Finding 09 — The chance node's two random quantities are welded together, and unwelding them is worth a whole ply

Status: exploratory. Evidence tier development for the depth-3 arms, pilot for the runtime projections. Positive result, with a monotone dose-response and one arm interrupted. Namespace: approaches/lifetime-objective/reveal-sampling, run runs/RUN-A525-reveal/, build build/reveal-sampling/. No existing file was modified. The frozen reference and the single-knob parameterized search are consumed unmodified through generated copies that differ from the originals by exactly one entry-point line, verified at build time. Files were created only under approaches/lifetime-objective/reveal-sampling/, build/reveal-sampling/, runs/RUN-A525-reveal/ and docs/exploratory/. Seed lease: SEEDLEASE-A52-REVEAL = 0xa5250000–0xa5257fff for CHECK probes and pilots. The headline arms play the fixed shared evaluation cohort 0xa51d1000–0xa51d103f (64 games) that every arm in finding-05 used, so the comparison is direct.

Summary

The frozen fair search has one knob, chance_samples, that governs two different random quantities — the next visible disc and the covered-disc reveals — and indexes both with the same sample counter, so they are perfectly rank-correlated. This document builds a search that factors the chance node into --disc-samples N × --reveal-samples M, proves it decision-identical to both existing arms at M = 1, and measures the reveal axis.

The reveal axis pays, and it pays monotonically. Holding the disc exact at seven strata and holding depth at three:

Score, moves, numbered clears, covered reveals and (downward) occupancy all move monotonically with M, and the ordering tracks the fraction of the chance node's joint atoms that receive non-zero weight. M = 3 alone is not significant; M = 6 is, at 36–0–28.

The headline comparison is against the frozen reference. Depth 3 with six reveal samples beats frozen depth-4 fair D4 by +79,115 points (95% lower bound +30,242, 35–0–29) — and it is statistically indistinguishable from finding-05's depth-4 seven-strata arm (−22,056, 95% bound −89,867, 30–0–34, a tie) at 4,244,020 logical work per move against that arm's 4,956,614.

So the corrected statement of finding-05 is not "depth needs an exact chance estimator". It is symmetric: at fixed work, a fourth ply and a better reveal estimate buy roughly the same thing, and the reveal estimate is slightly cheaper. finding-05 found one half of that; this finding supplies the other.

The mechanism check passes. Flow rates move in the required direction and by an amount comparable to finding-05's positive result: +0.0598 numbered clears and +0.0422 covered reveals per move from M = 1 to M = 6, with mean occupancy falling from 23.88 to 23.49 of 49 cells.

1. The coupling, in the frozen code

The frozen chance node draws two different random quantities from one knob.

Reveal values. cfpi::detail::StratifiedRandom (src/core/native/public-behavior.hpp:595–607) is constructed as StratifiedRandom{state_seed, sample, chance_samples, 0} and its nextDisc() is called once per revealed cover inside the cascade (public-behavior.hpp:655–657), advancing an event counter each time. A wave can expose a variable number of cells and each takes an independent uniform 1..7 value.

The successor's visible next disc. cfpi::detail::sampledNextDisc (public-behavior.hpp:736–742), domain kDiscSampleDomain, always event = 0.

Both go through the same quadrature, stratifiedUnit (public-behavior.hpp:579–593):

event_seed = mix32(seed ^ domain ^ (event+1)*kDepthMultiplier)
rotation   = event_seed % count
stratum    = (sample + rotation) % count
jitter     = mix32(event_seed ^ (sample+1)*kSampleMultiplier) / 2^32
return (stratum + jitter) / count

The coupling is that both consumers are passed the same sample index and the same count. In the frozen driver this is approaches/fair-expectimax/reference/fair-only-horizon.cpp:211–237 (for (int sample = 0; sample < kChanceSamples; ++sample), then StratifiedRandom{state_seed, sample, kChanceSamples, 0} at :214 and sampledNextDisc(state_seed, sample, kChanceSamples) at :231), repeated for the depth-4 driver at approaches/fair-expectimax/reference/fair-only-depth4.cpp:137–163 (:140 and :157) and in the shared library search at public-behavior.hpp:824–846 (:826 and :840). The parameterized single-knob search inherits it verbatim at approaches/lifetime-objective/risk-calibration/search.cpp:114–133 (:116 and :132).

Two consequences follow, and the second is the one that turned out to matter:

1. chance_samples cannot be raised for one quantity without raising it for the other, so finding-05's 5 → 7 change moved both.
2. Because sample i's disc value and sample i's reveal values are both deterministic functions of i, they are perfectly rank-correlated. Across the seven samples the pair (first reveal value, next disc) takes only 7 of its 49 joint outcomes.

Two things in the frozen code are not defects and were checked before changing anything. state_seed is column-independent (scenarioSeedForState, public-behavior.hpp:721–734, hashes board, next disc and moves remaining but not the column), so all siblings at a node see the same unit values — textbook common random numbers, which reduces comparison variance without biasing the difference. And playMoveSampled draws its own next disc internally (public-behavior.hpp:711–712) which evaluateAction immediately overwrites; that is deliberate, and makes the successor's disc independent of how many reveals a column's cascade happened to consume.

2. The factorization

approaches/lifetime-objective/reveal-sampling/search.cpp replaces the single loop with N·M scenarios indexed by a disc index d ∈ [0,N) and a reveal index r ∈ [0,M), total T = N·M:

scenario s = r * N + d
reveals:   StratifiedRandom{state_seed, s, T, 0}
next disc: sampledNextDisc(state_seed, d, N)

Two properties make this a strict generalization rather than a new policy.

At M = 1 it collapses to the single-knob search exactly. s = d and T = N, so every draw, the iteration order, the work accounting and the divisor are bit-identical to chance_samples = N. That is what makes N=5, M=1 the frozen reference and N=7, M=1 the existing seven-strata arm, rather than something that merely resembles them.

For M > 1 the disc keeps exactly N equally weighted strata while the reveal marginal gets all T. The stride s = r·N + d means one disc branch's M reveal draws are spaced 1/M apart in the unit interval rather than occupying a contiguous block, so each disc branch also sees a properly stratified reveal sample.

What raising M actually fixes

approaches/lifetime-objective/reveal-sampling/chance-coverage.py reimplements mix32 and stratifiedUnit in Python and measures, over 20,000 synthetic chance nodes, the fraction of joint atoms that receive non-zero weight:

This sharpens the premise of the experiment and partly corrects it. At seven strata each individual reveal event's marginal is already exact, not just the next disc's — seven strata over seven atoms is exact for any single stratifiedUnit consumer. The residual error is not in any marginal. It is in the joint: 7 samples cover 14.3% of the two-reveal grid, 2.0% of the three-reveal grid, and 14.3% of the (disc, reveal) grid.

Raising M attacks the (disc, reveal) joint efficiently — 14.3% → 42.9% → 85.7% → 100% at M = 3, 6, 12 — because it decouples the disc index from the reveal index. It attacks the reveal-by-reveal joint only weakly — 14.3% → 23.8% → 26.2% → 27.4% — because all reveal events within one scenario still share the sample index s and therefore stay rank-correlated with each other. M is mostly a disc/reveal decoupling knob, and only secondarily a reveal-joint knob.

The measured strength ordering tracks the (disc, reveal) column and not the reveal-by-reveal column. Mean score goes 312,327 → 337,306 → 376,442 as that coverage goes 14.3% → 42.9% → 85.7%, while two-reveal joint coverage barely moves. That is a specific, falsifiable attribution of which part of the chance node was broken: it was the correlation between the disc and the reveals, not the correlation among the reveals.

3. CHECK gate

No gameplay arm was started before these passed. All are move-by-move comparisons on whole games from SEEDLEASE-A52-REVEAL.

The gate-B oracle is not a copy of the existing arm's logic — it is the existing arm's own source, approaches/lifetime-objective/risk-calibration/search.cpp, compiled into the same binary through a generated copy whose only difference from the original is the entry-point line (build.sh refuses to build if more than one line differs). Logical work is compared as well as the chosen column, which is a much stronger equality than action agreement alone.

End-to-end identity, 64 whole games each. The two M = 1 depth-3 arms were run on the full shared cohort and compared field-by-field against the artifacts the existing risk-calibration binary produced in finding-05:

Score, moves, censor flag, rises, board clears, level/clear/chain points, numbered clears, covers revealed, maximum chain depth and mean occupancy agree exactly on all 64 seeds, and the summed logical work is equal to the unit. The factored search at M = 1 is the same policy, not an approximation of it.

4. The work bound

Worst-case iterative-deepening work is the frozen arithmetic (fair-only-depth4.cpp:52–62) with branching generalized from 7 × chance_samples to b = 7 × N × M:

Σ_{i≤d} [ Σ_{l≤i} b^l + b^i ]

The frozen bound of 3,200,000 was sized to sit just above five-stratum depth 4 (3,134,950, a 2.1% margin). A larger branching factor left at that bound does not error — it silently completes a shallower depth and reports it as a chance-sampling result. Every arm here therefore declares --max-work explicitly, and the binary refuses to run a gameplay cohort without it.

The two M = 1 rows keep the bounds finding-05 used, because reproducing those arms bit-for-bit is the CHECK gate. The others are set to their own worst case. Cache size is a pure performance knob here and cannot change a decision: the declared bound assumes no cache at all, so cache eviction can only make the search do more of the work the bound already allows, never less depth.

Evidence that the bound never bound. Every gameplay decision records its completed depth; the binary counts decisions that finished below the target depth and work-limit events, and reports the maximum work any single decision consumed.

Not one decision in 30,865 completed below its target depth, and the busiest single decision in the whole study used 77% of its bound (the N=5, M=3 arm; the M=6 arm's busiest used 68%). Every arm below is a genuine depth-3 result.

5. The arms

Shared evaluation cohort 0xa51d1000–0xa51d103f, 64 paired games, 2,000-move cap, corrected 17,000-point Hardcore scoring. Score-decomposition identity violations: 0 in every arm. Censored games: 0 in every arm.

The two depth-4 rows are finding-05's arms on the same cohort, reproduced here for comparison; they were not re-run.

Paired deltas over whole games

One-sided 95% lower bounds by percentile bootstrap, 20,000 resamples, RNG seed 0xb0075eed, resampling whole games.

The dose-response is the result. Each individual step (M=1→3, M=3→6) is inside noise on 64 games; the endpoints are not. Score, moves, clears, reveals and occupancy all order correctly across three settings of one knob, which is a much stronger signal than any single pairwise test, and the ordering matches the mechanism table in §2.

The control does its job and its answer is now informative. If the reveal axis were simply substituting for disc exactness, (5,3) would recover most of what (7,1) gains over (5,1). It gains +7,506 against (7,1)'s +7,276 — nominally identical, both inside noise. But the large effect at (7,6) is measured with the disc already exact, so it cannot be disc exactness in disguise. The two axes are not substitutes at small M; the reveal axis simply needs M ≥ 6 to show.

Reveal sampling substitutes for the fourth ply, slightly more cheaply. Depth 3 at M = 6 ties finding-05's depth-4 seven-strata arm (−22,056, 95% bound −89,867, 30–0–34) using 4,244,020 work per move against 4,956,614 — 86% of the cost. That reframes finding-05: the fourth ply and a decorrelated reveal estimate are, at this budget, two ways of buying the same thing.

6. Flow rates — the mechanism that had to move, and did

finding-01 establishes that Hardcore score is very nearly 3,400 × lifetime, and finding-06 that indefinite survival requires ≥ 2.400 numbered clears and ≥ 1.400 covered reveals per move. If reveal sampling improved the policy, these are the quantities that would have to move.

The mechanism moved, monotonically, and by a material amount. From M = 1 to M = 6 the clear rate rises 0.0598 and the reveal rate 0.0422 — respectively 14% and 14% of the distance from the 7-strata arm to the requirement, and about a third of the distance from fair D4 to finding-07's legal sampling planner. Mean occupancy falls 0.39 cells, in the direction finding-07 associates with longer life. Mean lifetime rises 17.19 moves.

Two calibrations worth stating plainly. The M = 6 arm reaches exactly fair D4's reveal rate (1.1423) while running a shallower search; and its clear rate, 2.0447, is within 0.002 of fair D4's 2.0467. A depth-3 search with a decorrelated chance node reproduces the flow profile of a depth-4 search with a correlated one.

None of this reaches the 2.400 / 1.400 requirement. Every arm here still fills its board and dies; no arm was censored at the move cap.

7. Work per move, and whether the gain is bought with compute

Stated plainly: yes, this buys strength with compute — at very nearly the same exchange rate as the fourth ply, and no better. M = 1 → 6 at depth 3 costs 27.1× the logical work per move for +64,116 points. Score per unit work falls monotonically with M, as it does with depth. Under benchmarks.md this is a fixed-depth comparison that owes a fixed-work account, and the account says the cheapest arm in the table is still the best value per unit work, exactly as finding-05 found.

What is new is the substitution: at a matched work budget of roughly 4.2–5.0 million per move, depth 3 with M = 6 and depth 4 with M = 1 are indistinguishable in strength (30–0–34), and the depth-3 arm is 14% cheaper. That is a genuine second axis on the frontier, not merely a second way to spend compute — and it is available at depth 3, where a decision costs a third of a depth-4 decision's worst case.

Measured work per move stays far below the worst-case bound because the transposition cache collapses the many reveal scenarios that produce identical successor boards — most moves cause no cascade at all, so their M reveal variants are the same state. Predicted worst-case ratio for M = 1 → 6 at depth 3 is (294/49)³ = 216×; the measured ratio is 27.1×. That is why the depth-3 M sweep was affordable at all.

8. Verdict

• Validity: valid. The candidate is decision-identical to both existing arms at M = 1 by explicit move-by-move comparison (562 decisions, 0 action mismatches, 0 logical-work mismatches) and by exact 64-game artifact identity against the existing binary's own output. Work bounds were computed per configuration, passed explicitly, and verified never to bind: 0 of 30,865 decisions completed below target depth, busiest decision at 77% of bound. Score-decomposition identity holds on every game; 0 censored games.
• Outcome: pass, with the caveat that the effect needs M ≥ 6 to clear noise on 64 games. The hypothesis was that at seven strata the residual chance-node error is reveal sampling and that this axis is under-sampled. Both are supported: the residual is specifically the (disc, reveal) joint, and decorrelating it is worth +64,116 over the 7-strata arm (95% bound +7,475) and +79,115 over the frozen reference policy (95% bound +30,242), with flow rates moving monotonically in the required direction.
• Evidence tier: development. 64 paired games per arm on a previously-read fixed evaluation cohort. No protected or final seed was opened, and none is justified. Mean 376,442 is far below the 1,050,000 the frozen protocol requires before any candidate may be frozen.

What this changes for the research program.

1. finding-05's conclusion is half of a symmetric statement. It found that the fourth ply is worthless with a biased chance estimator and worth 86,000 points with an exact one. The mirror is now measured: at fixed depth 3, a decorrelated chance estimator is worth as much as the fourth ply, slightly cheaper. Depth and chance-node quality are two exchangeable ways of spending the same budget, not a hierarchy.
2. The defect has a precise name. It is not "too few samples" and not "the disc marginal is biased" — at seven strata every marginal is exact. It is that the disc and the reveals are the same random variable in disguise, and a column whose value depends on the pair cannot be evaluated. That is a two-line change to how one index is computed.
3. This partially answers the question finding-07 left open. finding-07 bounded the fair planner's advantage over D4 at 0.1842 clears per move and attributed it to planning structure. Chance estimation alone now recovers 0.0598 of that at depth 3 — roughly a third — so the residual attributable to planning structure is smaller than finding-07's framing implied.
4. The obvious next arms are the two this session could not finish (§10), and the higher-value one is depth 4 at M = 2, because the whole point of §5 is that the two axes trade against each other and nobody has measured them together.

9. Limitations

1. Depth 3 only, for the completed arms. The depth-4 arm that would test whether the two axes compound rather than substitute was launched and killed (§10). Everything here is measured at one depth.
2. The individual M steps are not separately significant. (7,1)→(7,3) is +24,980 with a 32–0–32 split and (7,3)→(7,6) is +39,136 at 37–1–26; only the endpoints and the comparisons against the frozen configurations clear zero. The claim rests on the monotone ordering of five quantities across three settings plus the significant endpoint tests, not on any one pairwise result.
3. M is mostly a disc/reveal decoupling knob (§2). It raises (disc, reveal) joint coverage from 14.3% to 85.7% at M = 6, but the reveal-by-reveal joint only from 14.3% to 26.2%, because all reveal events in one scenario share the sample index. A design giving each scenario an independent reveal event stream would attack the reveal joint directly and is not tested here. The attribution in §2 is therefore that the (disc, reveal) correlation was the binding defect; the reveal-by-reveal correlation remains largely untouched and unmeasured.
4. 64 paired games. Score standard deviation is 55–62% of the mean, so a 64-game mean carries roughly ±22,000 to ±28,000 at one standard error.
5. One cohort, already read. These seeds were opened by finding-05 and are development data permanently. Nothing here can be confirmation evidence, and the result should be replicated on a fresh block before it is promoted.
6. No fixed-time comparison. Arms ran at 2–12 threads on a machine whose load average was 20–55 throughout, driven mostly by another contributor's jobs. Only work per move is comparable; the wall-clock column is provenance, not a performance claim.
7. --auto-cache raises the LRU cache above the frozen 60,000 entries for arms whose worst-case entry count exceeds it (d3 M=6 at 87,025). This cannot change a decision — the work bound assumes no cache — but it is a declared deviation and is recorded per arm in §4.
8. These are this repository's simulator semantics, including the two rise-boundary scoring discrepancies in audit-01.
9. 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/reveal-sampling/ and docs/exploratory/. The same debt is open for finding-02, finding-06 and finding-07.

10. Arms not completed in this session

Recorded rather than omitted, as the repository contract requires.

• depth 4, N=7, M=2 — INTERRUPTED, no result. Launched on the shared cohort with --max-work 187336114 --auto-cache --threads 8 and killed by the runtime after about one hour with no games written. This is a partial run, not a censored result, and it appears in no table above. It was authorized by a matched pilot rather than guessed: 2 games × 15 opening moves against an identically-run (N=7, M=1) pilot gave 21,943,224 work/move against 6,107,746 — a 3.59× logical-work ratio and a 3.29× wall ratio, far below the 15.75× worst-case bound ratio, because the cache collapses duplicate reveal outcomes. Max work in one decision was 53,154,318 (28% of bound) with 0 decisions below target depth. Whole-game work/move projects to about 17.8M, i.e. about 11.6 core-seconds per move, so 64 games project to 1.7–3.7 hours on 14 threads idle and perhaps 5–7 hours at the load average of 45–55 this machine carried — within the 12-hour budget, which is why it was started. This is now the single highest-value open arm, because §5 shows the two axes are exchangeable at fixed work and nobody has measured whether they compound.
• depth 3, N=7, M=12 (b = 588, worst-case work 407,634,528) — not run. This is the configuration that takes (disc, reveal) joint coverage to 100%, so it is the natural endpoint of the dose-response in §5, but it projects to roughly 5.7 hours on 14 threads and the compute budget closed first. A deliberate no-run, and now clearly worth doing.

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/reveal-sampling/build.sh
B=./build/reveal-sampling/reveal-sampling

# CHECK gate, before any gameplay
$B --parity --seed-start 0xa5250000 --parity-games 2 --parity-moves 25
$B --gate --depth 3 --disc-samples 5 --max-work  3200000 --seed-start 0xa5250010 --parity-games 4 --parity-moves 60
$B --gate --depth 3 --disc-samples 7 --max-work 16000000 --seed-start 0xa5250010 --parity-games 4 --parity-moves 60
$B --gate --depth 4 --disc-samples 5 --max-work  3200000 --seed-start 0xa5250010 --parity-games 3 --parity-moves 30
$B --gate --depth 4 --disc-samples 7 --max-work 16000000 --seed-start 0xa5250010 --parity-games 2 --parity-moves 15

# work bounds; --max-work is mandatory for a gameplay cohort
for M in 1 3 6 12; do $B --work-bound --depth 3 --disc-samples 7 --reveal-samples $M; done
$B --work-bound --depth 4 --disc-samples 7 --reveal-samples 2

# arms, shared evaluation cohort
R=runs/RUN-A525-reveal
$B --depth 3 --disc-samples 7 --reveal-samples 1 --max-work 16000000 --seed-start 0xa51d1000 --games 64 --threads 12 --quiet --output $R/d3-n7-m1.json
$B --depth 3 --disc-samples 5 --reveal-samples 1 --max-work  3200000 --seed-start 0xa51d1000 --games 64 --threads 12 --quiet --output $R/d3-n5-m1.json
$B --depth 3 --disc-samples 7 --reveal-samples 3 --max-work  6418461 --auto-cache --seed-start 0xa51d1000 --games 64 --threads 8 --quiet --output $R/d3-n7-m3.json
$B --depth 3 --disc-samples 5 --reveal-samples 3 --max-work  2348745 --auto-cache --seed-start 0xa51d1000 --games 64 --threads 2 --quiet --output $R/d3-n5-m3.json
$B --depth 3 --disc-samples 7 --reveal-samples 6 --max-work 51084852 --auto-cache --seed-start 0xa51d1000 --games 64 --threads 12 --quiet --output $R/d3-n7-m6.json

# the M = 1 arms must be identical to the existing single-knob artifacts
C=approaches/lifetime-objective/reveal-sampling/compare.py
python3 $C identity $R/d3-n7-m1.json runs/RUN-A51D-s7confirm/s7d3.json
python3 $C identity $R/d3-n5-m1.json runs/RUN-A51D-s7confirm/s5d3.json

# tables
python3 $C table \
  "d3 N5 M1=$R/d3-n5-m1.json" "d3 N5 M3=$R/d3-n5-m3.json" \
  "d3 N7 M1=$R/d3-n7-m1.json" "d3 N7 M3=$R/d3-n7-m3.json" \
  "d3 N7 M6=$R/d3-n7-m6.json"
python3 $C table "d4 N7 M1=runs/RUN-A51D-s7confirm/fresh-s7.json" "d3 N7 M6=$R/d3-n7-m6.json"
python3 $C table "d4 N5 M1=runs/RUN-A51D-s7confirm/fresh-s5.json" "d3 N7 M6=$R/d3-n7-m6.json"

# joint chance-atom coverage
python3 approaches/lifetime-objective/reveal-sampling/chance-coverage.py

Environment

AMD Ryzen AI Max+ 395 (16 physical / 32 logical cores), 125 GiB RAM, AMD clang 23.0.0git, -O3 -std=c++20 -pthread -Wall -Wextra. Shared with another contributor's jobs throughout; load average 20–55, at most 14 threads used by this work. Frozen source hashes recorded at build time in build/reveal-sampling/sources.sha256:

45e7c223…  approaches/fair-expectimax/reference/fair-only-depth4.cpp
8828379b…  approaches/fair-expectimax/reference/fair-only-horizon.cpp
d2c74210…  approaches/lifetime-objective/risk-calibration/search.cpp
14de83dc…  approaches/lifetime-objective/common/harness.hpp
8b4267af…  src/core/native/public-behavior.hpp
b6dcde5f…  src/core/native/engine.hpp

Continuation — the two arms §10 left open

Added 2026-08-21 by a later session. Nothing above this line was edited. This section completes depth 4, N=7, M=2 (the interrupted arm) and depth 3, N=7, M=12 (the deliberate no-run), both on the same fixed shared evaluation cohort 0xa51d1000–0xa51d103f, 64 paired games. New files were created only under approaches/lifetime-objective/reveal-sampling/, build/reveal-sampling/ and runs/RUN-A525-reveal/.

Headline: the two axes substitute, they do not compound, and the reveal dose-response does not continue to the 100%-coverage endpoint.

Neither difference clears zero, so neither is a demonstrated loss. But both point estimates are negative, both cost 3.2–4.1× the work of their comparator, and in both arms score, moves, numbered clears and covered reveals all move together in the negative direction — the same four-quantity co-movement that §5 used as evidence for the positive result, here pointing the other way.

11. CHECK gate, re-run before any gameplay

The binary was rebuilt with clang++ (the Makefile's CXX ?= clang++ loses to make's builtin CXX=g++, which trips a false -Werror=array-bounds) and the whole §3 gate was re-run. The frozen source hashes written to build/reveal-sampling/sources.sha256 are byte-identical to those recorded at the end of this document, so the oracles are the same code the original session compared against.

Identical counts to §3, all five CHECK OK. Logs: runs/RUN-A525-reveal/cont-gate*.log.

12. Chunked execution, and why pooling is not a new statistical object

The previous attempt at d4 (7,2) was killed after about an hour with no games written, because the artifact is only serialized when the whole cohort finishes. Both arms here were instead run as four sequential 16-game chunks at seed starts 0xa51d1000, +0x10, +0x20, +0x30, launched detached with setsid nohup so they survive the controlling session, with per-game progress to a log. runCohort maps game index i to seed seedStart + i, so the four chunks play exactly the 64 cohort seeds, once each.

Whole games are the statistical unit and each game is a deterministic function of its seed and the policy, so a pooled 64-game cohort is the same object as a single 64-game run. That is asserted rather than assumed, and it was checked:

pool.py recomputes every aggregate with the same formulas as harness.hpp:writeArtifact, including a Python reimplementation of the Mulberry32 percentile bootstrap, and combines chunk diagnostics conservatively (counts add, minCompletedDepth takes the minimum, maxDecisionWork the maximum). This also re-confirms worker-count independence: 1-thread and 12-thread runs produce identical per-game records.

13. The work bound, computed per configuration and verified never to bind

Both arms passed --max-work explicitly, taken from the binary's own --work-bound (the frozen arithmetic with b = 7·N·M). Leaving a bound sized for a smaller branching factor does not error — it silently completes a shallower depth — so this is load-bearing.

Evidence the bound never bound. Every decision records its completed depth; the binary counts decisions finishing below the target depth and work-limit events, and reports the busiest single decision.

Per chunk the minimum completed depth is 4, 4, 4, 4 for the depth-4 arm and 3, 3, 3, 3 for the depth-3 arm, and the busiest decision in each chunk is 73.1M / 76.7M / 81.7M / 71.4M and 97.3M / 128.4M / 190.2M / 135.7M. No decision in 13,156 completed below its target depth; the busiest used 44% and 47% of its bound. Both arms are genuine depth-4 and depth-3 results.

14. The arms

Shared evaluation cohort 0xa51d1000–0xa51d103f, 64 paired games, 2,000-move cap, corrected 17,000-point Hardcore scoring.

Paired deltas over whole games

One-sided 95% bounds by percentile bootstrap, 20,000 resamples, RNG seed 0xb0075eed, resampling whole games. For a null or negative result the upper bound is the informative direction, so both are given.

Stability across cohort halves. No fold was preregistered, so this is a post-hoc diagnostic, not a gate. Both headline deltas keep their sign and their losing win–loss split in both halves; the magnitudes differ substantially, which is what a 64-game cohort with sd ≈ 60% of the mean predicts.

15. Arm 1 — the axes substitute, they do not compound

The question §8 left open was whether the fourth ply and chance-node decorrelation are exchangeable (§5's finding) or additive. Adding the cheapest useful reveal decorrelation on top of the fourth ply gives −41,950 points at 4.07× the work, with a 95% upper bound of +17,541.

That upper bound is the useful number. Had the axes compounded even weakly — say by half of the +64,116 the same knob is worth at depth 3 — the data would have had to show it; instead the 95% one-sided ceiling on any compounding gain is about +17.5k, roughly a quarter of the depth-3 effect, at 4× the cost. The axes substitute in the strong sense. The remaining question §8 posed, the exchange rate, is answered by §7 and unchanged: at matched work d3 (7,6) and d4 (7,1) are a tie and the depth-3 arm is 14% cheaper.

Two things make this more than a null. First, every flow quantity moves with the score: −11.02 moves, −0.0265 clears/move, −0.0191 reveals/move, and mean occupancy rises 0.20 cells — the exact reverse of the co-movement §6 used to argue the depth-3 effect was mechanical. Second, the sign is stable across both cohort halves. The straightforward reading is that once the disc marginal is exact and a fourth ply is present, the (disc, reveal) joint is no longer the binding error, and paying 4× for it buys nothing.

The pilot in §10 projected the cost well: 3.59× predicted logical work against 4.07× measured, versus a 15.75× worst-case bound ratio.

16. Arm 2 — coverage saturation and flow-rate saturation do not coincide

d3 (7,12) is the configuration at which the (disc, reveal) joint reaches 100% coverage. If the §5 dose-response were driven by that coverage, this is where the curve should top out. It does not top out here — it turns over one step earlier:

Answer: no, coverage saturation does not match flow-rate saturation. Score, moves, numbered clears and covered reveals all peak at M = 6 with 85.7% coverage and fall back when the remaining 14.3% of joint atoms are filled in. Only mean occupancy keeps improving monotonically all the way to M = 12 (23.88 → 23.81 → 23.49 → 23.39), which is the one quantity that tracks coverage across all four points.

The M=6→12 step is −27,097 with a 95% interval of (−83,807, +31,209) and 28–0–36, so this is a saturation result, not a demonstrated regression: the data are consistent with the curve being flat from M = 6 onward, and are not consistent with it continuing to climb at the rate M = 1→6 showed. The strongest defensible statement is that the reveal axis is exhausted by M ≈ 6, and that filling the last of the joint grid buys at most +31k and plausibly nothing.

This weakens, without refuting, §2's attribution. The mechanism table predicted strength should track the (disc, reveal) column; it does over M ∈ {1, 3, 6} and then stops. Three readings are consistent with the data and this experiment does not separate them: (a) the effect saturates once most of the joint has weight and the last atoms are redundant; (b) the gain over M ∈ {1,3,6} was partly the extra scenario count acting as variance reduction rather than coverage per se, and that too saturates; (c) 64 games cannot resolve steps this size and the whole M ≥ 3 region is one plateau. Reading (c) deserves weight: the four M values span 312k–376k, and no adjacent step in the sweep is individually significant.

17. Work, and the shape of the frontier now

Measured work ratios again land far below their bounds because the transposition cache collapses duplicate reveal scenarios: 3.18× measured for M=6→12 against a 7.98× bound ratio, 4.07× for d4 M=1→2 against 15.75×.

The frontier has a knee and both new arms are past it. Everything above roughly 4–5 million work per move is indistinguishable from d4 (7,1) or worse, while costing up to 4× more. The two best-value points in this study remain d3 (7,1) at 156,834 work per move and, if strength is the objective, d4 (7,1) / d3 (7,6) at ~4–5 million. Nothing here moves the ceiling: the best arm on this cohort is still 398,498 against the 1,050,000 the frozen protocol requires.

18. Verdict, and what it revises above

• Validity: valid. Both arms passed the full CHECK gate on the rebuilt binary with identical counts to §3, on unmodified frozen sources with matching hashes. Work bounds were computed per configuration, passed explicitly, and verified never to bind (0 of 13,156 decisions below target depth, 0 work-limit events, busiest decision at 44–47% of bound). 0 censored games and 0 score-decomposition identity violations in both arms. The chunk-and-pool procedure was validated against an existing single-process 64-game artifact before being used.
• Outcome: two informative negatives. Neither arm produced a significant loss and neither produced a gain. Both were run to answer a question whose either-way answer was worth having, and both answered it.
• Evidence tier: development, on the same previously-read cohort as everything above. No protected or final seed was opened and none is justified.

What this revises in §8.

1. §8's open question is closed: the axes substitute, they do not compound. The 95% ceiling on any compounding gain from M = 2 on top of depth 4 is +17,541 points at 4.07× the work. The symmetric statement in §8 — that depth and chance-node decorrelation are two ways of buying the same thing — is supported and now has its complement: you cannot buy it twice.
2. The dose-response in §5 is not monotone beyond M = 6; it peaks there. §5's claim rested explicitly on "the monotone ordering of five quantities across three settings". Extending it to a fourth setting breaks the monotonicity for four of those five (occupancy is the exception). The positive result at M = 6 stands on its own paired test against M = 1 (+64,116, lower bound +7,475) and against the frozen reference (+79,115, lower bound +30,242), both unchanged — but the dose-response argument that was offered as corroboration no longer extends, and §5's framing should be read as "M = 6 is a local optimum" rather than "more M is better".
3. §2's mechanism attribution is weakened. Joint coverage was offered as the quantity strength tracks. It reaches 100% at M = 12 and strength is lower there than at 85.7%. Coverage may still be the right variable with a saturating rather than linear response, but this experiment cannot separate that from the alternative that the M ≥ 3 region is one noise-limited plateau.
4. §10's remaining agenda is now empty. Both arms it recorded as unfinished are complete and recorded here.

19. Limitations of this continuation

1. Neither headline delta is significant. Both are negative point estimates with intervals spanning zero. The claims made are the bounded ones — an upper bound on compounding, and saturation rather than regression — and nothing stronger is supported.
2. 64 paired games, sd 62–73% of the mean in the new arms. A separate session has shown a small cohort badly misstating a different quantity here (an 8-tape comparison put fair D4 at 117.75 mean moves against 93.78 on 160 tapes). These comparisons are paired on a fixed 64-seed cohort, so they are not exposed to that particular failure, but they are exposed to ordinary 64-game noise and the half-split shows magnitudes moving by 4–6× between halves.
3. One cohort, already read. The same permanently-development seeds as everything above. Nothing here can be confirmation evidence.
4. Only M = 2 was tested at depth 4. A larger M at depth 4 is not ruled out by this arm, only made unattractive: M = 2 already costs 4× and its ceiling is +17.5k. The d3 sweep needed M ≥ 6 to show any effect, and depth-4 M = 6 would cost roughly 4,500,000,000 worst-case work per decision.
5. The reveal-by-reveal joint remains untouched. As §2 notes, all reveal events within one scenario still share the sample index. A design giving each scenario an independent reveal stream is still untested, and the saturation in §16 is a fact about the (disc, reveal) axis only.
6. Wall times are provenance, not performance. Arm 1 ran across a machine whose load average moved between 5 and 72 driven by other contributors, at 8 threads for one chunk and 12 for the rest; only work per move is comparable.
7. A model contribution record under research/contributions/ is owed and was not written, the same open debt §9 records, because this work was scoped to the reveal-sampling namespace and docs/exploratory/.

Reproduce (continuation)

./approaches/lifetime-objective/reveal-sampling/build.sh   # clang++
B=./build/reveal-sampling/reveal-sampling
R=runs/RUN-A525-reveal

# CHECK gate, as §3
$B --parity --seed-start 0xa5250000 --parity-games 2 --parity-moves 25
$B --gate --depth 3 --disc-samples 5 --max-work  3200000 --seed-start 0xa5250010 --parity-games 4 --parity-moves 60
$B --gate --depth 3 --disc-samples 7 --max-work 16000000 --seed-start 0xa5250010 --parity-games 4 --parity-moves 60
$B --gate --depth 4 --disc-samples 5 --max-work  3200000 --seed-start 0xa5250010 --parity-games 3 --parity-moves 30
$B --gate --depth 4 --disc-samples 7 --max-work 16000000 --seed-start 0xa5250010 --parity-games 2 --parity-moves 15

# work bounds; --max-work is mandatory for a gameplay cohort
$B --work-bound --depth 4 --disc-samples 7 --reveal-samples 2    # 187,336,114
$B --work-bound --depth 3 --disc-samples 7 --reveal-samples 12   # 407,634,528

# the chunk-and-pool procedure, validated against an existing 64-game artifact
for c in 0 1 2 3; do
  printf -v s '0x%x' $((0xa51d1000 + c*16))
  $B --depth 3 --disc-samples 5 --reveal-samples 1 --max-work 3200000 \
     --seed-start $s --games 16 --threads 1 --quiet --output $R/cont-pooltest-c$c.json
done
python3 approaches/lifetime-objective/reveal-sampling/pool.py \
  $R/cont-pooltest-pooled.json $R/cont-pooltest-c{0,1,2,3}.json
python3 approaches/lifetime-objective/reveal-sampling/compare.py identity \
  $R/cont-pooltest-pooled.json $R/d3-n5-m1.json

# the arms, detached and chunked (CROWDED_THREADS bounds the shared-machine
# backoff: wait up to 15 min for load <= 26, then run at that many threads)
setsid nohup ./approaches/lifetime-objective/reveal-sampling/run-arms.sh arm1 \
  >> $R/cont-arm1-driver.log 2>&1 &      # d4 N=7 M=2
CROWDED_THREADS=12 setsid nohup ./approaches/lifetime-objective/reveal-sampling/run-arms.sh arm2 \
  >> $R/cont-arm2-driver.log 2>&1 &      # d3 N=7 M=12

python3 approaches/lifetime-objective/reveal-sampling/pool.py $R/d4-n7-m2.json  $R/d4-n7-m2-c{0,1,2,3}.json
python3 approaches/lifetime-objective/reveal-sampling/pool.py $R/d3-n7-m12.json $R/d3-n7-m12-c{0,1,2,3}.json

# tables
S=approaches/lifetime-objective/reveal-sampling/stats.py
python3 $S rows "d3 (7,1)=$R/d3-n7-m1.json" "d3 (7,3)=$R/d3-n7-m3.json" \
  "d3 (7,6)=$R/d3-n7-m6.json" "d3 (7,12)=$R/d3-n7-m12.json" \
  "d4 (5,1)=runs/RUN-A51D-s7confirm/fresh-s5.json" \
  "d4 (7,1)=runs/RUN-A51D-s7confirm/fresh-s7.json" "d4 (7,2)=$R/d4-n7-m2.json"
python3 $S delta  "d4 (7,2)=$R/d4-n7-m2.json" "d4 (7,1)=runs/RUN-A51D-s7confirm/fresh-s7.json" \
                  "d3 (7,12)=$R/d3-n7-m12.json" "d3 (7,6)=$R/d3-n7-m6.json"
python3 $S halves "d4 (7,2)=$R/d4-n7-m2.json" "d4 (7,1)=runs/RUN-A51D-s7confirm/fresh-s7.json" \
                  "d3 (7,12)=$R/d3-n7-m12.json" "d3 (7,6)=$R/d3-n7-m6.json"

Environment (continuation)

Same host as above: AMD Ryzen AI Max+ 395 (16 physical / 32 logical cores), 125 GiB RAM, AMD clang 23.0.0git, -O3 -std=c++20 -pthread -Wall -Wextra. Shared with other contributors' jobs throughout; 1-minute load average ranged from 5 to 72, at most 12 threads used by this work. Arm 1 took 36,813 s of wall across its four chunks (one chunk at 8 threads during a load-50 period, the rest at 12); arm 2 took 16,470 s at 12 threads. Frozen source hashes in build/reveal-sampling/sources.sha256 are unchanged from the list above.