Logbook — 2026-04-13 — S1 plots, with one new physics observation

Context. Plots for the S1 + R1 dataset of 2026-04-13-S1-and-R1.md, generated by ../numerics/plot_S1.py. One new observation surfaces from looking at the maps that was not visible from the carrier-only analysis: the φ_α = 0 slice is Doppler-blind for the contrast magnitude, with consequences for S2 priority.

1. Plot inventory

All written to ../plots/, 140 dpi PNG.

1.1 S1_carrier_summary.png — headline

Three-panel summary at δ₀ = 0:

Left: |C| vs |α| for full (η = 0.397) and R1 (η = 0.04). Both flat lines. Full at 0.917, R1 at 0.999. Visually demonstrates α-independence; the gap between the two is the η-driven spin–motion entanglement deficit (≈ 0.083), constant in α.
Centre: arg C vs |α|. Full engine wraps multiple times; R1 follows the linear guide +90° + 4.6° · α (overlay). The position- phase channel (dossier §1.3) is visible and is in the linear regime for R1, in the η-dressed nonlinear regime for the full engine.
Right: σ_z vs |α|. Flat at +0.128 for both engines. Confirms α-independence is not an artefact of how |C| is computed.

This is the figure to put in front of a reader who has 30 seconds.

1.2 S1_contrast_maps.png — Doppler signature… or its absence

Six-panel contrast figure:

Top row: heatmaps of |C|_full, |C|_R1, and Δ_η = full − R1 over (δ₀, |α|).
Bottom row: |C|(δ₀) line cuts at each α, plus a carrier zoom showing the −0.20 MHz/(2π) finite-δt bias.

What is striking. The full and R1 heatmaps each show a vertical band structure — every horizontal slice (i.e. every |α|) is identical. The line cuts confirm: the four α curves are pixel-overlapping. The Δ_η heatmap is also α-independent, alternating sign as a function of δ₀ but flat in α.

This is the new observation. See §2.

1.3 S1_phase_maps.png — position channel

Four-panel phase figure with phase masking at |C| < 0.1:

Top row: arg C heatmaps for full and R1, HSV cyclic colormap. Full shows rapid horizontal striping (fast wrapping of the position phase across δ₀ at every α) plus a slow vertical drift (the α dependence at fixed δ). R1 shows the same horizontal structure but without the vertical wrapping — the linear phase channel is intact.
Bottom row: scatter line cuts across the carrier zoom (|δ₀| < 2 MHz), masked. Full shows the four α traces fanning out into separate phase branches; R1 shows them all on the same near-linear ramp.

The phase masking is essential — without it, the Doppler tails dominate the visualisation with phase noise.

1.4 S1_eta_residuals.png — Δ_η = full − R1

Four-panel residual figure:

Top row: Δ_η|C| heatmap (alternating-sign vertical bands, α-independent) and Δ_η arg C heatmap (masked; shows triangular wedge pattern: |α| amplifies the phase residual roughly linearly for moderate |δ₀|).
Bottom-left: Δ_η|C|(δ₀) line cuts at each α — four traces overlap.
Bottom-right: carrier slice (δ₀ = 0). Δ_η|C| line is essentially flat at −0.083 (the carrier deficit); axis labelling is awkward (1e-14 − 8.26…e-2 units) because the residual is constant to ten decimals and matplotlib auto-zooms. Δ_η arg C(δ₀ = 0) climbs roughly linearly with α from 0° to ~150° before wrapping.

1.5 R1_convergence.png — Guardian flag 1

Two-panel convergence cross-check:

Left: |C|(δ₀) at η = 0.04 (solid) and η = 0.02 (dashed), for α = 0 and α = 3. Curves overlay completely.
Right: Difference |C|(η = 0.04) − |C|(η = 0.02) vs δ₀. Sub-10⁻³ oscillation, matches the predicted (η_a² − η_b²)/2 = 6 × 10⁻⁴ (dotted line). R1 is converged.

The S1 contrast maps show |C|(δ₀) is identical to four decimals across α ∈ {0, 1, 3, 5}. This is much stronger than "α-independent at δ₀ = 0"; the entire detuning lineshape is α-independent.

Mechanism. At φ_α = 0, the coherent state is prepared on the +X̂ axis: ⟨X̂⟩ = 2|α|, ⟨P̂⟩ = 0 (per the §2.2 convention of v0.3). The Doppler shift seen by the analysis pulse scales with ⟨P̂⟩ via δ_D = k_eff · v ∝ ⟨P̂⟩, which is identically zero at this preparation phase regardless of |α|. The classical motion has zero velocity at the turning point, and the stroboscopic pulses sample at exactly the turning point each cycle.

The position-phase channel (arg C) sees ⟨X̂⟩ ∝ |α|, which is why arg C does depend on α. So at φ_α = 0:

Position channel (arg C): bright. Full engine non-monotonic from η-dressing; R1 linear at +4.6° per unit α as predicted by k_eff x_0 per unit α at η_R1.
Velocity channel (|C|(δ₀) Doppler broadening): dark. The Doppler shift is zero by construction.

Consequence for the WP. The plan to "characterise the (δ₀, |α|) surface" implicitly assumed both channels would be active. They are not: at φ_α = 0 only the position channel is alive. To observe the Doppler-broadening signature predicted by dossier §1.4, S2 ((δ₀, φ_α) at fixed |α|) is required, not optional. Specifically:

At φ_α = π/2, ⟨P̂⟩ = 2|α| (maximum), and we expect maximum Doppler broadening of |C|(δ₀) scaling with α.
At intermediate φ_α, the broadening should scale as |α| · |sin φ_α|.

This recovers a check on dossier §1.4 that S1 alone cannot provide. S2 is therefore the load-bearing slice for the velocity-channel motivation, not just an extra dimension. Promote S2 priority.

Self-criticism. This was foreseeable from §2.2 of v0.3 (the convention defining φ_α = 0 → +X̂ axis), but I did not anticipate it when planning the slice ordering. The S1-first sequence still has value — it cleanly isolates the position channel without Doppler contamination, which is itself a useful slice — but the implicit expectation that S1 alone would address the velocity-channel motivation was wrong. Recorded honestly so the v0.4 README amendment can fold this in.

3. Δ_η phase residual structure (heatmap reading)

The Δ_η arg C heatmap (figure 1.4, top right) is the most physically informative single panel in the set. Reading it:

At δ₀ = 0, Δ_η arg C grows roughly linearly with α (~+50° per unit α) before wrapping. This is the η-driven nonlinear extension of the linear position-phase signal.
At |δ₀| ≳ 2 MHz/(2π), the residual flattens — both full and R1 are in the Doppler tails where |C| is small and arg C is poorly defined.
The wedge shape (broader at large α, narrow at small α) directly visualises the regime where η-nonlinearity dominates. A practical use case: this map will be the WP-E target for the WP-D perturbative linear regime — wherever |Δ_η arg C| < 5° (say), the η = 0.397 engine is approximately linear and BAE-compatible analysis is a candidate.

4. Plot conventions to carry forward

Codifying for S2 / S3 plots:

|C| heatmap: viridis 0–1; arg C heatmap: HSV cyclic, masked at |C| < 0.1 (PHASE_MASK_THRESHOLD in plot_S1.py).
Δ residuals: RdBu_r diverging colormap, centred at 0 via TwoSlopeNorm.
Detuning axis: MHz/(2π) (absolute units) on plots, det_rel internally.
Phase: principal branch (−180°, +180°], wrap_deg() helper.
α tick labels: integer-formatted, equispaced (heatmap rows are not on a uniform |α| grid, so y-extent is [-0.5, n_alpha - 0.5] with custom yticklabels; the visualisation is correct but spacing is symbolic, not metric — caveat for any quantitative read-off of slope along the |α| axis).

The α-axis caveat will matter when S2/S3 are plotted: their |α| grid is similarly sparse and non-uniform.

5. Outstanding actions (updated)

Reordered by priority after this entry:

[ ] S2 driver + execution: (δ₀, φ_α) at |α| ∈ {1, 3, 5}. Promoted to high priority — only S2 can show Doppler broadening (§2 above). Single cheap S2 sheet at |α| = 3 first to validate the φ_α-dependent broadening hypothesis before running all three.
[ ] R2 (instantaneous-pulse reference) implementation at driver level.
[ ] R12 follows from R1 + R2.
[ ] Plot conventions extended to S2/S3 once data lands.
[ ] v0.4 README amendments (preflight + S1 findings + Doppler-blind observation) — after S2.

6. Files added in this entry

../numerics/plot_S1.py — plotting driver.
../plots/S1_carrier_summary.png.
../plots/S1_contrast_maps.png.
../plots/S1_phase_maps.png.
../plots/S1_eta_residuals.png.
../plots/R1_convergence.png.

No edits to README.md. No engine modifications. Dossier untouched.

Next entry: S2 driver + first sheet (|α| = 3), to test the Doppler-broadening hypothesis from §2 of this entry.