PROG-R4 U3, GATE G1: seeded recovery of named Table IV RPOs at the compliant scale

Unit: PROG-R4 U3 (leg 380), ORCHESTRATION.md §3c programme lane. CLAIM unit: answers a gate. Mode: §3f SOLO. UNVERIFIED: produced and checked in one session, no paired verifier. Data of record: writeup/data/p2_prog_r4_g1_v1.json; ledger experiments/programme_r4/u3_g1_ledger.json. Figure: fig97, 22/22 checks. Pre-registration: experiments/journal/prog_r4_prereg.md; experiments/journal/prog_r4_u2u3_prereg_addendum.md §§3–3f.

1. Gate and answer

G1: DOES AT LEAST ONE NAMED TABLE-IV RPO RECOVER TO tol=1e-8?

UNDER-RESOURCED. n_recovered = 0, n_converged_to_tol = 14, n_attempts = 100, controls_fired_as_planted = true, control_failures = [].

Branch selection is mechanical and was fixed before the run:

answer = (("YES" if recovered else "UNDER-RESOURCED")
          if controls_fired else "UNANSWERED")

with "NO" removed from the branch set by AMENDMENT 3 on measured cost. §3d's stop does not fire; route 4 is not stopped on measurement.

2. The controls, which license the answer

A negative-side answer is admissible only from an instrument shown able to produce a positive. Three controls, pre-registered in addendum §3 before the recurrence stage and before any attempt:

FIRED AS PLANTED := P recovered AND N did not recover AND (R recovered, if R ran), satisfied.

Control P is the one that has already earned its keep: in the cap-fixing work it exposed a defect in newton_hookstep_rpo that discarded a converged iterate and returned success=False, which at G1 would have recorded a non-recovery on an attempt that recovered, a fabricated no on the one gate whose no stops route 4. Documented separately in TECHNICAL_P2_PROGR4_CONTROL_CATCH.md.

P is also run at T_P = 2.65, not the pre-registered T = 19.33, per AMENDMENT 1: at T = 19.33 the equilibrium's Lyapunov exponent (2.88) amplifies by ~1e24, which is not a control, it is a different problem. T_P is defined as the multiple of dt at which the equilibrium's measured amplification equals the attractor's 8.73e2, obtained by log-interpolating the measured amplification table between T=2 (99.9) and T=3 (2.81e3).

3. Resourcing as run

P(0 successes in 100) at the sourced rates: 1.2e-2 (C&K 4.3%), 3e-5 (L&K ~10%). Leg 353 baseline: 5 attempts at T = 2000, line search, any_converged = false.

4. Convergence behaviour

Rate. 14/100 = 14% per attempt, above both sourced rates.

Magnitudes. Best final ‖R‖ = 8.673594e-12; the 14 convergences span 8.67e-12 to 8.64e-09; the next-best non-convergence is 2.80e-02. The distribution is cleanly bimodal with a gap of six orders of magnitude and nothing in it.

Epochs. Convergences: min 10, median 16, max 29, all well inside max_newton = 52. Non-convergences: all at the cap (51 recorded steps).

Krylov. Max dimension over all attempts 25, p95 = 24, against max_gmres = 140. Clause (a) of the cap rule was satisfied by 5.8×.

Stall diagnostic on the 86 non-convergences, reduction in ‖R‖ over the final 10 epochs:

They are stalled, not descending.

5. The eight converged solutions

Shifts wrapped to (−π, π]; ±s are the same orbit under the flow's reflection symmetry.

The replications are across different snapshots and different anchors, agreeing to 0.012 in T and 0.002 in |s|. That is internal replication of a genuine solution, not a repeated number.

n_recovered = 0. The gate's predicate is convergence to tol and a match to a named row within MATCH_T_TOL = MATCH_S_TOL = 0.05 on both T and s. Nothing satisfies it. fig97 asserts this separation explicitly:

check("no attempt is marked recovered without matching a named row on BOTH T and s",
      all((not a["recovered_named_orbit"])
          or (a["delta_T_from_published"] <= 0.05
              and a["delta_s_from_published"] <= 0.05) for a in att))

6. The second selection bias

Converged |s| ∈ [0.073, 0.317], 12/14 below 0.14. Published |s| ∈ [0.295, 0.707]. Disjoint but for one point.

Measured on the library's 1153 m = 0 candidates: rank correlation corr(|s|, R) = 0.50, and admission by R < R_thres_window = 0.25 is monotone decreasing in shift:

Admitted pool median |s| = 0.112; 21% of it exceeds the smallest published shift. All candidates regardless of window: median |s| = 0.824. The population exists; the window removes it.

Newton converges near its seed, so the seed pool's shift distribution propagates to the solution set. The observed 14 solutions are the expected image of the admitted pool under the solver, not an anomaly of the solver.

Structural identity with AMENDMENT 4. U2 found the same failure in the period coordinate: a global top-N ranked by R_red left exactly 1 of 400 candidates near a published period, because short intervals score better for reasons unrelated to being an orbit. Both are: a scalar score used as an admission filter, systematically monotone in a coordinate on which the targets are selected. The period case was fixed by stratifying the budget; the shift case is unfixed, because it was not anticipated.

7. Errata against AMENDMENT 3 (addendum §3e)

1. Envelope arithmetic ignored Pool.map chunking. Sized as ceil(100/10) = 10 balanced rounds; actual chunksize ceil(100/(4·10)) = 3 gives 34 chunks and a 12-attempt tail worker. Errs safe.
2. Per-epoch cost underestimated ~1.7×. Probe 54.7 s/epoch; run ~95 s/epoch (134.45 core-hours / 100 attempts / ~51 epochs). Probable cause: the probe did not run at ten-way concurrency. Cost successors at 95.
3. The max_newton_required = 238 model is wrong about the mechanism. It assumed slow convergence (2× a censored median of 119 epochs to convergence-or-stall). Observed behaviour is bimodal: ≤29 epochs or flat. Neither cap bound any attempt that converged.

UNDER-RESOURCED remains correct (the run was below the named budget and no was correctly unavailable) but the named cost (36.2 h at 10 workers, 15.1 core-days) is the wrong purchase. Buying 238 epochs extends 86 flat sequences.

8. Lesson 91 (what this negative names

Realization: first-order in time (Lie–Trotter: RK4 on the nonstiff part then the exact viscous factor; measured local ratio 4.00, global ratio 2.00) the splitting error, not RK4, sets the order). Its periodic orbits are O(dt) perturbations of the continuous flow's.

Trial space: the extended residual R(w₀, T, s) = σ_s Φ_T(w₀) − w₀ carries a continuous x-shift only; there is no m unknown, so 345 in-window candidates cannot be expressed as seeds. All eight named rows have m_published = 0, so nothing named is lost, but the sentence must carry the limitation.

Seed supply: an admission filter monotone in |s|, per §6.

Full form of the negative: no named Table IV row was recovered by a first-order-in-time realization, from a seed pool whose admission filter is monotone in the coordinate that separates the targets, inside an iteration budget below the one the cap rule named. Not "the orbits are not there"; not "Newton cannot find them"; it found eight others, reproducibly.

9. Successor ordering

1. Stratify the seed budget by shift. 395 m=0 candidates already lie in the published band, 11% already inside the window. Cheapest untried action, targets the measured cause.
2. Replace or condition the R < 0.25 admission test, which is confounded with |s|.
3. Add an m unknown to the residual: a realization change, invalidates M1's reproduction, needs its own milestone.
4. Buy iterations last. The bimodality says this is the lowest-yield spend.

Not started here; selecting one would be choosing the unit's own next task.

10. Obligations

CLAY_OBLIGATIONS §6 items 1 and 2 OPEN. §4 OPEN and NOT discharged pending leg 386's pre-registered δ mode. Ceiling TIER 2. No L1 → L4 link moved. Clay ~0.05%.

Gate G2 (U4, basin radius, brick B5 c-iii) UNANSWERED, not opened: it requires a recovered named orbit to perturb.