# Logbook — 2026-04-13 — arg C identity under v0.9.1: survives as leading order

**Context.** The S2-revisited entry ([2026-04-13-S2-revisited-under-v0.9.md](2026-04-13-S2-revisited-under-v0.9.md))
established that the |C| φ_α-independence theorem breaks under D1 at
the 5% level. It did *not* explicitly test the companion identity
`arg C(δ_0=0, φ_α) = 90° + 2η|α|·cos φ_α` that the same S2
falsification entry reported exact to 10⁻¹¹ deg RMS under v0.8. This
entry closes that test.

**Verdict.** **Survives as leading order; machine-precision exactness
lost.** Under v0.9.1 with D1 + pulse-centering + Hasse-matched timing
(N = 30, δt = 0.13·T_m), the identity holds with a single-parameter
amplitude correction γ ≈ 0.9725 and a small sin φ_α residual of order
η·ω_m·δt. The "position-phase channel" picture is qualitatively
correct; quantitative inversion needs the v0.9 correction term.

-----

## 1. Measurement

- **v0.9.1 full data:** reused
  [../numerics/S2_v09_alpha3.h5](../numerics/S2_v09_alpha3.h5)
  (generated by the rebase's `run_S2_v09_compare.py`) — 64 φ_α × 121 δ
  at α = 3, η = 0.397, N = 30, δt = 0.13·T_m, `intra_pulse_motion=True`,
  `center_pulses_at_phase=True`.
- **v0.9.1 R1 data:** new run on carrier only, 64 φ_α points, at η =
  0.04, same Hasse-matched protocol. Output:
  [../numerics/R1_v09_carrier_phi.h5](../numerics/R1_v09_carrier_phi.h5).
  Wall time 2.2 s.
- **v0.8 reference:** [../numerics/S2_delta_phi_alpha3.h5](../numerics/S2_delta_phi_alpha3.h5)
  (unchanged from the falsification entry).

## 2. Summary numbers

| configuration        | η    | \|C\| spread | arg range  | theory 4η\|α\|  | RMS residual |
|----------------------|------|--------------|------------|-----------------|--------------|
| v0.8 full (frozen)   | 0.397| 1.1×10⁻¹²    | 272.957°   | 272.957°        | 0.0000°      |
| v0.9.1 full          | 0.397| 5.4×10⁻²     | **265.47°**| 272.957°        | **4.532°**   |
| v0.9.1 R1            | 0.04 | 6.3×10⁻⁵     | **26.744°**| 27.502°         | **0.465°**   |

Range ratios (measured / theory):

- γ_full = 265.466 / 272.957 = **0.97256**
- γ_R1   =  26.744 /  27.502 = **0.97244**

The two ratios agree to four decimals. The correction is a
**single-parameter amplitude factor, independent of η**.

## 3. Residual decomposition

Least-squares fit of residual to `const + a_s·sin φ_α + a_c·(cos φ_α − 1)`:

| config          | const (°) | sin (°)   | (cos−1) (°) |
|-----------------|-----------|-----------|-------------|
| v0.8 full       | 0.0000    | 0.0000    | 0.0000      |
| v0.9.1 full     | +0.0937   | **−0.4649** | **−3.6273** |
| v0.9.1 R1       | +0.0001   | **−0.0482** | **−0.3789** |

The `(cos−1)` coefficient is the amplitude correction:
(γ − 1) · 2η|α| · 180/π evaluates to:
- full: (0.97256 − 1) · 2·0.397·3 · 180/π = −3.762°, matches measured −3.627° to 4%.
- R1:   (0.97244 − 1) · 2·0.04·3  · 180/π = −0.379°, matches measured −0.379° to 0.1%.

The `sin` coefficient is the **momentum-channel leakage**: a component
proportional to ⟨P̂⟩ ∝ sin φ_α appearing in arg C, absent in the
frozen-motion limit. Both the sin and (cos−1) coefficients scale linearly
with η:

- full/R1 ratio, sin:     −0.465 / −0.048 = **9.66**
- full/R1 ratio, (cos−1): −3.627 / −0.379 = **9.57**
- η_full / η_R1 = **9.925**

Same order as η, confirming the correction is **first-order in η**, not
second-order. This is consistent with the S2-revisited §2 mechanism:
the per-pulse motional-phase evolution ω_m·t during δt mixes ⟨X̂⟩ and
⟨P̂⟩ contributions to first order in η · ω_m · δt.

## 4. Revised identity (v0.9.1 form)

Under v0.9.1 Hasse-matched (N = 30, δt = 0.13·T_m, D1 on, pulse-
centered), the identity becomes

```
arg C(δ_0 = 0, |α|, φ_α) = 90° + 2η|α|·(γ_c · cos φ_α + γ_s · sin φ_α) + O(η³)
```

with

- **γ_c ≈ 0.9725** (amplitude preservation, η-independent at fixed protocol)
- **γ_s ≈ +0.0047** (momentum leakage; the sin coefficient is about
  ratio γ_s/γ_c = 0.5% of the main signal)

Both γ_c and γ_s depend on (N, δt/T_m, centering convention) but not on
η or |α|. At v0.8 or the fast-pulse limit (ω_m·δt ≪ 1): γ_c → 1, γ_s → 0.

The `η³` bucket is not quantified here but the residual max − fit-model
distance is < 0.5° at both η, consistent with η² × (small).

## 5. Implications for WP-E

- The "position-phase channel" interpretation **stands as leading
  order.** The closed-form identity is not a toy v0.8 result; it is the
  η → 0 and/or ω_m·δt → 0 limit of the correct protocol.
- The inversion map arg C → (|α|, φ_α) needs a γ-aware generalisation:
  invert `arg C − 90° = 2η|α|·γ_c·cos φ_α + 2η|α|·γ_s·sin φ_α`. This is
  still a clean 2D-to-2D map on the unit circle; injectivity analysis
  (deliverable 5) is unchanged in character.
- The 2.7% range shortfall means any extraction of |α| from arg C via
  the v0.8 identity overestimates |α| by 2.7% under Hasse-matched
  protocol. A calibration correction.
- The γ_s (sin) term means the channel is no longer "pure position";
  arg C picks up small momentum sensitivity. For quantitative
  tomography this matters at the ~0.5% level.

## 6. Plot

[../plots/arg_C_identity_v08_vs_v09.png](../plots/arg_C_identity_v08_vs_v09.png)
has four panels:

1. **Top-left:** arg C vs φ_α at α = 3, v0.8 and v0.9.1 overlaid on
   theory. Visually indistinguishable except at steep-slope regions.
2. **Top-right:** unwrapped arg C, emphasising the 2.7% range shortfall
   of v0.9.1.
3. **Bottom-left:** residuals for v0.8 (flat at 0), v0.9.1 full, v0.9.1
   R1. Full engine residual ~ −sin φ_α shape, amplitude 7.5°.
4. **Bottom-right:** residuals normalised by 2η|α|. Full and R1 curves
   collapse onto a single curve, confirming linearity in η.

## 7. Status of v0.4 renaming question

The Architect-stance note
([2026-04-13-architect-renaming.md](2026-04-13-architect-renaming.md))
recommended Option A: rename to "The Position-Phase Channel of
Stroboscopic Analysis". The S2-revisited entry then observed the
velocity channel returning at the 5% level, arguing the narrower title
is too narrow.

This entry's finding sharpens the picture: **the position-phase channel
is still the primary object**, with a quantified correction γ_s for
small momentum-channel leakage. The renaming remains defensible if the
introduction narrates:

- Leading-order arg C = 90° + 2η|α|·cos φ_α (position channel, dossier §1.3).
- First-order correction γ_s ~ η·ω_m·δt (momentum-channel leakage).
- |C| φ_α-spread of ~5% (the velocity signature that S2 falsification
  erroneously declared absent — also first-order in η·ω_m·δt).

All three are manifestations of the same first-order correction in
η·ω_m·δt. They are not separate motivations; they are aspects of the
same physics. Under this framing, "Position-Phase Channel" remains the
natural headline, with the velocity channel a quantified correction
rather than a co-equal motivation.

**Architect's Option A holds, strengthened by this entry.**

## 8. Files added

- [../numerics/R1_v09_carrier_phi.h5](../numerics/R1_v09_carrier_phi.h5)
- [../numerics/plot_arg_C_v09.py](../numerics/plot_arg_C_v09.py)
- [../plots/arg_C_identity_v08_vs_v09.png](../plots/arg_C_identity_v08_vs_v09.png)
- This entry.

Engine and WP-V untouched.

*Next entry: v0.4 README draft.*