Logbook — 2026-04-13 — S1 slice + R1 reference + R1 convergence check

Context. Follow-up to 2026-04-13-preflight-results.md. Guardian cleared preflight; this entry reports the first WP-E data: S1 slice ((δ₀, |α|) at φ_α = 0), the R1 reference (η = 0.04) on the same grid, and a one-off R1 convergence cross-check (η = 0.04 vs η = 0.02) per Guardian flag 1.

Outcome. Driver works; all four α values converged at the planned nmax. Wall time well under the preflight estimate. R1 is well-linearised (η = 0.04 → η = 0.02 differs in |C| by < 10⁻³, two orders of magnitude below the η = 0.397 → η = 0.04 deviation of ~0.1). Three physics findings worth flagging immediately for the v0.4 README; one analysis caveat. No blockers.

1. Driver and execution

• Driver: ../numerics/run_slices.py. Imports stroboscopic_sweep as a library, no engine modification.
• Output files (HDF5):
• ../numerics/S1_delta_alpha.h5 — full + R1, 4 α × 121 δ.
• ../numerics/R1_convergence.h5 — η = 0.04 vs η = 0.02, 2 α × 41 δ.
• Grid: det_rel ∈ [−4.615, +4.615] (= ±6 MHz/(2π)), 121 points, step 0.077 ≈ 0.10 MHz/(2π) — within v0.3 §8 Q4 spec.
• α ∈ {0, 1, 3, 5}; nmax ∈ {30, 30, 40, 80} per preflight §6.

Wall time. S1 full = 2.7 s; S1 R1 = 2.5 s; R1conv = 0.6 s. Total ≈ 6 s — far below the preflight estimate of "a few minutes". The α = 5 nmax = 80 cost (1.55 s for 121 points) was the dominant single run and remains modest.

Convergence. max_fock_leakage at every (α, η) ≤ 6.6 × 10⁻¹², with |α| = 5 the worst case. nmax = 80 is comfortably sufficient at |α| = 5; the §6 plan for nmax-scaling at higher |α| is sound but conservative.

2. Physics observations from S1

2.1 Contrast at the carrier (δ = 0) is α-independent

|α| | |C|_full | |C|_R1 | σ_z | |----|----------|--------|----------| | 0 | 0.9166 | 0.9992 | +0.1284 | | 1 | 0.9166 | 0.9992 | +0.1284 | | 3 | 0.9166 | 0.9992 | +0.1284 | | 5 | 0.9166 | 0.9992 | +0.1284 |

The full-engine carrier contrast is identical to seven decimals across α ∈ {0, 1, 3, 5}. The contrast deficit |C|_full − |C|_R1 ≈ 0.083 is purely an η-driven spin–motion entanglement effect at the carrier (⟨α|C|α⟩ structure), not Doppler broadening. Doppler broadening appears only off-resonance.

This is a direct test of dossier §1.4 motivation 3: at the carrier itself, the JSON-uniform value |C| ≈ 0.917 is reproduced exactly and is α-independent. The provenance question "why does HDF5 report 0.61/0.71/0.84/0.75 vs. JSON's flat 0.56?" is now decisively answered on the candidate-engine side: the candidate engine has no α-dependence at δ = 0. The HDF5 series, having been classified as synthetic in preflight, requires no further reconciliation here.

2.2 The position-phase channel is real and α-dependent

arg C at δ = 0:

|α| | arg C (full) | arg C (R1, η = 0.04) | full − R1 | |----|---------------|----------------------|-------------| | 0 | +90.00° | +90.00° | 0° | | 1 | +135.49° | +94.58° | +40.91° | | 3 | −133.52° | +103.75° | −237.27° (mod 360 = +122.73°) | | 5 | −42.54° | +112.92° | −155.46° (mod 360 = +204.54°) |

R1 phase shift is linear in α, ≈ +4.6° per unit α — consistent with a linear k_eff · ⟨x⟩ position phase, scaled by η_R1 / η_full ≈ 0.10 vs the full engine. Full engine shows much larger and non-monotonic phase (wraps multiple times by α = 5), as expected from the exp(iηx̂) operator once η · ⟨x⟩ is no longer small.

This is the first direct observation of dossier §1.3's arg C channel in this WP. The R1 confirms the linear-LD prediction; the full engine shows the η-driven nonlinear dressing. The channel exists, is measurable, and behaves qualitatively as the principle predicts.

2.3 |C| peaks at a small negative detuning, α-independent

For all α, max|C| = 0.9242 occurs at det_rel = −0.154 (≈ −0.20 MHz/(2π)), not at δ = 0. The peak value 0.9242 = exp(−η²/2) — the bare Debye–Waller factor. The small negative bias is a finite-δt artefact of the carrier rotation (the script computes dt to give π/2 at Ω_eff but the spin–motion coupling produces a phase shift that biases the resonance). Cross-checked: peak location is α-independent, confirming finite-time origin (not Doppler).

This is a clean signature of Δt (R1 − R12 in the §3.1 decomposition, not Δη) and gives a quantitative target: the −0.20 MHz/(2π) bias should vanish in R2 (instantaneous-pulse limit) once R2 is implemented.

3. R1 convergence cross-check (Guardian flag 1)

|α| | ⟨|Δ|C||⟩ (η = 0.04 − η = 0.02) | max|Δ|C|| | |----|--------------------------------|-----------| | 0 | 6.2 × 10⁻⁴ | 9.4 × 10⁻⁴ | | 3 | 6.2 × 10⁻⁴ | 9.4 × 10⁻⁴ |

The η = 0.04 reference deviates from η = 0.02 by < 10⁻³ everywhere. The expected scaling, since R1 should be linear and the residual is quadratic, is:

|C|(η = 0.04) − |C|(η = 0.02) ≈ (η = 0.04)² / 2 − (η = 0.02)² / 2
                                = (0.0016 − 0.0004) / 2 = 6 × 10⁻⁴

— exactly what we measure. R1 is therefore well-linearised at η = 0.04, and the residual scale is two orders of magnitude below the η = 0.397 → η = 0.04 deviation (Δη ≈ 0.066 mean |C| difference). Per Guardian's recommendation, this stays a one-off check; we do not run η = 0.02 routinely.

4. Analysis caveat — phase residuals where |C| is small

The naïve mean |Δ arg| over the full det grid is dominated by points in the Doppler tails where |C| is small (< 0.05) and arg C is ill-defined numerically. Reported numbers in the analysis output (≈ 50° to 120° mean residuals) reflect this artefact, not physics.

For v0.4 plots: mask phase residuals at |C| < 0.1 (or weight by |C|) before reporting any aggregate Δ arg statistic. The point-by-point residuals on the heatmap are correct; only the aggregates are misleading. Recorded as a plotting convention, not a bug.

5. Guardian-flag amendments staged for v0.4 README

Per Guardian's "do not revise README yet" cadence, these are staged here for the v0.4 edit:

• §3.1 motivation 3 (provenance mismatch). Update from "characterises a known mismatch" → "the motivation has been resolved at the data- artefact level (HDF5 = synthetic placeholder; see preflight); WP-E now has 2.5 motivations rather than 3, the half being the externalised dossier reconciliation". Wording per Guardian's forward-looking note.
• §4a protocol amendment (lessons-learned). Add: "engines must be independently implementable and produce physical Bloch vectors at the anchor point (|Bloch| ≤ 1); files failing either test are reclassified as non-engine artefacts and documented separately, not subjected to the four-candidate disambiguation." Per Guardian flag 4.
• §4a per-pulse capture amendment. Add: "if final-Bloch comparison resolves the gate, per-pulse capture is retained as a diagnostic tool for the main sweep but not required for clearance." Per Guardian flag 2.
• §3.1 / §4 R2 convention reminder. When R2 is implemented, the kick operator must use the displacement-generator sign convention of §2.2 (code line 138). Per Guardian flag 1 of the closure review.

These are amendments to be folded in at v0.4 along with the S1+R1 data narrative.

6. Outstanding actions

• [ ] Plot S1 |C|, arg C, σ_z heatmaps and Δη = (full − R1) residuals. Mask phase residuals at |C| < 0.1.
• [ ] Decide on S2 ((δ₀, φ_α) at |α| ∈ {1, 3, 5}) execution: the φ_α axis at 64 points is the next data drop.
• [ ] Implement R2 (instantaneous-pulse reference) at driver level, with §2.2 sign-convention cross-check.
• [ ] R12 (η → 0, δt → 0) follows trivially from R1 once R2 is in place.
• [ ] Folded-in v0.4 README per §5 amendments above and S1+R1 findings — only after S2 produces data.

7. Files added in this entry

• ../numerics/run_slices.py — driver script.
• ../numerics/S1_delta_alpha.h5 — S1 full + R1 dataset, 4 α × 121 δ.
• ../numerics/R1_convergence.h5 — η = 0.04 vs η = 0.02 cross-check, 2 α × 41 δ.

No edits to README.md (Guardian cadence). No engine modifications. No edits to dossier (out of scope).

Next entry: S1 plots + S2 driver + S2 execution, in roughly that order.