Finding 07 — Separating planning from information: at most 41% of the clairvoyant advantage is hidden knowledge

Status: exploratory, evidence tier development/pilot. Built and measured in this checkout on 2026-08-20. Namespace: approaches/lifetime-objective/flow-ceiling/, run runs/RUN-FLOW-044da902f8e8/, leases SEEDLEASE-A52-FLOW = 0xa5230000–0xa5233fff (game seeds, deliberately reused so every cohort is paired with finding-06) and SEEDLEASE-A52-FLOW2 = 0xa5234000–0xa5237fff (the hidden-board sampler). Nothing in docs/research/, artifacts/, research/, runs/RUN-20260820T*, or any existing approach source was modified by this work. Files were created or edited only under approaches/lifetime-objective/flow-ceiling/, build/flow-ceiling/, runs/RUN-FLOW-*/ and docs/exploratory/.

CORRECTION NOTICE (added after publication). The eight master tapes this document is measured on are unrepresentative. They favour long games, and the bias scales with how long a policy survives. finding-12 re-ran every arm on 128 fresh tapes: fair depth 4, whose configuration never changed, drops from 117.75 to 93.56 mean moves (the fresh figure matching finding-01's independent 94.06 and a separate agent's 93.78), while the clairvoyant planner — censored at the move cap on both cohorts — is unaffected.

Consequently the central quantitative claim of this document is withdrawn as stated: "fair planning closes 58.8% of the clairvoyant-minus-D4 gap" becomes 27.1% at H = 7 on a matched fresh cohort (22.2% at H = 5), because D4's baseline was inflated while the clairvoyant ceiling was not, making the true gap 21% larger and the fair planner's share of it roughly half what is reported below. The summary line "at least 59% planning, at most 41% information" is withdrawn; the corrected reading is that at most about 27% is recovered by fair planning of this kind.

Also superseded: §1's reading that the K series was "still climbing" at K = 256. K = 1024 is worse on five of six paired tapes — the series has an interior maximum near K = 256, not an asymptote.

What stands unchanged: the method, the information-boundary gate, the arm A result that the future disc tape is worth nothing, the finding that every fair game ends (reinforced — 0 of 160 fresh-tape fair games survived), and the qualitative conclusion that the residual is concentrated where the hidden state is large. The original text below is left exactly as written.

What this corrects in finding-06

finding-06 §3 ends with this sentence, which is left in place there and is not supported by that document's evidence:

The gap between the two is a control gap.

finding-06's planner is privileged twice over. It plans exactly over a nine-move window and it reads latent[], so it knows which covered disc holds which number and therefore knows exactly which covers are worth cracking. finding-06 never separated those two privileges, so it could not attribute the gap.

This document separates them, and the sentence turns out to be directionally right but unproven, and the correct statement is narrower: at a matched horizon a legal planner closes 58.8% of the clairvoyant-minus-D4 gap, so at most 41.2% of it is hidden information — and that 41.2% is an upper bound that was still falling when the compute budget ran out. Knowing the future disc tape is worth nothing.

What is genuinely weakened is a different finding-06 claim: that its ceiling is a target. No legal policy tested here reached flow balance. All 87 fair-planner games ended, none reached the move cap, and every fair arm's board filled monotonically. finding-06's other conclusions — the conservation arithmetic, the clairvoyant ceiling, the occupancy-equilibrium mechanism, the score composition and the wave-depth results — stand as measured.

The question

Two research programs follow, and they are incompatible:

• if the gap is mostly planning, a learned evaluator plus search can chase it and finding-06's ~2.40 clears per move is a reachable target;
• if the gap is mostly hidden information, the ceiling for any legal policy is far below 2.400 and the survival program is capped no matter how good the search gets.

Method

The fair planner

approaches/lifetime-objective/flow-ceiling/fair-planner.hpp is finding-06's rh-clears with exactly one thing removed: access to hidden state. At each move it draws K independent completions of everything it is not allowed to know, solves the horizon-H window exactly against each completion, and plays the column with the best mean value. Common random numbers: every candidate column inside one decision is scored against the same K completions, via solveWindowRoot, which returns the exact value of every legal root move under one completion.

Three arms, so the privileges come off one at a time:

Arm B reads exactly what docs/methodology.md permits a deployable policy to read: the visible board, the visible next disc, and the moves remaining before the next rise. Structurally it is never handed the future — sampleWindow only touches the master tape when tape_known is set.

Everything else is held fixed against finding-06: same objective (maximize numbered discs cleared), same exact window solver, one move committed per decision, and the same master tapes, so every comparison below is paired at the seed.

What this bounds, and what it does not

The fair planner is a determinized / hindsight-optimization (PIMC) controller. Inside a sample it assumes the drawn hidden values will turn out to be true, which makes its value estimates optimistically biased and exposes it to the two classical PIMC pathologies, strategy fusion and non-locality. It does not compute the true POMDP optimum and is not claimed to. Consequently:

• its flow rate is a lower bound on what a legal policy can sustain;
• the share of the gap it closes is a lower bound on the planning share, and the residual is an upper bound on the information share;
• sampling variance is a first-order effect, not a nuisance. finding-05 found that at depth 4 moving from five to seven chance strata is worth +101,171 points while at depth 3 it is worth nothing measurable — chance-estimator exactness and planning depth are complements. K is the same axis, so it is swept over three orders of magnitude and the saturation point is reported rather than assumed. That precaution mattered: see §1.

Validation

flow-run --self-test grew three gates for this work on top of finding-06's four. All seven pass.

flow-run --cross-check over suite-h9-v1 (128 scenarios, H = 9) still reports 0 mismatches against the frozen exact solver after the solver changes.

1. The K sweep — and why a small-K answer would have been wrong

Arm B, eight master tapes, seeds 0xa5230000–0xa5230007, 400-move cap. Pooled clears per move over all moves of all games.

Three things this table settles.

A small-K answer would have inverted the finding. At K = 8 in a pilot, and at K = 16 here, the fair planner is worse than fair depth 4 (1.9279 against 2.0467) and the natural conclusion would have been "the gap is information". By K = 256 it is 2.2309, comfortably better than D4 and 93.0% of the requirement. Between K = 1 and K = 256 at H = 7 the planner moves 0.694 clears per move — more than twice the entire clairvoyant-minus-D4 gap being decomposed.

H = 5 saturates; H = 7 has not. The H = 5 row is flat from K = 256 to K = 1024 (2.1680 vs 2.1415, eight games each, difference well inside noise), so that horizon is planning-limited rather than sample-limited above K ≈ 256. The H = 7 row is still climbing at K = 256 (+0.091 over K = 64, against +0.212 from 16 to 64 — decelerating, not flat). The fair planner's true ceiling was not reached. H = 7 at K = 1024 would have cost roughly six hours on the available threads and was not run; H = 9 at any non-artifactual K was far out of budget (§Limitations 3).

Chance-estimator exactness and planning depth are complements, exactly as finding-05 found on the strata axis. At K = 16, H = 5 (2.0413) beats H = 7 (1.9279): with coarse sampling, more lookahead makes strategy fusion worse. By K = 64 the ordering reverses (2.1403 against 2.0373) and by K = 256 H = 7 leads by 0.063. Deeper fair planning is only worth having once the chance estimate can support it.

2. The arms, and the decomposition

All at H = 7 on the same eight master tapes unless noted. steady excludes the first 25 moves, the sparse opening no steady state contains (finding-06 §3).

The decomposition

Against fair depth 4 as baseline and the clairvoyant planner at the same horizon as ceiling, in whole-game clears per move:

The same decomposition at H = 5, where both planners are saturated in their respective compute axes, gives fair planning 0.1213 of a 0.3029 gap — 40.0% closed, 60.0% unattributed. The two horizons bracket the answer: the information share is at most 41% at the horizon where fair planning was pushed hardest, and the number falls as the fair planner is given more compute.

Knowing the future is worth nothing. Arm A is handed the exact future disc tape and the exact hidden values of every future risen row. At K = 64 it sustains 2.1361 against arm B's 2.1403 — a difference of −0.004, i.e. nothing. At K = 256 arm A is worse: 2.0761 against arm B's 2.2393 on the same seven seeds (arm A's eighth game was still running after 1 h 51 m of wall clock when the resource budget for this work closed and was stopped; it is excluded from both sides of that paired comparison. The excluded game is the slowest-running one, so the arm A figure may be pessimistic). This is not a claim that future knowledge is harmful; it is the well-known result that a non-Bayesian determinized planner is not monotone in information. Fixing the tape removes diversity from the K completions, which sharpens strategy fusion: every sample plans against the same future, so the planner commits to a specific line instead of choosing a move that is good on average. The safe reading is that the future disc tape is not the source of the clairvoyant planner's advantage; the board under the covers is.

What is not attributable, stated plainly. The 41.2% residual is an upper bound on the value of hidden information, not a measurement of it. It contains two things this experiment cannot separate:

1. genuine hidden information — value no legal policy can ever recover; and
2. the suboptimality of hindsight optimization itself. A PIMC planner is structurally blind to the value of revealing a cover in order to learn, because averaging over determinizations assumes the answer is already known. In a game whose second conservation requirement is 1.400 covered reveals per move, that is likely a material part of the residual.

Since the fair planner had not saturated at H = 7, item 2 is known to be non-zero and the honest statement is "at least 59% planning, at most 41% information, and the planning share can only go up with better fair methods."

3. The occupancy trend — no fair planner found a fixed point

finding-06 §2's decisive diagnostic, applied to the fair arms.

Every fair arm has a strictly positive occupancy slope and every fair game ended. Across 87 fair-planner games — 72 arm-B games at nine (H, K) settings plus 15 arm-A games — not one reached the move cap alive. The longest single fair game was 332 moves; the clairvoyant planner ran 6 of 8 games into a 400-move cap and, in finding-06, 6 of 6 into a 1,000-move cap.

But the slope is falling fast and monotonically with sampling: +1.269 → +0.833 → +0.311 as K goes 16 → 64 → 256 at H = 7, a factor of four over the range, with mean lifetime rising 76 → 155 → 201 moves. The trend has not flattened. This document therefore does not establish a flow-balance ceiling for legal policies; it establishes a measured floor of 2.2309 clears per move (93.0% of the requirement) and a trajectory that was still improving when the budget ran out.

4. Clears per move by occupancy — the requested column

finding-06 §3's table with the fair arms added. n is the number of moves observed in that band; bands are not matched samples across policies.

This is the clearest statement of the result in the document.

Fair planning buys a real restoring force, in exactly the band that matters. At 20–24 cells arm B at K = 256 clears 2.49 against D4's 2.10; at 25–29 it clears 3.00 against 2.59. Both are above the 2.400 requirement, so a legal planner does have a negative-feedback region around its 21.4-cell operating point. That is the mechanism behind its 200.88-move mean lifetime.

But the restoring force is an order of magnitude weaker than the clairvoyant's, and the shortfall grows with occupancy. Clairvoyant minus best-fair, by band: +0.35 at 15–19, +0.82 at 20–24, +1.46 at 25–29, +3.96 at 30–34. A fuller board holds more covered discs, so knowing their values is worth more — the advantage scales with exactly the quantity that hidden information should scale with. It is the one qualitative signature in this data that points at information rather than search.

And the fair planner's basin is too shallow to hold. Its margin over the requirement is 0.09–0.60 clears per move in the bands it occupies, against the clairvoyant's 0.91–3.93. Above 35 cells it collapses to 1.21 and cannot get back. A bad sequence walks it out of the basin and it does not walk back — which is why every fair game ends and no clairvoyant game did.

5. For the record: fair D4 is not too height-averse

The coordinator asked for this, because it closes a hypothesis cheaply. Mean occupied cells over every move of every game, identical master tapes:

Fair D4 runs a board six cells fuller than the clairvoyant planner's equilibrium and still extracts less at every occupancy band. It is not dying from excessive height aversion — it is not conservative, it is ineffective. Any candidate built on "make D4 keep the board emptier" is addressing the wrong variable: D4 would have to clear more at the height it already plays at. Note also that better fair planning moves occupancy in the right direction on its own — 23.45 at K = 64 down to 21.38 at K = 256 — without any height term in the objective.

6. Verdict

Cannot yet determine whether a legal policy can reach flow balance. The evidence moved substantially toward "yes", and the information share of the gap is bounded above by 41%.

• The best legal planner measured sustains 2.2309 clears and 1.2782 reveals per move — 93.0% and 91.3% of the 2.400 / 1.400 requirement — against fair depth 4's 2.0467 / 1.1423 and the clairvoyant planner's 2.3601 / 1.3785 at a matched horizon.
• Fair planning closes 58.8% of the clairvoyant-minus-D4 gap at H = 7 and 40.0% at H = 5, where both sides are saturated. Knowing the future disc tape closes none of it. The residual — at most 41.2% — is hidden board plus the known suboptimality of hindsight optimization, and this experiment cannot split those two.
• Every fair game ended. 87 games across nine arm-B settings and two arm-A settings, none reached the move cap, best mean lifetime 200.88 moves against the clairvoyant's 366.25 with 6 of 8 censored alive.
• Every fair arm's board filled, but the slope falls fast with sampling: +1.269 → +0.833 → +0.311 cells per cycle as K goes 16 → 64 → 256 at H = 7, against the clairvoyant's +0.014. The trend had not flattened when compute ran out.

So finding-06's ceiling is not obviously out of reach for a legal policy, but it was not reached here, and the honest position is that the fair-planner frontier is still moving and its limit is unmeasured.

What this changes for the research program

1. Report K — or any chance-estimator budget — as a first-class result. At H = 7 the difference between K = 1 and K = 256 is 1.5370 vs 2.2309 clears per move, more than twice the entire gap under study. A fair sampling planner reported at small K is reporting an artifact, and this experiment would have reached the opposite conclusion had it stopped at K = 16.
2. finding-06's distillation plan is weakened but not dead. Up to 41% of the teacher's margin is not transferable in principle, so a student fitted to clairvoyant labels is partly being asked to learn a function of hidden state — the sibling extrapolation failure docs/research/status.md records, by a new route. Distilling the fair planner instead has no such defect and is now the better-posed target: it is legal by construction, it already beats D4 by 0.18 clears per move, and it costs 256 exact window solves per move, which is exactly the kind of expense a learned evaluator exists to amortise.
3. The residual has a name, and it is actionable. The one thing a PIMC planner structurally cannot do is act to reduce its own uncertainty. Drop7 demands 1.400 covered reveals per move by conservation, and every fair arm here under-delivers on reveals by about the same fraction as on clears. A fair planner that values cracking covers in order to learn — an explicit information term, or a belief-state search — is a bounded change to the objective rather than a new family, and it targets precisely the part of the residual that is not genuine hidden information.
4. Do not spend compute on making D4 keep a lower board. §5.

Limitations

1. Hindsight optimization is not the optimal fair planner, and this is the dominant limitation. Every "information share" number is an upper bound and every "planning share" a lower bound.
2. The K axis is not saturated at H = 7. The headline 2.2309 and 58.8% were still improving at the budget boundary. H = 5 is saturated and gives 40.0%; the truth for a fully-resourced fair planner is at or above the larger of the two.
3. The fair planner could not be run at H = 9, the horizon of finding-06's headline clairvoyant number. A fair decision costs K window solves and an H = 9 window costs 2.92 s against H = 7's 0.079 s, so H = 9 at a non-artifactual K was far outside budget. All decompositions are stated at a matched horizon and never against finding-06's H = 9 figure.
4. Eight games per arm, 87 fair games in total. Small cohorts. Paired master tapes remove most between-game variance from the comparisons, but per-arm means carry wide intervals — visible in the K = 16 vs K = 64 pairing, where seven of eight seeds improved and one collapsed from 75 moves to 35.
5. Arm A is a diagnostic, not a clean measurement of tape value. A determinized planner is not monotone in information, so arm A's result supports "the tape is not the source of the advantage" and not a signed valuation of tape knowledge. Its K = 256 cohort is also 7 of 8 games: the eighth was stopped, still running, when the resource budget closed, and the arm A vs arm B comparison at that K is therefore reported paired over the seven seeds both arms completed.
6. Objective mismatch with fair D4. The comparator maximizes its frozen terminal utility over a depth-4 expectimax; the fair arms maximize numbered discs cleared over an H-move exact window. They differ in more than information, so "fair planning closes 58.8%" describes these planners.
7. The latent randomness model is not the base engine's model (audit-01 M2), inherited from finding-06. Scores are not comparable to any ledger figure.
8. The occupancy-band table is descriptive, not causal. Counts are given for that reason.
9. The timings are not timing-grade. Shared machine, load averages 20–40, at most 12 threads. No performance claim is made.
10. 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/flow-ceiling/ and docs/exploratory/. The same debt is open for finding-02 and finding-06.

Reproduce

./approaches/lifetime-objective/flow-ceiling/build.sh

# gates, including the information-boundary gate
./build/flow-ceiling/flow-run --self-test
./build/flow-ceiling/flow-run \
    --cross-check approaches/lifetime-objective/scenario/data/suite-h9-v1.jsonl \
    --threads 12

RID=RUN-FLOW-044da902f8e8

# arm B, fully fair: the K sweep at H = 7 and H = 5
for K in 1 4 16 64 256; do
  ./build/flow-ceiling/flow-run --policy fair-rh --games 8 \
      --seed-start 0xa5230000 --sampler-seed 0xa5234000 --horizon 7 \
      --samples $K --max-moves 400 --threads 6 --time-limit 20 \
      --jsonl runs/$RID/fair-rh-h7-k$K.jsonl
done
for K in 16 64 256 1024; do
  ./build/flow-ceiling/flow-run --policy fair-rh --games 8 \
      --seed-start 0xa5230000 --sampler-seed 0xa5234100 --horizon 5 \
      --samples $K --max-moves 400 --threads 6 --time-limit 20 \
      --jsonl runs/$RID/fair-rh-h5-k$K.jsonl
done

# arm A: the future disc tape is known, the hidden board is not
for K in 64 256; do
  ./build/flow-ceiling/flow-run --policy fair-rh --games 8 \
      --seed-start 0xa5230000 --sampler-seed 0xa5234200 --horizon 7 \
      --samples $K --tape-known --max-moves 400 --threads 6 --time-limit 20 \
      --jsonl runs/$RID/fair-rh-armA-h7-k$K.jsonl
done

# the tables
python3 approaches/lifetime-objective/flow-ceiling/compare.py \
    "fair-d4=runs/$RID/fair-d4.jsonl" \
    "armB H7 K64=runs/$RID/fair-rh-h7-k64.jsonl" \
    "armB H7 K256=runs/$RID/fair-rh-h7-k256.jsonl" \
    "armA H7 K64=runs/$RID/fair-rh-armA-h7-k64.jsonl" \
    "armB H5 K256=runs/$RID/fair-rh-h5-k256.jsonl" \
    "clairvoyant H7=runs/$RID/rh-clears-h7-c1.jsonl"

Seed lease

Game seeds are 0xa5230000–0xa5230007 from SEEDLEASE-A52-FLOW, deliberately reused from finding-06 so both documents' cohorts play the same master tapes and every comparison is paired. Those seeds were already opened by finding-06 and are development data permanently.

The hidden-board sampler is a separate stream from SEEDLEASE-A52-FLOW2 = 0xa5234000–0xa5237fff, one sub-stream per game, partitioned by arm: 0xa5234000+ arm B at H = 7, 0xa5234100+ arm B at H = 5, 0xa5234200+ arm A. flow-run refuses to start with a sampler seed outside FLOW2 or a game seed outside FLOW.

Environment

AMD clang 23.0.0git, -O3 -std=c++20 -pthread -Wall -Wextra. 16 physical cores / 32 logical, 125 GiB RAM, shared with other jobs throughout; at most 12 threads were used. Frozen sources consumed unmodified, hashes recorded at build time in build/flow-ceiling/sources.sha256.