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) 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 (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. Wall time 2.2 s.
• v0.8 reference: ../numerics/S2_delta_phi_alpha3.h5 (unchanged from the falsification entry).

2. Summary numbers

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):

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 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) 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/plot_arg_C_v09.py
• ../plots/arg_C_identity_v08_vs_v09.png
• This entry.

Engine and WP-V untouched.

Next entry: v0.4 README draft.