Part IX-A · The Alphabet of Orbifold Boundary Conditions: a boundary condition is a representation, and the alphabet is forced by the fixed-point counts (Part IX-A)

Each count of orbifold boundary conditions has so far been a separate piece of work, because the boundary conditions were a list of matrices to be found per orbifold. They are not a list.

An orbifold boundary condition is an action on the gauge fields; in the inner, zero-flux sector it admits a linear lift, and after the lift it is a representation of the space group Λ ⋊ Z_m. Its letters are the irreducibles induced from characters of Λ whose stabiliser fixes only finitely many characters — the rest are the Wilson lines; the weight of a letter is its dimension; and how many letters of each weight there are is a Möbius count over |det(I − A^k)|, the number of fixed points of the k-th power of the rotation.

From that one identification follow the twelve closed-form counts published for this programme, from a single argument with no free parameter, with the degree of each read off the cone signature; the class count of S¹/Z₂ × S¹/Z₂ — the orbifold of Parts VI to VIII — which is 2809 at rank 4, with the gauge transformation that closes it written down; and the symplectic column, which the programme had not reached because it is not another orbifold but another type of defining space: alphabets, counts, closed forms and all twenty-one labels built as matrices.

Theorem 1 states the alphabet for Z^r ⋊_A Z_m in any rank, identifies the hypothesis as the condition that every eigenvalue of A have order exactly m, and gives the fixed-point counts in closed arithmetic form — so the alphabet depends on the order of the rotation and the rank of the lattice and on nothing else.

What is not claimed. The completeness of the local datum is proved here in the generality the counts need and no priority is claimed for it. The trace conservation laws, the local gauge groups, the classification of Z_N embeddings and the identification of the surviving class with the ’t Hooft flux are inherited. And the whole of the arithmetic section except one line is inherited: Latimer–MacDuffee (1933), Diederichsen–Reiner, Charlap (1965), Montgomery–Uchida (1971) and Hiller (1985). A Ledger table says which is which, line by line.

Paper in English and Spanish; every ancillary script and its archived receipt travel with the record, and the script list is read off the paper itself.