PROG-R4 U5, MILESTONE M3: stratifying the seed budget by SHIFT

Unit: PROG-R4 U5 (leg 380), ORCHESTRATION.md §3c programme lane. BUILD unit: answers a milestone, not a gate. Under §3c the worker is not terminated on landing it, and U0's novelty pass binds (no fresh pass run). Mode: §3f SOLO. UNVERIFIED: produced and checked in one session, no paired verifier. Data of record: writeup/data/p2_prog_r4_m3_v1.json; ledger experiments/programme_r4/u5_m3_ledger.json; seed library experiments/programme_r4/u5_shift_library_admissible.json. Evidence: experiments/p2_prog_r4_m3_evidence.py, 27/27, re-derives every number below from the banked JSONs. Figure: fig107, 16/16 checks. Pre-registration: experiments/journal/prog_r4_u2u3_prereg_addendum.md §3g (AMENDMENT 5), commit 4f6f33f, before either stage ran. Journal: experiments/journal/prog_r4_u5.md.

G1 IS NOT RE-ANSWERED HERE. It stays UNDER-RESOURCED as banked in writeup/data/p2_prog_r4_g1_v1.json, which this unit does not write to. n_recovered = 0 is a count, pre-registered as such; it is not a no at G1, and no G1 re-open is raised.

1. Milestone and answer

M3: THE SEED BUDGET IS STRATIFIED BY SHIFT, the anchored reservoir is exhausted, the published |s| band is filled to a pre-committed quota, and the per-stratum yield is measured against U3's banked baseline.

DELIVERED, on all five pre-committed clauses:

n_attempts = 100, n_converged_to_tol = 9, n_recovered = 0, best final ‖R‖ = 8.67e-12, median final ‖R‖ = 3.14, 42,228 Jacobian evaluations, 7.13 h at 8 workers.

Ceiling: TIER 2. No L1 → L4 link moved. Clay stays ~0.05%.

2. Why the previous fix does not transfer

AMENDMENT 4 stratified the mining budget by period, because period is a coordinate of the (t, T) plane the mining searches. Shift is not. R_red is built from amplitude spectra and is shift-invariant by construction: the same property that makes it a lossless pre-filter (R_red ≤ R pointwise, so R_red ≥ 0.25 provably cannot hide R < 0.25). |s| does not exist until the full minimisation over the continuous x-shift and discrete y-shift runs.

U5 therefore exhausts at the mining stage and stratifies downstream:

MINING RULE. Take every anchored strict local minimum of R_red with R_red < 0.25. Nothing ranked, nothing truncated, no per_anchor, no researcher degree of freedom in the take.

3. Resourcing as run

Costed at 95 s/epoch (U3 erratum 2), not the probe's 54.7.

Options explicitly not taken. U3 §8's (b) relax the admission test, (c) carry an m unknown in the residual, (d) buy iterations. (c) would change the realization and invalidate M1's reproduction; it needs its own milestone. (d) is the opposite of what happened, see §3.1.

3.1 The stall exit, validated before it was used

max_newton_hit was replaced by an exit rule (after epoch 20, stop if ‖R‖_k > 0.5·‖R‖_{k−10}) and the rule was replayed against U3's banked residual histories before U5 ran:

This is a spend reduction, not an iteration purchase, and clause 3 is checked in that direction.

4. The supply side

The take. 913,301 strict local minima → 103,844 anchored (within T_ANCHOR_TOL = 1.0 of a named period) → 75,873 with R_red < 0.25. U2 took 2,014: a 37.7× deeper take.

The funnel. 75,873 mined → 575 in the Newton window R < 0.25 → 241 with m = 0 (all 575 already anchored) → 100 spent. 334 dropped for m ≠ 0 (58.1% of the window).

U2's library offered 35 in-band admissible candidates; U5's offers 72. That is the whole supply-side intervention.

Per-row in-band supply (the per-row cap of 8 bound on UPO37, UPO22, UPO9, which is what it is for): UPO37 21, UPO9 16, UPO22 11, UPO32 9, UPO17 6, UPO35 5, UPO20 4, UPO34 0. UPO34 has no in-band anchored admissible candidate anywhere in the exhaustive pool.

4.1 The admission table, remade on 38× the data

TECHNICAL_P2_PROGR4_G1.md §5 measured admission against |s| on 2,014 globally-ranked candidates. That measurement is superseded here by the same measurement on the exhaustive pool, same phenomenon, no ranking step in it:

Admission falls monotonically with |s| to 0.9, then the pattern breaks: the window rate recovers sharply while the m = 0 rate collapses. Those cells are the m ≠ 0 class the extended residual cannot express. So "large shift is hard" and "large shift is a symmetry class this realization is blind to" are separated by the m = 0 column, and above |s| = 0.9 it is overwhelmingly the second.

Read with its confound. For a candidate that is not a genuine near-recurrence, the minimising shift is a nuisance parameter and its |s| is close to meaningless, so the high-|s| bins mix "shift is hard" with "this was never a recurrence". The m = 0 column is the part that survives.

4.2 Containment, and a number that reads like a defect

U2's library contains 81 candidates with R below U5's pool minimum of 0.0685, global best 0.0165. All 81 are unanchored, at T ∈ [1.75, 2.5]: trivial short-time near-recurrences far below the smallest named period, outside an anchored-only rule by design. The check that binds: all 133 of U2's anchored admissible candidates are inside U5's 241, none disagreeing on (R, s, m) to full float precision. Asserted in u5_reduce_library.py, not claimed.

Parallel regeneration was checked before it was trusted, and the check fired. The first block-parallel implementation walked from pos = block · CKPT_EVERY and produced R = 2.50754 / 2.15963 / 2.06460 against U2's serial 2.28409 / 1.99639 / 2.03410. Cause: run_dns writes the checkpoint before the steps producing snapshot k, so ckpt[b] is the state at index b·CKPT_EVERY − 1. Fixed; re-tested to max|diff| = 0.000e+00 over 80 sampled snapshots, and 40 banked U2 candidates re-mined to exact equality in (R, s, m).

5. Per-stratum yield, and the pre-registered fork

Every interval overlaps its U3 counterpart; not one stratum's change resolves at 95%. The in-band rate rose from 1/31 to 5/60 and that is inside noise. Overall 9/100 against U3's 14/100, not distinguishable, and the seed pools are not the same anyway, which is what §5's matched-R comparison is for.

AMENDMENT 5 named two rival readings of U3's in-band drought and what would kill each.

H-supply (REFUTED as the operative explanation. Its premise was repaired (supply 35 → 72, spend 31 → 60) and its prediction) more in-band seeds produce a recovery: failed at 60 in-band attempts and 0 recoveries.

H-hard: FAVOURED, NOT RESOLVED. A raw rate comparison confounds "the band is hard" with "these seeds are worse", which is the same shape of confound this programme is about, so the comparison is made at matched seed R, pooled over both runs' 200 attempts in common quartiles:

Fisher exact two-sided p = 0.073. Above 0.05. This is a lean, reported as a lean. (Fisher is implemented from scratch in the evidence script; no scipy dependency was added for one number.)

6. The result: convergences leave the band

Of the 9 convergences, 5 were seeded in-band. 4 of those 5 finished outside it.

8 of 9 convergences finished at |s| < 0.15; range 0.0729–0.5867.

This is a replication, not a post-hoc reading. The addendum recorded from U3's banked data, before U5 ran, that 13 of its 14 convergences also finished at |s| < 0.15. (The addendum's wording "moved |s| toward zero or stayed there" is loose, only 9 of 14 strictly decreased; the measured invariant is the endpoint, and that is the form U5 replicates on an independent 60-attempt arm.) Separately, commit c67179e, landed by a concurrent session while U5 was still in flight and before any U5 number was visible, fixed that what decides U5 is where the converged orbits sit in |s|. Its branch (b) fired: the bias is in the BASIN STRUCTURE, not only in the seed supply.

Not separated by this data: "the low-|s| basins are larger" vs "this hookstep globalisation drifts that way". Both are claims about the solver's destination rather than its menu.

Closest approach to a named row across all 100 attempts: attempt 0 (UPO37), ΔT = 0.127, Δ|s| = 0.0099 (inside tolerance on s, outside on T) at final ‖R‖ = 2.20, i.e. not a solution. No attempt came close on both coordinates and converged. No near-miss is manufactured.

7. What the stratification bought, against U3's solution set

Measured after the run, not pre-registered; changes no milestone clause. Clustering at the matching predicate's own tolerance (0.05 on both T and |s|, s wrapped to (−π, π]), U5's 9 convergences give 5 distinct solutions and U3's 14 give 8.

The one new solution is the only solution either unit has found inside the published band, and it is the single in-band convergence that did not leave (§6, first row). Read narrowly: it is not a named Table IV row, n_recovered is still 0, G1 is still UNDER-RESOURCED.

R0's metric (distinct orbits per core-hour, which R0 fixed as the reported metric before this landed, because per-attempt rate counts re-finds as successes): U5 5 / 57.04 = 0.0877 against U3 8 / 144.69 = 0.0553. Two cautions owed to R0, neither resolved here: (i) R0's baseline quotes 134.45 core-hours for U3, where p2_prog_r4_g1_v1.json gives 52,087.95 s × 10 workers = 144.69; (ii) R0 disputes U3's distinct count of 8, and at 7 the baseline is 0.0484. Independent re-clustering here reproduces 8. U5 is above U3 on every variant, but the comparison stays provisional until R0 lands.

The case for deflation, sharpened. 4 of 5 re-finds is cross-unit: a second 100-attempt budget rediscovered what the first already had. This is what R2 is queued against.

8. Option (c), priced

Carrying an m unknown in the extended residual would unlock 334 of the 575 in-window candidates (58.1%). But of those 334, exactly 1 lies in the published |s| band. Option (c) is not a band fix; it is the |s| > 0.9 fix. Measured, not argued, evidence script §6.

9. Controls

Unchanged from U3's addendum §3, plus two structural controls added in §3g.6 that could have fired against this unit: L positive on the pipeline (the low-|s| stratum must still converge, or the stratification broke the seed pipeline) and H against the monotone-in-|s| reading.

FIRED AS PLANTED := P recovered AND N did not recover AND (R recovered, if R ran), satisfied, failures = []. A null from an instrument with a dead control would be uninterpretable.

10. Errata against this unit's own pre-registration

Full list in experiments/journal/prog_r4_u5.md §7. The one a reader most needs:

(i) "Before either stage ran" is imprecise. AMENDMENT 5 (4f6f33f, 12:31:10) quotes the realised take (75,873 candidates, 98,413 snapshots, 3,980 blocks) but the mining stage did not finish until 13:29:33. Those three numbers come from the prefilter prune alone (913,301 → 103,844 anchored → 75,873, in 50.3 s), which performs no minimisation and computes no |s|. No outcome-dependent number (the 72/89/67/13 supply, the 575, the 241) appears in that commit.

Solo session, §3f. UNVERIFIED: self-checking is never recorded as verification, and every number above is re-derivable by a fresh session from the banked JSONs via experiments/p2_prog_r4_m3_evidence.py.