PROG-R4 R0 + R1, the metric re-derived, and the flatness abort measured

Unit: PROG-R4 R0 + R1, Lane R (reformulation for scale and solver competitiveness), wall W7. Dispatched as one worker in CONDUCTOR mode (ORCHESTRATION.md §3g), wave 1. Mode: §3f SOLO. UNVERIFIED: produced and checked in one session, no paired verifier. Kind: MEASUREMENT over banked artefacts. ≈0 new compute: no solver attempt, no DNS step, no mining, nothing re-run. U5 was not re-run (standing prohibition) and no additional seed supply is proposed. Data of record: writeup/data/p2_prog_r4_r0r1_v1.json, written by experiments/programme_r4/r0_metric.py (20/20) and experiments/programme_r4/r1_flatness.py (10/10). Read-only inputs: writeup/data/p2_prog_r4_g1_v1.json (U3), writeup/data/p2_prog_r4_m3_v1.json (U5), experiments/programme_r4/u5_m3_converged_orbits.npz, and experiments/p2_prog_r4_m3_evidence.py §5 as the arbiter of "distinct". SHA-256 of all three data artefacts is banked in r0.provenance. Evidence: experiments/p2_prog_r4_r0r1_evidence.py, 66/66, rebuilds every number below from the JSON and cross-checks the load-bearing ones back against the two unit JSONs. Figure: fig108, 12/12 checks. Journal: experiments/journal/prog_r4_r0r1.md.

NO GATE IS RE-ANSWERED HERE. G1 stays UNDER-RESOURCED as banked in p2_prog_r4_g1_v1.json; M3 stays DELIVERED as banked in p2_prog_r4_m3_v1.json. Neither file is written to by this unit.

TIER 2. CEILING CLAUSE. No L1→L4 link moved. A best-in-field orbit finder does NOT move a Clay link: it makes the questions affordable, which is a DIFFERENT AND LESSER THING. Clay stays ~0.05%.

1. Why the unit exists, and why the obvious answer was not acceptable

WALLS.md §LANE R fixes the reported metric as DISTINCT ORBITS PER CORE-HOUR, with per-attempt rate demoted to a secondary diagnostic, on the ground that per-attempt rate "is trivially inflated by feeding the solver easier seeds … and it counts a re-find of a known orbit as a success."

Two figures behind that metric were in dispute: U3's core-hours (134.45 vs 144.69) and U3's distinct count (8 vs 7). A concurrent session (commit 5c6d495) had already written both resolutions into WALLS.md's prose. Those resolutions are a transcription of U5's own numbers, not an independent measurement: the same move this unit exists to catch. R0 therefore re-derived both figures from the artefacts under ORCHESTRATION.md §3e ("re-derive every number from writeup/data/*.json and the banked ledgers, NEVER from prose") and reports agreement or disagreement. No figure in this document is cited from WALLS.md.

2. R0(i), core-hours

All four quantities from named fields of the two unit JSONs:

Verdict: AGREE on the numbers, DISAGREE on the diagnosis.

WALLS.md's current 144.69 (U3) and 57.04 (U5) are confirmed: 52087.95185184479 × 10 / 3600 = 144.6888 and 25665.83147907257 × 8 / 3600 = 57.0352.

WALLS.md further states: "This file first quoted 134.45 for U3 … Reconcile against the JSON, not against prose." That characterisation is wrong. 134.4475 is Σ attempts[].wall_seconds / 3600 over the 100 attempt rows of the same JSON, the attempt CPU time, and it is the figure U3's own row in writeup/INDEX.md already carries ("14.47 h, 134.45 core-hours"). Nothing here was derived from prose. The two figures are the pool reservation and the attempt CPU; the 10.2413 core-hour difference is straggler idle at 92.92% pool utilisation, against U5's 98.20% under imap_unordered with chunksize=1. Both are correct measurements of different quantities and both should be retained, with a convention label.

Missing artefact, reported not estimated (§3d). The denominator is strictly worker-hours: U3 reserved 10, U5 reserved 8. The machine's physical core count appears in no numeric field of either JSON; it exists only inside the prose string resourcing.workers_note in U5's file, which §3e forbids using as a number. A machine-core-hour figure is therefore not derivable from what is banked, and is not estimated. What would have to be banked: magnitudes.physical_cores, and ideally per-attempt CPU time (time.process_time) alongside wall_seconds.

3. R0(ii): the distinct count

3.1 The rule, verbatim from the arbiter

experiments/p2_prog_r4_m3_evidence.py §5 is the arbiter named in the commission. Its rule, banked verbatim in r0.cluster_rule_verbatim:

# Lucas & Kerswell 2015 Table IV: the |s| band the eight named rows live in.
BAND = (0.295, 0.707)
# The matching predicate of record (Ban 2): a convergence is a recovery only if
# it matches a named row on BOTH T and s to within this. Reused here as the
# clustering tolerance, so "distinct" is never finer than "recovered".
TOL = 0.05

def wrap_abs(s):
    """|s| on the circle: the domain is 2*pi periodic, so |s|=6.18 is 0.10."""
    a = abs(float(s)) % (2.0 * math.pi)
    return min(a, 2.0 * math.pi - a)

def cluster(attempts):
    """Group convergences into distinct solutions at the matching tolerance."""
    out = []
    for a in sorted(attempts, key=lambda a: a["T_converged"]):
        for c in out:
            if (abs(c[0]["T_converged"] - a["T_converged"]) <= TOL
                    and abs(c[0]["S"] - a["S"]) <= TOL):
                c.append(a)
                break
        else:
            out.append([a])
    return out

It is a greedy leader rule: each point is compared only against c[0], the cluster's first member. That is not an equivalence relation in general, so reproducing its output is not by itself enough. The rule was re-implemented independently twice (in r0_metric.py and again in the evidence script) rather than imported, so agreement is a check and not a tautology.

3.2 The result

U3: 14 convergences → 8 distinct. U5: 9 → 5. AGREES with WALLS.md on both.

3.3 Robustness

The partition is a genuine equivalence class here, so the rule's greediness costs nothing on this data. Tolerance sweep (U3): 0.01 → 11, 0.02–0.04 → 9, 0.05–0.10 → 8, 0.15–0.20 → 7, 0.30 → 6. The "7" reading requires TOL = 0.15, three times the predicate of record.

Where the count is decided. Widest accepted merge: attempts 12 & 43, Chebyshev distance 0.047969 = 0.959 × TOL. Narrowest failed merge: attempts 12 & 99, 0.064414 = 1.288 × TOL, decided by T alone (their |s| differ by 7.13e-5). The count 8 lives in the band [0.0480, 0.0644].

3.4 The prose error, located

WALLS.md §R0's argument for doubting the 8 was: "10 convergences over 3 replicated solutions, then 4 remaining cannot yield 5 more." Measured: 9, not 10, cluster sizes 4 + 3 + 2, with 5 singletons. 3 + 5 = 8 and the arithmetic closes. The same "10 of U3's 14 convergences landed on three solutions" sentence is still live in WALLS.md §R2 and in STATE.md. R2's conclusion is unaffected (9 of 14 is still the majority).

3.5 Instrument limit

U3's converged states are not banked anywhere in this repository. Only U5's are, in u5_m3_converged_orbits.npz (9 fields, each (24,24) float64). So U3's 8, the headline numerator, can be tested only in the (T, |s|) invariant pair, never in state space, and nothing banked can adjudicate the 1.288 × TOL pair.

Even for U5 the npz cannot validate the rule: it banks one snapshot per converged attempt at an unspecified phase along the orbit, so two snapshots of the same orbit need not be close. Measured shift-invariant |FFT2| relative distances: attempts 64 and 76 (same cluster) 0.0014, attempts 40 and 52 (same cluster) 0.1992, attempts 52 and 45 (different clusters) 0.0888. A same-cluster distance exceeding a cross-cluster one is reported here as an instrument limit, not as support for or against the count. Settling it would need U3's 14 states plus a phase-aligned distance minimised over the continuous shift and over time-translation, i.e. the stepper, i.e. new compute. Not proposed.

4. R0(iii): the seed overlap, and what it does to the metric

Not previously measured anywhere. Seeds matched on (T_seed, s_seed, R_seed) at full float precision:

• 57 of U5's 100 seeds had already been spent by U3.
• 5 of U5's 9 convergences sit on those seeds, and all 5 are bit-identical re-executions, same T_converged, s_converged, final_residual and n_iters as the U3 attempt. U5 att40 = U3 att80; att64 = U3 att12; att74 = U3 att39; att76 = U3 att43; att77 = U3 att44. All five U3 attempts had themselves converged.
• On U5-only seeds the distinct count is 4, not 5. One of U5's five solutions was reached only by re-running seeds U3 had already spent.
• Clustering U5's five against U3's eight under the same rule, four are re-finds. U5's contribution new to the programme is one orbit: T = 20.4175, |s| = 0.5867, attempt 7, anchor UPO37.

The metric therefore still counts a re-find as a success: one level up. It removes the defect within a run and not between runs, which is the level at which WALLS.md's comparison is made.

The retraction

WALLS.md's "U5 is above U3 on every variant) Lane R's first measured improvement" is RETRACTED AS AN INFERENCE. The arithmetic is confirmed: U5 is above U3 under all three derivable core-hour conventions, 1.27× to 1.59×, direction robust, magnitude not. What does not stand is reading that as Lane R's machinery having improved. The metric is per-run; the two runs are not independent samples (57% seed overlap, 5 of 9 convergences bit-identical); and on the reading that tracks what the programme actually gained, U5 is 3.15× worse. Lane R may claim a cheaper run. It may not claim a better orbit finder on this evidence.

Retracting this was pre-committed as an allowed and expected outcome (branch (a)), and it fired. Branch (b) (the prohibition on reporting a correction that flatters the programme as a win) did not arise (the count did not resolve to 7) and is honoured anyway: nothing in this unit is reported as strengthening the result.

5. R1 (early abort on flatness

WALLS.md's name for the unit is "R1) early abort on flatness", so the commissioned filename r1_flatness.py already matches and was not renamed.

5.1 The premise, checked rather than taken

All of WALLS.md §R1's stated numbers are CONFIRMED from residual_history directly:

• all 14 U3 convergences finished in ≤ 29 epochs, median 16 (n_iters = 10,10,10,10,12,14,15,17,17,22,23,26,28,29);
• all 86 non-convergences ran to n_iters = 51 (52 residuals, the cap);
• over their final 10 epochs, 53% reduced ‖R‖ by <1%, 83% by <10%, 3% were still halving.

Only 243 of U3's 4,629 epochs (5.2%) sit inside a convergence.

5.2 The criterion family, and its determinism

abort at epoch k iff k ≥ K and ‖R‖_k > θ · ‖R‖_{k−W}

Three fixed constants and a comparison of two banked residuals, evaluated by replaying residual_history for all 200 banked attempts.

Leg 349 ban: compliance confirmed explicitly. No learned or evolved seed-scoring fitness appears anywhere in this unit. The criterion has no fitness, no scoring of seeds, no stochastic component, and does not touch seed selection at all: it changes only the stopping time of an already-launched attempt. Replaying it is exactly reproducible. R0's only randomness is 20,000 shuffles (seeded 20260813) used to stress a deterministic clustering rule, and the result is invariant. Nothing here is a GA, is fitted against seed quality, or yields a score a future unit could rank seeds by.

5.3 The controls

1. Zero false kills (asked for by WALLS.md): the rule must abort none of the 23 banked convergences.
2. Safety margin factor: θ divided by the worst window ratio ‖R‖_k / ‖R‖_{k−W} any banked convergence actually exhibited where the rule looks. Zero false kills over 23 examples is a weak statement on its own; this quantifies how much worse a future convergence could behave.
3. Hold-out both ways (added by this unit): select on one unit's ledger, score on the other's.

5.4 The incumbent already exists

U5's banked resourcing.stall_exit.rule is "from epoch 20, stop if ‖R‖k > 0.5 · ‖R‖(k−10)", exactly (K, W, θ) = (20, 10, 0.5).

Replayed independently, it reproduces U5's banked u3_epochs_under_rule = 2083 to the epoch and its banked margin_factor = 6.901609656627281 to nine decimals (θ = 0.5 against a worst banked convergence window ratio of 0.072447).

So the "cheapest competitive win in the repository" was already collected, by U5, before R1 was commissioned.

5.5 The sweep, and the headroom

12,597 rules (K ∈ [2,40] × W ∈ [2,20] × 17 thresholds), scored on both banked ledgers. 9,760 kill no banked convergence. Best U3 recovery at each safety floor:

HEADROOM over the deployed rule, at no loss of safety margin: +0.45 percentage points = +0.48 core-hours on a 134.45 core-hour run.

The unconstrained optimum is rejected: zero false kills but a 1.09× margin, i.e. a future convergence behaving 9% worse than the worst banked one dies under it. Taking the maximum of a statistic over 12,597 rules against 23 convergences is exactly how a stopping rule is overfitted.

5.6 Hold-out both ways, the control that decides it

At the deployed margin floor (≥ 6.9016 on the fitting set):

The asymmetry is the result. Nine convergences are not enough to fix a stopping rule, and the direction that fails is the one with the smaller fitting set. Any future tuning of this family must select on the larger ledger and score on the smaller, never the reverse.

5.7 Lesson 91, the negatives, named

Negative 1 (the retraction, §4). Realization: PROG-R4's Newton–GMRES–hookstep solver on the DSS/Kolmogorov flow at the banked Re and resolution, extended residual carrying T and continuous shift s. Trial space: the two banked 100-attempt runs, U3's globally-ranked seed budget and U5's |s|-stratified one, drawn from the same anchored admissible pool. Basis: the metric is per-run and the runs overlap in 57 seeds, so a between-run comparison of per-run distinct-orbit yield does not measure a change in the finder. A negative about the comparison, not about either run.

Negative 2 (the hold-out failure, §5.6). Realization: the abort family k ≥ K ∧ ‖R‖_k > θ‖R‖_{k−W} replayed on banked residual histories. Trial space: 12,597 rules, selected on one unit's ledger and scored on the other's. Basis: selection on U5's 9 convergences yields a rule that kills one of U3's 14. The family is usable; selection on the smaller ledger is not.

6. THE GATES

R0 gate: Are the programme's two headline figures, total core-hours and distinct-orbit count, > re-derived independently from the banked artefacts, each with the derivation shown, and each reported as agreeing or disagreeing with the figure currently in WALLS.md?

YES.

• Total core-hours: AGREE with WALLS.md's 144.69 (U3) and 57.04 (U5); DISAGREE with its account of the 134.45 figure, which is Σ attempts[].wall_seconds/3600 from the same JSON and not prose. Derivation in §2.
• Distinct-orbit count: AGREE: 8 (U3) and 5 (U5), under the arbiter's own rule, robust to linkage convention, to 20,000 orderings and to TOL over 0.05–0.10; DISAGREE with the prose premise "10 of U3's 14 convergences landed on just three solutions", measured as 9. Derivation in §3.

The corrected metric, in the gate's required form:

U3 achieved 0.0553 distinct orbits per core-hour. Numerator 8 = the number of clusters of attempts[].{T_converged, s_converged} in writeup/data/p2_prog_r4_g1_v1.json, under experiments/p2_prog_r4_m3_evidence.py §5's rule at TOL = 0.05 with s wrapped to the circle. Denominator 144.6888 = magnitudes.wall_seconds (52087.95185184479) × magnitudes.workers (10) / 3600.

U5 achieved 0.0877 distinct orbits per core-hour. Numerator 5, same field and same rule, from writeup/data/p2_prog_r4_m3_v1.json. Denominator 57.0352 = magnitudes.wall_seconds (25665.83147907257) × magnitudes.workers (8) / 3600.

Both denominators are strictly worker-hours. The physical core count is not a banked numeric field and is not estimated; see §2.

R1 gate: Is R1, as WALLS.md's Lane R defines it, executed to a measured number on the banked data, with its saving expressed as a compute quantity?

YES. Measured on U3's banked ledger: writeup/data/p2_prog_r4_g1_v1.json, the residual_history of all 100 attempts, replayed, nothing re-run. The criterion "from epoch 20, stop if ‖R‖_k > 0.5 · ‖R‖_(k−10)" recovers 55.00% of U3's 4,629 epochs (4,629 → 2,083) = 74.87 of its 134.4475 attempt-CPU core-hours (55.69%), at zero false kills over 23 banked convergences and a 6.90× safety margin.

With the correction that this saving is U5's, not R1's. R1's own measured contribution is the search that closes the family: 12,597 deterministic rules, 9,760 admissible, and the best rule that does not degrade the deployed safety margin buys +0.45 percentage points = +0.48 core-hours. R1 CLOSES; no further compute should be spent tuning this criterion family.

7. The projection, labelled

U3 replayed under the deployed rule finds the same 8 distinct orbits in 59.5736 rather than 134.4475 attempt-CPU core-hours:

PROJECTED 0.1343 distinct orbits per core-hour (attempt-CPU convention), against the measured 0.0595. PROJECTED, factor 2.26×.

The zero-false-kill control supports the "same 8" clause for the 14 convergences, but this remains a replay, not a run, and it is labelled PROJECTED in r1.projection, in fig108 panel C, in the journal and in the blog. Pre-committed branch (c) fired.

No recycled-budget orbit yield is reported. The assumption that recovered compute converts into further orbits at the observed rate is optimistic and is not made here: R0 measured that U5 spent 57.04 core-hours to gain one orbit new to the programme.

8. Corrections raised, and applied nowhere

Six corrections to WALLS.md / STATE.md are written verbatim in experiments/journal/prog_r4_r0r1.md under CORRECTIONS FOR THE CONDUCTOR and are applied nowhere: those files are Conductor-owned. In summary: (1) "10 of 14" should read 9; (2) the 134.45 figure is not prose and both conventions should be kept with labels; (3) the denominator is worker-hours and magnitudes.physical_cores should be banked; (4) the metric counts cross-run re-finds, so both the per-run and cumulative rows should be reported and the "first measured improvement" claim withdrawn as an inference; (5) §R1's own numbers are confirmed but R1 should be marked CLOSED: collected by U5, headroom +0.45 pp; (6) U3's converged states are not banked, which caps any future audit of the distinct count at the (T, |s|) proxy.

9. What this unit did not do

• No solver attempt, no DNS step, no mining. Nothing was re-run. U5 was not re-run.
• No proposal for additional PROG-R4 seed supply.
• No edit to STATE.md, WALLS.md, DIRECTION.md, plan_of_record.py, any experiments/programme_r4/u*.py, any banked ledger, p2_prog_r4_g1_v1.json, p2_prog_r4_m3_v1.json, u5_m3_converged_orbits.npz, or p2_prog_r4_m3_evidence.py. DIRECTION.md was not read.
• No gate answer changed. No estimate substituted for a missing artefact.
• No external outreach (standing user hold).

Scale is not evidence. A faster solver is not a result. The gate is the deliverable: and the ceiling stands: this makes the questions affordable, which is a different and lesser thing than progress on the Clay statement.

Blog version: BLOG_P2_PROGR4_R0R1.md. Figure: fig108 (writeup/figures/fig108_prog_r4_r0r1.py). Evidence: experiments/p2_prog_r4_r0r1_evidence.py, 66/66.