DspRecovery

Phase 2 gate: pilot phase recovery and the xi it leaves behind

18/18 passed in 5460ms

Test	What it does	Result
test_bracket_placement	A described DAC raises ${\xi }_{phase}$ over the channel-only term by $\mathrm{row}(e^v-1)$, 3.2286e-06 SNU at 8 bits and v_err = 0.05, an Alice-plane row sitting inside budget.phase's $(V_A+\xi )$ bracket.	✅ pass
test_estimator_domain_warns	The default form is silent at PHASE_LIMIT, at v_err = 1.5 and at 12, and budget has no WRAP_LIMIT.	✅ pass
test_estimator_form_gate	The default form tracks xi_hat - xi_ideal to 2% at v_err = 0.054, 0.27, 0.60 and 2.81, where the literature form sits 6%, 24%, 58% and 944% below.	✅ pass
test_estimator_recovers_variance	$-2\ln E[\cos \phi ]$ returns a wrapped normal's variance within five deviations of its sampling law at 4e6 draws from ${\sigma }^2=0.05$ to 12 rad^2, while the wrapped second moment saturates at ${\pi }^2/3=3.2899$: 3% low at 1.5, 22% at 3.13, 48% at 6, 73% at 12.	✅ pass
test_estimator_slope_ratio	The excess over the oracle run matches $(V_A+\xi )(1/{\kappa }^2-1)$ at the run's own slope ratio $\kappa$ to 0.2% over v_err = 0.0025 to 0.60 -- Kish, Quantum 8, 1382 (2024), Eqs. (80)-(82).	✅ pass
test_link_names_the_form	explain() names the ${\xi }_{phase}$ form, estimator on every q.Link run, derived, pinned or dry.	✅ pass
test_link_wiring	The derived path's xi_input is the pinned channel $\xi$ plus budget.phase(v_err) on $(V_A+\xi )$ and nothing else, a trusted $v_{el}$ staying in ${\chi }_{det}$.	✅ pass
test_phase_domain_edge	The literature form, lacking Kish Eq. (82)'s renormalisation, misses the measured excess by the closed ratio $(e^v+e^{v/2})/2$ -- 5.5% at v_err = 0.054, 23% at 0.27, 58% at 0.60, 940% at 2.81 -- and holds to 10% only inside budget.PHASE_LIMIT, a range Marie-Alleaume Eqs. (10)-(11) do not state.	✅ pass
test_phase_domain_warns	Past PHASE_LIMIT the literature form warns naming v_err and still returns the closed form, inside it is quiet, and assemble() inherits the warning.	✅ pass
test_phase_noise_costs_key	The derived path's key rate lands strictly below the same link with v_err pinned at 0.	✅ pass
test_pinned_costs_rate	Charging the noise half against an unattenuated $T$ over-claims the metro rate by 0.34% at v_err = 2e-3, 0.88% at 0.005, 1.89% at 0.01 and 4.98% at 0.02.	✅ pass
test_pinned_short_circuits	A pinned q.PilotPhase(v_err=...) returns the closed-form ${\xi }_{phase}$ for ${10}^8$ symbols in under 0.1 s, simulating none.	✅ pass
test_pinned_transmittance	A pinned v_err charges both halves of Kish App. E Eq. (78): T_claimed and res.budget.inferred equal budget.infer(T, v_err), and $T_{claimed}(V_A+{\xi }_{total})=T(V_A+{\xi }_{channel})$ at every variance.	✅ pass
test_sampled_not_twice	The fitted slope already carries $\kappa$, so the sampled path's T_claimed is the clamped res.est.T with no second budget.infer, within 1% of budget.infer(T, v_err) at 200k symbols.	✅ pass
test_v_err_grows_with_linewidth	v_err rises with linewidth at fixed block size, 100 kHz reaching over 10x the noiseless run.	✅ pass
test_v_wrap_falls_with_pilot	With the lasers noiseless, 12 dB buys ${10}^{1.2}$ in v_wrap once the block phase estimate is linear, 20 to 32 dB, and 5% more across 0 to 12 dB, where it is not.	✅ pass
test_wrapped_is_not_the_budget_input	v_err and v_wrap agree to 0.1% while narrow, and at 10 MHz, 2.81 against 2.34, only v_err reproduces the measured 78.76 SNU, to 1.0%, where v_wrap gives 46.77.	✅ pass
test_xi_from_phase_noise	Measured excess noise matches Marie-Alleaume's $2V_A(1-e^{-v_{err}/2})$, form="literature", to 25% at 1, 10 and 100 kHz, inside PHASE_LIMIT.	✅ pass