CrossModulation

The discrete-modulation engine reduced to the Gaussian key-rate layer

5/5 passed in 69ms

Test	What it does	Result
test_alice_marginal	M-PSK of amplitude $\alpha$ has $V_A=2{\alpha }^2$ for every M, so the discrete penalty lives in z alone.	✅ pass
test_dm_receiver	dm_rate on the true channel with a real receiver equals dm_rate on the merged channel with an ideal one, pinning heterodyne's mu = 2.	✅ pass
test_gaussian_limit	dm_holevo at $z=\sqrt{T{V}_A(V_A+2)}$ with an unbounded alphabet equals cv_rate at ideal heterodyne to 1e-10 over 336 points, $V_A$ 1e-3 to 1e3, $T$ 1e-4 to 0.99, $\xi$ 0 to 1.	✅ pass
test_transparent_seam	At $T=1$, where both eigenvalues sit within sqrt(eps) of vacuum, dm_holevo and cv_rate agree to 1e-12 on I_AB and 2e-6 on chi_BE, twice an offline 1.2e5-point worst of 9.6e-7, cv_rate reading Eve low.	✅ pass
test_untrusted_detector	With $\eta$ and $v_{el}$ folded into the channel by hand, dm_holevo reproduces cv_rate's untrusted mode to 1e-13.	✅ pass