Entanglement is lost fastest to imperfect records: an exact solution for a repeatedly recorded qubit

A qubit maximally entangled with an inaccessible reference is recorded, once per round, by a fresh environment qubit whose two conditional states overlap with amplitude c, and is then rotated by an angle theta about an axis not commuting with the recorded observable. We solve the coherent information I_k = I(R > S_k) exactly for every c, theta and number of rounds k: the Choi spectrum of the k-round channel is Bell-diagonal with eigenvalues (1 +- c^k +- s_1 +- s_2)/4, an even number of minus signs, where s_1 s_2 = c^k and s_1^2 + s_2^2 is a Chebyshev polynomial of the second kind in (1+c) cos(theta) / (2 sqrt c). The consequence we put forward is that a perfect record is not the fastest way to take a qubit's entanglement away. The asymptotic loss rate is maximal at an interior record strength c* = tan^2(pi/4 - theta/2), where it equals artanh(sin theta), the inverse Gudermannian of the rotation angle, against -ln cos(theta) for a perfect record: a factor 2/theta in rate and 0.36/theta in half-life, a factor 7 in rounds at theta = 0.05. The mechanism is critical damping. The damping regimes of this model were classified by Lema et al. (arXiv:2512.14583) in the classical information of the record; what is added here is the reading in coherent information, whose sign decides who can carry the entanglement, with the critical line and both rates in closed form at every angle. Two further exact statements. At theta = pi/2 consecutive records measure complementary observables and the balance is exactly additive, I(R:E) = g(S_Z) + g(S_Y) with g(s) = h_2((1 + e^{-s})/2), so an environment of witnesses becomes an heir once it has spent half a nat of record strength, S = 0.497, split evenly between the two observables. Weak records give a curve that depends on the accumulated strength alone, and the whole family collapses onto three scaling functions with time scales 1/theta^2, 1/theta and 1/s.