Can Miller–Rabin Hear the Shape of a Carmichael Number? Isospectrality, Inversion, and the Intrinsic Information Limit of the Strong Test

The Miller–Rabin strong primality test, run at random bases against a Carmichael number n, emits a pattern of square-root-of-unity events whose statistics we characterize completely. We show that this pattern is a complete invariant of the 2-adic spectrum T(n) = {v₂(pᵢ − 1) : pᵢ | n} and of nothing finer. The distribution of the reduced ladder signature over uniform bases determines T(n) exactly — including the number of prime factors — via an explicit convex inversion of its cumulative law (Theorem A); and two Carmichael numbers with the same spectrum are statistically indistinguishable to the test, their observation laws coinciding at every query budget (Theorem B). Isospectrality is typical: among the 255 Carmichael numbers below 10⁸, the 57 distinct spectra leave 229 numbers in non-singleton classes, the largest holding 41 mutually indistinguishable composites. Completeness comes at an exact price: recovering the spectrum behind a depth-D boundary costs Θ̃(2ᴰ) queries, with matching upper (Theorem C) and lower (Theorem D) bounds; the one exception is a shallow factor, whose presence leaks through the alignment bit at a cost set by the leaked alignment mass rather than by depth. We give the exact law of the factor bipartitions induced by observed roots — the splitting spectrum — and the expected number of bases to shatter n completely (Theorem E), and we extend the whole theory beyond the Carmichael regime: over the Fermat-liar subgroup of any squarefree odd composite, the instrument measures the truncated spectrum {min(v₂(pᵢ − 1), v₂(n − 1))} (Theorem F), turning a scope limitation of the companion paper into a positive result. The intrinsic leakage of the strong test — present in every correct implementation, independent of any side channel — is thus exactly the 2-adic spectrum, delivered at an exponential price in depth. All closed forms are verified in exact rational arithmetic against exhaustive enumeration, including the induced factor split on every one of 37,768 misaligned units across six Carmichael numbers.