedassessedevidence: mechanics-onlydiagnosticTH-20260822-entombed-disc-hazard-61faa529
No explanation has been written for this theory yet. Add web/content/research/TH-20260822-entombed-disc-hazard-61faa529.mdx and it will appear here. The registered record is shown below.

The registered record

Claim

A numbered disc of value n whose column already holds more than n discs and whose horizontal run already exceeds n (an 'entombed' disc; for n >= 3 unpriced by the frozen fair leaf, which has such terms only for 1s and 2s) is an early predictor of death for the fair depth-4 policy: it is present at most deaths, it typically appears well outside the four-ply horizon (more than two rises before death), and it carries lifetime information beyond the eighteen leaf features and occupancy.

Mechanism

In a gravity-packed column the vertical run through every disc equals the column height, so a disc n in a column taller than n can only clear horizontally, and if its row run already exceeds n it cannot clear at all until neighbours are removed. Such a disc also blocks vertical cracking of every cover beneath it. The harm (unrevealed covers, a column that only grows, rise pressure) arrives many moves later, outside a four-ply search, and the leaf's height terms are value-blind (a 7 and a 3 at height 5 score the same) while its dead-disc terms stop at value 2. The search is therefore not short of fear of death (finding-04: the terminal utility is saturated) but of foresight: the decision that entombs the disc looks harmless at depth 4. A trigger that fires when such a disc appears is a candidate for the 'emergency mode' the owner asked about, and a leaf term for it is a candidate 19th coordinate.

What would prove it wrong

  • In complete depth-4 behaviour games of the corpus, fewer than half of deaths have an entombed disc of value >= 3 on the board at death: the disc is not the usual way the policy dies.
  • The median lead time between the first persistent entombed >= 3 disc and death is under 5 moves (one rise): the signal arrives too late to act on and is not foresight.
  • On a whole-origin held-out split, the partial correlation of the entombed count with log remaining lifetime, after the eighteen frozen leaf features, occupancy and the rise clock, is above -0.02, or the incremental held-out R^2 is below 0.005: the leaf already prices it.
  • Out of scope: whether pricing it changes play. That is a separate preregistered gameplay experiment.
registered 2026-08-22T05:12:38Z by Claude Code / claude-fable-5

Experiments that test it

Results recorded against it

valid run · outcome: failnot-supported-as-testedmechanics-onlyRS-20260822T051732Z-ceb257d7

Corpus analysis of entombed numbered discs (value n with column height > n and horizontal run > n) over 4,863,627 non-explored positions. Prevalence: an entombed disc of value >= 3 is on the board in 19.1% of depth-4 behaviour states overall, rising from 7.6% more than 40 moves before death to 69.8% within 5 moves of it. At death: 579 of 768 complete depth-4 games (75.4%) end with one on the board (criterion 1 holds). Lead time from first persistent appearance to death: median 11 moves, quartiles 5.5-15, 57% at least 10 moves, 8% at least 20 (criterion 2 holds, narrowly). Incremental signal: held-out R^2 for log1p(moves to death) is 0.6952 with the 18 frozen leaf features + occupancy + rise clock and 0.6955 with the entombed features added (+0.0003); the held-out partial correlation of the entombed count is -0.023 (threshold -0.05). Criterion 3 fails. Occupancy-matched, an entombed >= 3 disc shortens mean remaining life by about two moves (e.g. 35.0 vs 37.4 at 24-27 occupied cells; 24.6 vs 26.3 at 28-31). Reading: the disc is a reliable marker of the policy's last two rises, but the leaf's value-blind height and cover terms already carry almost all of its lifetime information; as a static foresight term it is not what the search is missing. Gate: FAIL (not-supported-as-tested).

What it had to pass
  • ✓analysis completes on all non-explored records with finite leaf values — observed: 4863627 records analysed, 393554 explored records excluded, 0 invalid leaf rows
  • ✓depth-4 fracAtDeath >= 0.5 — observed: 0.7539
  • ✓depth-4 lead-time median >= 10 moves — observed: median 11.0, q25 5.5, q75 15.0
  • ✕held-out partial correlation <= -0.05 AND incremental R^2 >= 0.005 — observed: partial -0.0234; incremental R^2 0.00031; depth-4-only incremental R^2 0.00043
Recorded metrics
prevalence
depth1
states
664,432
fracWithEntombedGe3
0.2291
fracWithEntombedLe2
0.7340
byMovesToDeath
death<=5
states
70,086
fracWithEntombedGe3
0.6572
meanEntombedGe3
1.5275
meanOccupancy
38.0101
fracWithEntombed34
0.6525
death6-15
states
142,831
fracWithEntombedGe3
0.4339
meanEntombedGe3
0.7169
meanOccupancy
31.2209
fracWithEntombed34
0.4324
death16-40
states
323,577
fracWithEntombedGe3
0.1177
meanEntombedGe3
0.1483
meanOccupancy
22.1048
fracWithEntombed34
0.1174
death>40
states
127,938
fracWithEntombedGe3
0.0477
meanEntombedGe3
0.0580
meanOccupancy
17.3375
fracWithEntombed34
0.0477
depth2
states
3,840,470
fracWithEntombedGe3
0.2056
fracWithEntombedLe2
0.7731
byMovesToDeath
death<=5
states
302,037
fracWithEntombedGe3
0.6640
meanEntombedGe3
1.5443
meanOccupancy
38.6116
fracWithEntombed34
0.6588
death6-15
states
609,783
fracWithEntombedGe3
0.4542
meanEntombedGe3
0.7616
meanOccupancy
31.7454
fracWithEntombed34
0.4527
death16-40
states
1,472,663
fracWithEntombedGe3
0.1487
meanEntombedGe3
0.1933
meanOccupancy
23.8524
fracWithEntombed34
0.1484
death>40
states
1,455,987
fracWithEntombedGe3
0.0640
meanEntombedGe3
0.0787
meanOccupancy
19.2009
fracWithEntombed34
0.0639
depth3
states
295,894
fracWithEntombedGe3
0.1981
fracWithEntombedLe2
0.7871
byMovesToDeath
death<=5
states
19,429
fracWithEntombedGe3
0.6915
meanEntombedGe3
1.5718
meanOccupancy
39.2730
fracWithEntombed34
0.6840
death6-15
states
39,219
fracWithEntombedGe3
0.4741
meanEntombedGe3
0.8055
meanOccupancy
32.2350
fracWithEntombed34
0.4727
death16-40
states
95,924
fracWithEntombedGe3
0.1716
meanEntombedGe3
0.2273
meanOccupancy
24.5149
fracWithEntombed34
0.1714
death>40
states
141,322
fracWithEntombedGe3
0.0716
meanEntombedGe3
0.0883
meanOccupancy
19.7004
fracWithEntombed34
0.0716
depth4
states
62,831
fracWithEntombedGe3
0.1912
fracWithEntombedLe2
0.7960
byMovesToDeath
death<=5
states
3,709
fracWithEntombedGe3
0.6983
meanEntombedGe3
1.5980
meanOccupancy
40.0768
fracWithEntombed34
0.6945
death6-15
states
7,461
fracWithEntombedGe3
0.4832
meanEntombedGe3
0.8128
meanOccupancy
32.7330
fracWithEntombed34
0.4801
death16-40
states
18,403
fracWithEntombedGe3
0.1783
meanEntombedGe3
0.2425
meanOccupancy
24.8813
fracWithEntombed34
0.1777
death>40
states
33,258
fracWithEntombedGe3
0.0763
meanEntombedGe3
0.0941
meanOccupancy
20.0137
fracWithEntombed34
0.0761
occupancyMatched
occ12-15
with
513
without
186,855
meanMovesToDeathWith
63.0546
meanMovesToDeathWithout
61.1056
medianWith
57
medianWithout
53
occ16-19
with
6,583
without
658,931
meanMovesToDeathWith
55.4624
meanMovesToDeathWithout
54.8682
medianWith
48
medianWithout
47
occ20-23
with
43,236
without
856,842
meanMovesToDeathWith
45.5290
meanMovesToDeathWithout
47.7573
medianWith
38
medianWithout
40
occ24-27
with
142,959
without
845,929
meanMovesToDeathWith
35.0183
meanMovesToDeathWithout
37.3720
medianWith
28
medianWithout
30
occ28-31
with
233,244
without
541,605
meanMovesToDeathWith
24.6265
meanMovesToDeathWithout
26.3040
medianWith
18
medianWithout
20
occ32-35
with
232,362
without
235,132
meanMovesToDeathWith
14.6345
meanMovesToDeathWithout
15.7091
medianWith
11
medianWithout
12
occ36-39
with
168,268
without
78,623
meanMovesToDeathWith
8.2391
meanMovesToDeathWithout
8.4068
medianWith
7
medianWithout
7
leadTime
depth3
completeGames
4,096
gamesWithEntombedGe3AtDeath
3,011
fracAtDeath
0.7351
leadTimeMoves
median
10
q25
6
q75
15
mean
11.0651
fracAtLeast10
0.5739
fracAtLeast20
0.0973
meanGameLength
75.4805
depth4
completeGames
768
gamesWithEntombedGe3AtDeath
579
fracAtDeath
0.7539
leadTimeMoves
median
11
q25
5.5000
q75
15
mean
10.8377
fracAtLeast10
0.5665
fracAtLeast20
0.0794
meanGameLength
84.0182
incrementalSignal
leafScalar+occupancy
heldOutR2
0.6593
heldOutPearson
0.8120
train
3,891,854
test
486,819
leafFeatures18+occupancy
heldOutR2
0.6952
heldOutPearson
0.8338
train
3,891,854
test
486,819
leafFeatures18+occupancy+entombed
heldOutR2
0.6955
heldOutPearson
0.8340
train
3,891,854
test
486,819
entombedOnly+occupancy
heldOutR2
0.5588
heldOutPearson
0.7476
train
3,891,854
test
486,819
depth4only
leafFeatures18+occupancy
heldOutR2
0.6483
heldOutPearson
0.8062
train
50,003
test
5,528
leafFeatures18+occupancy+entombed
heldOutR2
0.6488
heldOutPearson
0.8065
train
50,003
test
5,528
depth3only
leafFeatures18+occupancy
heldOutR2
0.6657
heldOutPearson
0.8162
train
235,370
test
28,612
leafFeatures18+occupancy+entombed
heldOutR2
0.6662
heldOutPearson
0.8165
train
235,370
test
28,612
partialCorrelationHeldOut
ent_count_ge3
-0.0234
ent_mass_ge3
-0.0156
ent_covers_ge3
-0.0162
ent_top_ge3
-0.0241
ent_count_34
-0.0234
featureMeans
ent_count_ge3
0.3518
ent_mass_ge3
8.1043
ent_covers_ge3
1.1874
ent_top_ge3
0.1959
ent_count_le2
2.3535
ent_count_34
0.3465
Limitations
  • Offline, teacher-labelled diagnostic on already-played behaviour games; says nothing about whether avoiding the entombing move would have extended life (a counterfactual replay is the natural follow-up).
  • Linear incremental test: a nonlinear interaction with height could still carry signal a linear leaf cannot use; the NNUE leaf sees cell/pair features and could learn it.
  • The structural definition ignores release readiness (whether a neighbouring clear could free the disc); dead_low_numbers discounts by readiness and this feature does not.
  • Depth-4 behaviour games are 768 of the corpus's 86,784 games; depth-3 (4,096 games) agrees on every quantity.
  • The corpus seeds are training-role data; the numbers are development diagnostics and not strength evidence.