# Logbook — 2026-04-13 — S2 revisited under engine v0.9.0 **Context.** Sequel to [2026-04-13-S2-falsification.md](2026-04-13-S2-falsification.md). That entry proved the matrix-element-magnitude theorem `|⟨α|C|α⟩| = exp(−η²/2)` and concluded that |C| is φ_α-independent for coherent states under the stroboscopic protocol. Falsification was at machine precision (worst Δ|C| = 6.6 × 10⁻¹³). WP-V's engine audit ([../../wp-hasse-reproduction/logbook/2026-04-13-engine-audit.md](../../wp-hasse-reproduction/logbook/2026-04-13-engine-audit.md)) identified that the v0.8 engine *drops* `ω_m a†a` from the per-pulse Hamiltonian (audit item D1). Engine v0.9.0 added the `intra_pulse_motion=True` toggle that restores it. Question: does the matrix-element-magnitude theorem survive D1? **Answer: no — it survives only at the level of the v0.8 approximation.** ----- ## 1. Quick numerical re-check at α = 3 Driver: ad-hoc φ_α scan at three engine configurations, α = 3, η = 0.397, ω_m = 1.3, n_thermal = 0. | Configuration | |C| min | |C| max | spread | |-------------------------------------------------|-----------|-----------|--------------| | v0.8-equivalent (intra_pulse_motion=False) | 0.916566 | 0.916566 | **3 × 10⁻¹⁵** | | v0.9 D1 on, default WP-E params (N=22, δt auto) | 0.915912 | 0.927576 | **1.17 × 10⁻²** | | v0.9 D1 on, Hasse-matched (N=30, δt=0.13·T_m) | 0.894060 | 0.948126 | **5.41 × 10⁻²** | The first row reproduces the published S2 falsification number: the theorem holds at machine precision under v0.8. The second and third rows show the theorem broken at the **1 % and 5 % levels** when intra-pulse motion is included. The 5 % case is the more physically faithful one (it matches Hasse's Appendix D protocol regime). Motivation 3 of the original WP-E framing (Doppler-broadening signature in |C|) is therefore **not falsified at the protocol level** — it was falsified only against the v0.8 stroboscopic-frame approximation, where ω_m·a†a was dropped from the per-pulse Hamiltonian. ----- ## 2. Why the theorem fails under D1 The displacement-operator identity `D(α)†(a + a†) D(α) = (a + a†) + 2 Re(α)` is unchanged. The proof of the theorem in the S2 falsification entry §3 used: ``` ⟨α|exp(iη(a + a†))|α⟩ = exp(−η²/2) · exp(i · 2η|α| cos φ_α) ``` i.e., the matrix element of a single, instantaneous coupling operator C between displaced ground states. This is exact. The v0.8 engine evolves each pulse by exp(−i δt · H_eng) with H_eng = (Ω/2)(C σ_- + C† σ_+) (no ω_m·a†a). For coherent input |α⟩, the leading-order spin response per pulse is set by ⟨α|C|α⟩ — which has the φ_α-independent magnitude. Higher-order corrections involve matrix elements between |α⟩ and shifted states, but these contribute only at O((ηΩδt)²) above the leading O(ηΩδt) term, and the per-pulse budget here is (Ω_eff·δt) ≈ π/(2N) ≈ 0.071 rad — small. Hence the theorem is *protected by the smallness of per-pulse rotation* in the v0.8 frame. Under v0.9 D1, the per-pulse evolution becomes H_v0.9 = ω_m·a†a + (Ω/2)(C σ_- + C† σ_+). Now during δt the motional state rotates by ω_m·δt (~0.34 rad in the default regime, ~0.82 in the Hasse-matched regime). The coupling operator effectively averages over this rotation: ``` C̄(δt) = (1/δt) ∫₀^δt exp[iη(a e^{−iω_m t} + a† e^{+iω_m t})] dt ``` `|⟨α|C̄(δt)|α⟩|` is **no longer φ_α-independent**: the phase under the integral mixes ⟨X̂⟩ = 2|α| cos φ_α and ⟨P̂⟩ = 2|α| sin φ_α contributions in a φ_α-asymmetric way. This is exactly the regime where Hasse's velocity-encoded contrast comes from. In our numerics: the v0.9 spread is about (η · |α| · ω_m · δt) ≈ 0.397·3·1.3·0.628 ≈ 0.97 rad → contrast variation of order 1 − cos(0.97)/cos(0) ≈ 47 %; the actual measured spread of 5.4 % is reduced by the Bessel-like averaging over δt, but not to zero. Order of magnitude consistent. ----- ## 3. Implications for WP-E The S2 falsification entry §4 reduced WP-E to "1.5 motivations" by eliminating motivation 3 (Doppler signature). Under v0.9 D1, that elimination must be **partially reversed**: | Motivation | v0.8 status | v0.9 D1 status | |-------------------------|-------------------|---------------------------------------| | 1: arg C as position | ✓ confirmed | ✓ confirmed (theorem on phase intact) | | 2: φ_α as lock probe | ½ alive | ½ alive — same role | | 3: |C|(δ₀) Doppler sig | ✗ falsified (10⁻¹³)| **¼ — small (1–5 %) but real signal** | Motivation 3 is now: "|C|(δ₀) carries a small (1–5 %) φ_α-dependent signature whose magnitude is set by η·|α|·ω_m·δt; that signature is the Doppler velocity channel as predicted by the dossier §1.4 analytical sketch, but suppressed by stroboscopic time-averaging relative to a non-stroboscopic continuous Rabi reading." So the dossier's velocity-channel intuition was *qualitatively right*. The v0.8 falsification was a quantitative artefact of the missing ω_m·a†a term — exactly the audit item D1 that WP-V identified independently. ----- ## 4. What this changes operationally ### Existing S1, S2 (|α|=1,3,5), R2, H1 datasets Were generated under v0.8 (intra_pulse_motion=False, default). **They remain valid** — they correctly characterise the engine's v0.8-mode behaviour, which is what the published WP-E plots and deliverables describe. They do **not** characterise the corrected Hasse-protocol behaviour. ### WP-E v0.4 README Must address this gap directly. Two options: (a) **Re-run S1, S2 sheets under v0.9 D1** and replace the published plots. Larger compute (~factor 2 over v0.8 since the per-pulse propagator is now 2nmax × 2nmax with non-zero diagonal, but expm scaling unchanged); preserves WP-E's data products as canonical. (b) **Keep v0.8 plots, add a v0.9 comparison panel.** Cheaper. A single sheet at α = 3 under v0.9 D1 + Hasse-matched is enough to document the 5 % φ_α-dependence and bound it; the rest of the WP-E claims about arg C remain valid as published. Option (b) is the lighter path; option (a) is the more rigorous one. **Recommend (b)** because: - The S2 falsification's primary finding (theorem at machine precision) is a clean v0.8 result that should not be overwritten — the v0.8 → v0.9 contrast is the *interesting* finding. - WP-V already demonstrated the difference at the panel level (Fig 6/8). Re-deriving it inside WP-E adds little. - Compute budget: option (a) is roughly 30 min wall; (b) is ~2 min. ### Deliverable 5 (Jacobian + cond-J) The reframed simplification in the S2 falsification §8 ("J = diag(0, J_arg) on (|α|, φ_α) for |C|") needed the theorem. Under v0.9 the J_|C| row is no longer identically zero — it has small but non-zero entries ∂|C|/∂|α| and ∂|C|/∂φ_α. Conditioning analysis must include these. Adds one row to the cond(J) heatmap; doesn't change the deliverable scope. ### S3 (|α|, φ_α at fixed δ₀) Was deferred indefinitely as redundant after S2. Under v0.9 it is **no longer redundant**: the (|α|, φ_α) → |C| map has 5 %-level structure that S3 is the cleanest probe of. Recommend **un-deferring S3** for execution under v0.9 D1. ----- ## 5. Recommendations 1. Add a §3.1bis paragraph to WP-E v0.4 README naming the v0.8 → v0.9 motivation reversal and citing this entry. 2. Run S2 |α|=3 under v0.9 D1 + Hasse-matched as a single comparison sheet (option (b) above), tagged S2_v09. 3. Un-defer S3 in v0.4 §8 outstanding actions; queue it for execution. 4. Update Deliverable 5 to read "J on (|α|, φ_α) under both engine modes" so the v0.8/v0.9 contrast is preserved. 5. Footnote in the short note (Deliverable 4) acknowledging that WP-V's audit findings retroactively change WP-E's motivation 3. None of this invalidates published WP-E results. It adds context. ----- ## 6. Update (engine v0.9.1 — pulse-centering convention) Subsequent exchange raised the question: is pulse 1 timed to the motional position extremum when φ_α = 0? Under v0.9.0, pulse 1 *starts* at φ_α = 0 but spans into negative phase territory during δt (ω_m·δt ≈ 47° at Hasse-matched parameters). The physically natural convention is to have pulse 1 *centered* on φ_α = 0 — a symmetric reading of "pulse timed at position max". Engine v0.9.1 adds `center_pulses_at_phase` (opt-in, default False). When True, the engine shifts the prepared motional phase by `+ω_m·δt/2` internally so that every stroboscopically-synchronous pulse (including pulse 1) is centered on the user-specified `alpha_phase_deg`. Backward compatibility preserved at defaults. ### S2 v0.9 comparison rerun under centered convention Driver: [run_S2_v09_compare.py](../numerics/run_S2_v09_compare.py) (updated to set `center_pulses_at_phase=True`). Output: same H5 path (overwritten) + updated plot [S2_alpha3_v09_compare.png](../plots/S2_alpha3_v09_compare.png). | Quantity | v0.8 | v0.9.0 D1 Hasse-matched | v0.9.1 +centered | |----------------------------------------|---------|-------------------------|-------------------| | worst \|Δ\|C\|(δ, φ)\| vs φ=0 | 6.6×10⁻¹³ | 6.93×10⁻² | **7.01×10⁻²** | | carrier \|C\|(δ=0, φ_α) spread | — | 5.40×10⁻² | **5.43×10⁻²** | **Verdict.** The 5 % |C| spread is **real physics**, not a timing artefact. Pulse-centering shifts *where* in φ_α the min/max of |C| occur, but not the spread magnitude — the |C|(φ_α) envelope is a closed loop over [0, 2π), and both conventions traverse the same loop from different starting angles. This is the cleaner version of the S2 v0.9 revisited finding: WP-E's motivation 3 (Doppler signature in |C|) is partially restored at ~5 % under the corrected protocol, invariant under the pulse-centering convention. The canonical number for WP-E v0.4 is **worst |Δ|C|| = 7.0 × 10⁻²** — eleven orders of magnitude above the v0.8 falsification. ----- ## 7. Status Engine: v0.9.1 (adds `center_pulses_at_phase`; v0.9.0 behaviour default). Files added/updated: this entry + the comparison driver + HDF5 + plot. WP-E published deliverables: unchanged; reinterpreted. *Next entry: WP-E v0.4 README, incorporating S2 falsification, v0.9 re-evaluation, S3 un-deferral, and WP-V cross-references.*