The Mirror Decides the Gamma Factor: A Parity Dictionary from the 16-Cell Modulo 30 to Dirichlet L-Functions

The eight residue classes coprime to 30 sit on the vertices of a 16-cell, and the four diagonals of the 16-cell join r to 30−r. We show that this single geometric mirror, r ↦ −r, controls every structural feature of the Dirichlet L-functions modulo 30 that depends on the sign χ(−1). A character that is constant on each diagonal is even. It is then an eigenvector of the 16-cell with eigenvalue −2 (or 6 for the trivial character), a factor of the Dedekind zeta function of the real field Q(cos(π/15)), and its completed L-function carries the gamma factor π−s/2Γ(s/2). For χ ≠ 1 its trivial zeros lie at 0, −2, −4, … and its non-zero values at negative integers at −1, −3, −5, …. A character that changes sign along each diagonal is odd, has eigenvalue 0, belongs to the imaginary part of Q(ζ30), carries π−(s+1)/2Γ((s+1)/2), and has its non-zero left values at 0, −2, −4, …. We collect these facts as one theorem (the parity dictionary), explain why the factor Γ((s+1)/2) is forced by a cancellation that is visible on the 16-cell, show that the choice of gamma factor is invisible high in the critical strip but decisive on the real axis, prove that the lifts of the 16-cell to the primorial levels 210, 2310, … read exactly the even left values, with the Euler factors of the new primes removed, while the individual odd values need a half system, and follow the dictionary to the class number one of Q(ζ15)−, to the golden ratio on the two tetrahedra of the cube, and to the Coxeter element of E8, whose eigenvalues are the eight vertices of the 16-cell. Every individual statement is classical; what is new is the single geometric reading and its systematic use. All numerical statements are checked by preregistered computations at 50 and 100 digits.