Midy's Theorem for Quadratic Pisot Units: An Elementary Conjugate Proof

When do the two halves of a repeating expansion add to β raised to the half-period, minus one? This working paper treats every quadratic Pisot unit β > 1 satisfying β² = mβ + 1, with integer m ≥ 1. A simple enclosure for a remainder and its conjugate turns arithmetic congruences into exact dynamical returns. The paper proves an exact rational-return formula and gives a self-contained proof of the uniform Midy criterion: a reduced denominator q > 2 qualifies exactly when a power of the companion matrix is −I modulo q. The complete denominator test excludes multiples of four and requires the matrix orders at the odd prime divisors to have one common 2-adic valuation, at least two. Odd-prime exponents do not affect the test. The least digit period equals the matrix order; every Midy period is divisible by four, and its half-block sum has finite expansion (m0)ʳ. Prime tests, boundary examples, and explanatory vector figures make the criterion explicit. The treatment addresses the extension proposed by Masáková and Pelantová and explains its implication from earlier results of Qu, Rao, and Yang. Its contribution is an elementary proof and arithmetic synthesis. All proofs are analytical; exact computations are supplementary. Quadratic nonunits and higher-degree Pisot bases remain outside the classification.