TECHNICAL, PROG-R4 U2/U3: the planted positive control, the defect it caught, and the cap arithmetic

Companion to BLOG_P2_PROGR4_CONTROL_CATCH. Pre-registration: experiments/journal/prog_r4_u2u3_prereg_addendum.md (§2 cap rule, §3 controls, §3a amendment 1, §3b amendment 2).

Status at time of writing. MILESTONE M2 and GATE G1 are both UNANSWERED. The T=1e5 DNS is in flight. Nothing below is evidence on G1 in either direction. Everything below is UNVERIFIED under ORCHESTRATION.md §3f (solo mode): it was produced and checked in the same session, and is carried into G1's own answer with that label.

1. What G1 asks, and why its no needs an instrument warrant

Gate G1, immutable wording:

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

• yes → U4 proceeds; ceiling TIER 2.
• no → a resourced null; §3d's stop fires; route 4 stops.

Because the negative branch has consequences, the pre-registration binds the negative branch to the controls:

CONTROLS FIRED AS PLANTED := P recovered AND N did not recover (AND R recovered, if R ran). If the controls do not fire as planted, G1 is NOT answered no. G1 answers UNANSWERED.

UNANSWERED is stricter than no, not softer: it withholds a stop that a no would fire.

2. Control P: construction

P must be recoverable by construction, exactly known, and require no search of ours that could itself fail. It is the discrete relative equilibrium: the exact fixed point of the map the code iterates, not of the PDE.

For the Lie–Trotter step (RK4 on the nonstiff part, then the exact viscous factor), a state with no nonlinear self-interaction satisfies, mode by mode,

w*_hat = dt * forcing_hat * decay / (1 - decay)

with decay = exp(-dt k² / Re). Implemented in experiments/programme_r4/u3_controls.py:

def discrete_relative_equilibrium(solver):
    d = solver.decay
    den = 1.0 - d
    ok = np.abs(den) > 1e-300          # masked division: k=0 gives 0/0 under np.where
    w_hat = np.zeros_like(solver.forcing_hat)
    w_hat[ok] = solver.dt * solver.forcing_hat[ok] * d[ok] / den[ok]
    return np.fft.ifft2(w_hat).real

Verified residuals of the planted point: ||Φ_dt(w*) − w*|| = 7.6e-14, and ||Φ_T(w*) − w*|| = 2.35e-13 at T=19.33. The extended residual there is an exact zero, so P's seed residual is not an estimate.

P exercises the state block only. At a relative equilibrium ||∂R/∂s|| = 0 exactly and ∂R/∂T is negligible (this is M1's recorded rank deficiency of 2). The (T, s) directions are control R's job, not P's.

3. Amendment 1, P failed as pre-registered, and the re-scaling rule

At the pre-registered T=19.33, P failed: trust_region_collapsed at epoch 1, residual 236 → 223, state unmoved.

It was not retuned to pass. The cause was measured:

Double precision spans ~1e16. At T=19.33 the equilibrium's Jacobian is numerically empty: the linearisation carries no recoverable information, and no globalisation can descend. That condition is a property of the equilibrium, not of the attractor the real attempts are seeded from, so P at full period was testing something the unit does not do.

The re-scaling rule was therefore fixed from the target: T_P := the multiple of dt at which the equilibrium's measured amplification equals the attractor's amplification over one Table IV period, 8.73e2. The equilibrium's amplification was measured in the linear regime (eps = 1e-13 relative):

Log-interpolating between T=2 and T=3 for a target of 8.73e2 gives T_P = 2.65.

Verified after the change: the plant is still exact at the new period (||Φ_T(w*) − w*|| = 8.03e-14) and its amplification is 8.35e2 against the 8.73e2 target.

The number comes from a property of the attractor measured before P was re-run: derived from the quantity being controlled, not from P's outcome. That is the distinction that keeps it a control rather than a tuned pass. The original failure remains in the record (u3_control_P_trial.json).

Consequence carried forward: P is now a conservative positive; it proves the state block can be recovered at attractor-like conditioning, not at arbitrary conditioning.

4. Amendment 2, the defect

4.1 The impossible pair

Sweeping P over GMRES caps returned, at max_gmres ∈ {80, 120}:

reason = "converged"     converged_to_tol = False     final_residual = 1.63e-8     tol = 1e-8

Both cannot hold if the reporting is correct.

4.2 Mechanism

newton_hookstep_rpo (solver/kolmogorov2d_nkbasin.py) drives newton_hookstep (solver/hookstep_newton.py) with max_newton=1 per epoch, because the residual's two phase rows are evaluated against a reference that is rebuilt at each accepted iterate (a moving Poincaré section). Inside newton_hookstep, with a budget of one iteration, "converged" is reachable from two places:

# hookstep_newton.py, top of the loop
if r < tol:
    reason = "converged"          # (i) AT ENTRY: hist == [r], x unmoved
    break
...
x, F, r = x_try, F_try, r_try
hist.append(r)                    # step taken, post-step residual appended
# after the loop:
if reason == "max_newton_hit" and r < tol:
    reason = "converged"          # (ii) ON THE STEP: hist == [pre, post], x MOVED

The outer routine read, unconditionally:

r = float(out["residual_history"][0])     # PRE-step in case (ii)
hist.append(r)
if out["reason"] == "converged":
    reason = "converged"
    break                                  # ...before the block that adopts out["x"]

In case (ii) this (a) reports the pre-step residual, (b) never adopts out["x"], so w0, T, s stay at the previous iterate, and (c) returns success = bool(r < tol) = False on a solve that had reached tol.

4.3 Impact

At G1 this records recovered_named_orbit = False on an attempt that recovered: a fabricated no on the gate whose no stops route 4 under §3d. It fails silently: no exception, no non-finite value, a plausible residual, and the exit reason and the success flag disagreeing in a field nothing was reading.

4.4 Fix and blast radius

The converged branch is moved after the adopt block and re-reads the converged pair:

        w0, T, s = w0_try, T_try, s_try

        if out["reason"] == "converged":
            r = float(out["final_residual"])
            hist.append(r)
            reason = "converged"
            break

Case (i) is preserved unchanged: when the inner call converges at entry it runs no iteration, so its ledger is empty and it exits through the pre-existing if not out["ledger"] branch with r already correct.

Blast radius: the branch is unreachable for any solve that never reaches tol.

• test_hookstep_newton.py (32/32) and test_kolmogorov2d_nkbasin.py (12/12) pass under the fix, and scripts/merge_gate.sh origin/main prints MERGE GATE: PASS. Neither suite detected the defect beforehand: the branch only executes on a success this problem had never produced.
• M1 re-run against the fix reproduces bit-identically: every field of u1_m1_ledger.json except wall-time fields. M1 stalls at ||R|| = 0.0794 and never enters the branch. M1 is not re-opened.

5. The cap sweep, and clause (a)

Pre-registered CAP RULE, clause (a): max_gmres ≥ 2 × the 95th percentile of Krylov dimensions actually consumed, so that GMRES is stopped by its rtol=1e-3 early exit and not by the cap.

Control P, max_newton=60, T=2.65, perturbation 1e-3:

Readings:

• At 80 the cap is binding in 13 of 28 epochs, clause (a) fails outright, independently of anything else.
• The 120 row is a cap failure and a reporting failure compounded, and is not a verdict on P. Only the 140 row is a clean measurement.
• Clause (a) requires max_gmres ≥ 131.6. max_gmres = 140 is the smallest round value meeting it, and is also the only cap at which P recovers. The two criteria agree.

5.1 Why the cost probe alone was not allowed to set this

The first cost probe reported p95 ≈ 20 Krylov directions, which would have justified max_gmres = 40. That p95 was measured over three epochs only, all far from the solution, where the linear solve converges easily. P collapses outright at max_gmres = 50.

Had the caps been set from that probe, a no at G1 would have been an artefact of the budget rather than a fact about orbits: precisely the under-resourced null §3d forbids. Clause (a) is therefore read against P's measured floor, not against a probe p95 alone, and P doubles as the cap-adequacy test: if P cannot recover an exact planted solution at the caps the attempts use, those caps are too small.

6. Measured cost, and the envelope

Second cost probe, run at the fixed max_gmres = 140 on seeds anchored to named Table IV periods (T ≈ 14.75 and 17.0), two attempts concurrent with the DNS:

• ≈ 40 s per epoch per attempt, measured over a 180 s window.
• Both probe attempts stall rather than converge: residuals flatten at ‖R‖ ≈ 2.31 and 1.10 with the trust-region radius decaying geometrically.

The stall is not evidence on G1: the probe's seeds come from a T=2000 DNS, which is leg 353's own under-resourced scale, the length whose null §3d already re-read as UNDER-RESOURCED rather than NO. What transfers from the probe is cost, not outcome. A Jacobian action is one integration over the orbit period and costs the same seconds whichever DNS the seed came from.

Envelope arithmetic, with N = max_newton and W workers over 100 attempts:

worst case  =  40 s × N × ceil(100 / W)      against the pre-registered 8 h = 28800 s

so W = 10 admits N ≤ 72 before contention, and less after it. Clause (b) of the cap rule requires max_newton ≥ 2 × the median epochs to convergence-or-stall. If clauses (a) and (b) cannot both be met inside the envelope, the pre-registration is explicit:

U3 does not report a no. It reports the shortfall as a cost and G1 answers UNDER-RESOURCED per §3d, naming the wall time a compliant attempt would need.

7. Realization note (lesson 91)

Carried into G1's answer because a negative names its realization. The stepper is Lie–Trotter split and therefore globally first order, not fourth: RK4 is applied to the nonstiff part only, then composed with the exact viscous factor. Confirmed by two independent measurements (local order ratio 4.00, global ratio 2.00).

This corrects M1's recorded mechanism without changing any M1 number: M1 attributed its ||R(w_lam)|| = 0.0794 floor to RK4 truncation at dt = 0.01, but ||rhs_physical(w_lam)|| = 7.8e-14, RK4 truncation where the RHS vanishes identically is exactly zero. The floor is splitting error. The stepper was not changed to Strang: that would alter the realization M1 was validated against and discard an in-flight DNS, and it is outside U2/U3's scope. It is recorded as the top successor item instead.

Bearing on G1's matching predicate: MATCH_T_TOL = 0.05 is 0.26% on T ≈ 19.334, against a measured 0.13% systematic period offset, covered by a factor of only ~2. Reported as a magnitude, not waved away.

8. What is and is not established

Established (UNVERIFIED, §3f): one false-negative path in newton_hookstep_rpo is closed and M1 is unaffected; max_gmres = 140 is fixed by clause (a) and corroborated by P's floor; P recovers an exact planted solution at those caps; per-epoch cost is ≈ 40 s.

Not established: G1 in either direction. M2. That any Table IV orbit is recoverable. That the solver has no other defects. Nothing here is a null result: a null requires an attempt resourced at the scale the question is posed at, and that attempt has not yet run.

Obligations. CLAY_OBLIGATIONS §6's two no-method obligations stay OPEN; §4 stays OPEN and NOT discharged. Ceiling TIER 2. No L1→L4 link moved. Clay ~0.05%.