PairingForward

The mode-pairing observables at the limits Zeng writes closed forms for

6/6 passed in 0ms

Test	What it does	Result
test_click_linear	pairing_click is linear in the single-arm transmittance, hence in the square root of the end-to-end one.	✅ pass
test_error_dark_only	pairing_sift gives a Z-basis error of exactly zero without dark counts, at every intensity and transmittance.	✅ pass
test_error_grows	With dark counts the Z-basis error rate rises with distance, under 1e-4 at 200 km and past a percent by 600 km.	✅ pass
test_sift_eighth	pairing_sift returns 0.125 to 5e-4 at 300, 400 and 500 km.	✅ pass
test_sift_from_bases	pairing_bases gives each party a Z share of 0.5, half of whose square is the 0.125 sifting rate.	✅ pass
test_single_fraction	pairing_single tends to exp(-2*mu) as the arm weakens, and is a fraction of sifted Z-pairs rather than a gain: a further Poisson prefactor charges the source twice.	✅ pass