Reciprocity by Design - The shape of the loss in space-time modulated media

A modulated medium whose permittivity depends on both position and time, and whose conductivity depends on position, may be invariant under an operation that reverses time and translates space at once. Two such space-time glide operations exist in one spatial dimension, and the literature gives both the same name; this paper treats the one that reverses time, and separates it from its sister at the outset. We show that invariance under a whole one-parameter family of these operations preserves reciprocity. Closure of the operation forces the spatial offset to half a unit cell and leaves the time origin free, so the family is a line rather than a point. Two conditions already in the literature — glide time-reversal on one side, symmetry of the modulation under plain time reversal on the other — are its two endpoints, and the interior is reachable from neither: a modulation is exhibited that satisfies the family condition exactly while failing the published time-reversal criterion for every choice of time origin, and it is reciprocal to the precision of the computation. The proof splits into three independent relations. The first uses the symmetry and, in a lossy medium, reverses the sign of the conductivity. The second uses only that the material functions are real, and reverses that sign back. The third uses nothing at all and holds for every medium. The split is what makes the lossy case provable rather than merely observed, and it yields a criterion on the shape of the loss rather than on its size: a static, spatially uniform conductivity is harmless at any strength, while a conductivity with the wrong spatial parity breaks the conclusion however small it is. The dual case of a static dielectric and a dynamic loss profile is also proven to preserve reciprocity. Modulating the loss together with the permittivity, the typical case in practice, breaks reciprocity even when both profiles carry the symmetry. Finally, the violation is quantified. Where the symmetry is broken by a controlled amount, the asymmetry of the band structure is linear in that amount over a range located from the data, which turns a statement about what cannot be done into a tolerance budget for what can. Numerical verification accompanies every claim; each measuring test is paired with an input on which the claim must fail, and the scripts are deterministic and deposited with this record. The condition is sufficient, not necessary, and the article does not claim otherwise.