TITLE:
Carrier-Resolved Burnside Data for CSS Lattice Codes: Incidence Complexes, Ground-Space Reduction, and X-Cube Sewing
AUTHORS:
Thomas Verdier
KEYWORDS:
CSS Stabilizer Codes, Effective Burnside Category, X-Cube Model, Fracton Order, Foliated Equivalence
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.9,
September
14,
2026
ABSTRACT: We attach carrier data to CSS stabilizer calculations by using the effective Burnside category of a finite support meet-semilattice. A cellular CSS Hamiltonian determines a literal support lattice and a Pauli coefficient; chosen generator-relation presentations determine carrier-labelled Burnside chain complexes. Their binary evaluations retain point, line, plane, tube, and leaf labels while reproducing the usual plaquette, Bacon-Shor, and periodic X-cube ranks. The microscopic coefficient detects a pinned local qubit. Ground-space compression removes this ancilla dependence and produces a monotone diagram of operator systems whose support changes by bounded thickening under finite-depth local circuits. For the X-cube code, the affine-leaf syndrome complexes recover the eight quotient sectors. We lift coordinate leaf insertion to an isomorphism of global CSS complexes and give its block action on directional logical sectors: the two toric-code classes enter the two transverse summands. This yields a
ℤ/2
sewing class for direction-framed presentations. Its scalar shadow
log
2
GSDmod2
is invariant when the free layers are two-dimensional topological Pauli-stabilizer codes, and a semion layer shows why the statement does not extend to unrestricted foliated resources. Thus the paper separates presentation-level carrier data from the two restricted equivalence invariants proved here.