# Logbook — 2026-04-13 — R2 reference reveals a frame-convention choice, not a simple limit

**Context.** Sequel to `2026-04-13-S2-expansion-and-H1.md`. R2 is the
instantaneous-pulse (δt → 0) reference per v0.3 §3.1 / §8 Q3. Implemented
as *impulsive kicks with explicit inter-pulse spin free evolution for
duration T_m at detuning δ* — the Monroe-like picture. Ran it. The
result is physically interesting but **not** the "small Δt correction"
the v0.3 decomposition implicitly assumed.

**Verdict.** R2 reveals a structural difference between two defensible
interpretations of the protocol, not a small finite-δt correction. The
v0.3 §3.1 decomposition Δt = R1 − R12 and cross = full − (R1 + R2 − R12)
needs reinterpretation. WP-E's physics findings are unchanged; the
decomposition framing is what needs an update (staged for v0.4).

-----

## 1. Execution

Driver: `run_R2_single` and `execute_R2` in
[../numerics/run_slices.py](../numerics/run_slices.py). Output:
[../numerics/R2_delta_alpha.h5](../numerics/R2_delta_alpha.h5).
Grid: 121 δ × 4 |α| × 2 engines (R2 at η = 0.397; R12 at η = 0.04).
Wall time: 0.5 s R2 + 0.4 s R12. Cheap.

Plot: [../numerics/plot_R2.py](../numerics/plot_R2.py) →
[../plots/R2_vs_full.png](../plots/R2_vs_full.png).

## 2. The R2 protocol implementation

Per the v0.3 spec, R2 is the δt → 0 limit of the full engine at fixed
pulse area. Two interpretations are defensible; they give different
results.

**Interpretation A (engine-consistent):** shrink δt while preserving the
engine's convention that detuning is a pulse-only term. In the strict
δt → 0 limit at fixed pulse area Ω · δt, the detuning term δ · δt → 0
and detuning drops out. R2 becomes *flat* in δ, |C|(δ) = constant. Not
useful as a reference.

**Interpretation B (Monroe-like):** impulsive kicks with detuning acting
on the spin via free evolution between pulses over duration T_m. Makes
the δt → 0 kick a clean SU(2)⊗motion rotation, with all δ-dependence
in the gap. This is the interpretation implied by dossier §1.1
("pulses much shorter than both motional and spin dynamics — the ion
is frozen during the pulse") and it is the one implemented.

The kick operator is

```
K = exp(−i · (π / (2N exp(−η²/2))) / 2 · (C ⊗ σ₋ + C† ⊗ σ₊))
```

and between pulses the motion rotates by 2π (identity at strobe) and
the spin rotates by exp(−i (δ/2) σ_z T_m).

## 3. Finding — periodic comb at δ = k · ω_m

The R2 and R12 |C|(δ₀) spectra are periodic in δ with period ω_m. At
each integer multiple δ = k · ω_m (for k = … −4, −3, … +4), the
spin free-evolution phase over one gap is exp(−i · k · π), and over
(N − 1) gaps accumulates to a product of phases that collapses back to
the δ = 0 value. Hence |C|(k · ω_m) = |C|(0) for all integer k.

Between teeth, |C| falls off sharply: the HWHM of each tooth is ~0.1
MHz/(2π), matching the resolution 1/((N − 1) · T_m) = 1/(21 · 769 ns)
= 62 kHz · (factor for oscillation of a cos interference) ≈ 0.1 MHz
when evaluated at our 0.1 MHz grid step.

Explicit values at |α| = 0:

| δ/(2π) (MHz) | \|C\|_R2 | \|C\|_R12 |
|:------------:|:--------:|:---------:|
| −5.2 (k = −4)| 0.9166   | 0.9992    |
| −3.9 (k = −3)| 0.9166   | 0.9992    |
| −2.6 (k = −2)| 0.9166   | 0.9992    |
| −1.3 (k = −1)| 0.9166   | 0.9992    |
|  0.0 (k =  0)| 0.9166   | 0.9992    |
| +1.3 (k = +1)| 0.9166   | 0.9992    |
| +2.6 (k = +2)| 0.9166   | 0.9992    |
| +3.9 (k = +3)| 0.9166   | 0.9992    |
| +5.2 (k = +4)| 0.9166   | 0.9992    |
| off-tooth (e.g. 0.1) | 0.229 | 0.232 |

All |α| values show the same comb. The α-independence theorem (matrix-
element-magnitude theorem, logbook `2026-04-13-S2-falsification.md` §3)
applies at each tooth individually, because at δ = k · ω_m the sequence
reduces to an effective pulse train with identity between pulses.

## 4. Why this is not a small Δt correction

The v0.3 §3.1 decomposition presumed

```
Δt = R1 − R12   (pure finite-time effect, η held at 0)
```

with R12 interpretable as "R1 with δt → 0". Under interpretation B
(Monroe-like R2, R12), R12 has the comb; R1 has a smooth single peak of
HWHM 0.9 MHz/(2π). The residual Δt = R1 − R12 is therefore *not small*
— between teeth R1 has smooth finite amplitude and R12 has ~0, and at
teeth R1 and R12 are comparable. The "residual" is dominated by a
structural mismatch, not a correction.

The structural mismatch is: the full engine and R1 omit inter-pulse
spin free evolution entirely; R2 and R12 include it. These are two
different physical models, not two limits of the same model.

**The full engine's convention matches the Hasse2024 experiment.** The
JSON legacy run's central-lobe HWHM of ~0.28 MHz (≈ Ω_eff) and single
peak at δ = 0 are what the engine reproduces and what is measured. If
the experiment were described by interpretation B, one would see a comb
at multiples of ω_m in the Rabi spectrum, which is not reported. The
engine's interpretation is correct for this protocol.

Why: in the Hasse protocol, between analysis pulses the laser is OFF
and the ion is not in any rotating frame — the spin simply sits in its
natural basis with no dynamics driven by the (absent) laser detuning.
Detuning is a *pulse-only* quantity, not a continuous interaction-
picture Hamiltonian. The engine captures this correctly.

## 5. Revised interpretation of the R1/R2/R12 decomposition

Given that R2 (interpretation B) models *different* physics from the
full engine, the Δη / Δt / cross decomposition as originally framed is
not well-posed. A repair:

**Repair 1.** Keep R1 as defined (η → 0, engine convention). Drop R2
from the main decomposition; note the comb result as a theoretical
sideline for the resolved-sideband limit, pointing toward WP-C and to
potential future comparison with alternative protocols.

**Repair 2.** Redefine R2 as "full engine with δt reduced to ε, at
fixed pulse area, otherwise identical". In this case R2 ≡ full in the
engine convention (interpretation A gives flat |C|, trivial). The only
finite-δt effect *within the engine* is the Debye-Waller Ω_eff ≠ Ω
and the small carrier-resonance shift noted in S1 (peak at det_rel ≈
−0.154). Both are already present in R1 at η = 0.04. So Δt within the
engine is negligibly small — the residual is essentially zero.

**Net for v0.4.** The interesting residual is Δη = full − R1 (the η-
dressing of the contrast and phase we've already measured across S1 and
S2). The "Δt" and "cross" pieces of the v0.3 decomposition do no
productive work. v0.4 should drop the three-baseline decomposition in
favour of a single η-decomposition.

## 6. Data values at carrier — closing one prediction

From `2026-04-13-S2-expansion-and-H1.md` §4.3, I predicted: "R2 should
put the |C|(δ₀) peak at δ_0 = 0 exactly (no −0.20 MHz/(2π) bias)". This
is now answered — but not quite as predicted:

- R2 has the |C| peak at δ₀ = 0 (central-lobe restriction): yes.
- But R2 also has peaks of equal height at δ₀ = k · ω_m for all integer
  k. The "peak is at 0" claim is technically correct but incomplete.

The bias in the full engine's peak at −0.20 MHz/(2π) is within the
0.1 MHz/(2π) grid step (the S1 grid has steps of 0.1 MHz/(2π); the
closest grid point to the true peak is −0.154 in det_rel units, which
rounds to −0.20 MHz). It may not even be real — plausibly the true peak
is at δ = 0 to machine precision and we are seeing grid aliasing. A
finer grid around δ = 0 would settle this.

Action (not gating): run a carrier-zoom S1 at high resolution (say, step
0.01 MHz/(2π)) in |δ| < 0.5 MHz to localise the full engine peak.
Deferred — minor.

## 7. Revised v0.4 amendments (additional staging)

In addition to prior-entry stagings:

- **§3.1 Δη / Δt / cross decomposition.** Drop. Replace with a single
  η-decomposition: Δη = full − R1, which is the only interpretive
  residual that does work in this protocol. Note the R2 comb result as
  a theoretical sideline for the resolved-sideband limit, pointing to
  future WP comparison.
- **§8 Q3 resolution.** "R2 = numerical instantaneous pulse" was an
  underspecified decision. The two interpretations (engine-consistent
  vs Monroe-like) give qualitatively different predictions, and only
  the engine-consistent one is internally consistent with the full
  simulation. v0.4 should state this and commit to R2 = engine-at-
  reduced-δt, if R2 is kept at all.
- **§4 deliverable 2.** Residual plots should be against R1 only, not
  "all three baselines". The cross-term panel of S1_eta_residuals and
  S2_combined already does the right thing.

## 8. Files added in this entry

- [../numerics/R2_delta_alpha.h5](../numerics/R2_delta_alpha.h5)
- [../numerics/plot_R2.py](../numerics/plot_R2.py)
- [../plots/R2_vs_full.png](../plots/R2_vs_full.png)
- This entry.

Engine
[../../scripts/stroboscopic_sweep.py](../../scripts/stroboscopic_sweep.py)
unchanged. README.md unchanged (Guardian cadence).

## 9. Outstanding actions for v0.4

All substantive WP-E physics is now delivered. The remaining work is
editorial:

- [ ] Architect consultation on WP-E renaming (Guardian closing note
      of the falsification review). "The Position-Phase Channel of
      Stroboscopic Analysis" is the natural new title if we accept
      that the forward map has cleanly factorised.
- [ ] Draft v0.4 README folding in: full history from v0.3 →
      preflight → S1 → S1-plots → S2-falsification → S2-expansion +
      H1 → R2-comb. Credit attribution for the 3 → 1 motivation
      reduction (½ off via preflight, 1 off via S2-falsification, ½
      off via R2's reinterpretation).
- [ ] Final figure set: probably `S2_combined.png` is the single
      headline plot. `S1_carrier_summary.png`, `H1_lock_tolerance.png`,
      and `R2_vs_full.png` are the supporting cast. S1 phase/residuals
      maps are diagnostic.

*Guardian cadence: still no README edit. v0.4 is the next (and
probably only remaining) major deliverable.*