The sublattice parity of a two-component resolvent chain is independent of dimension, with the mass as the weight of a reversal (Geometry of the vacuum, floor −1)

The higher-dimensional path picture stops, and this note is about what does not stop with it. On the bipartite root lattice A_d (cell Gram -1/d, coordination z = d+1) with the two-component nearest-neighbour operator H = [[0,f],[f-bar,0]] and a diagonal insertion m0*sigma_z, the resolvent resums exactly as G0 (I - H G0)^-1 with dispersion E^2 - m0^2 - |f|^2; that identity is machine-checked in Lean over a general commutative ring, with no division and no field or characteristic assumption. The series then carries a binary alternation: the off-diagonal block is non-zero exactly at an odd number of hops and the diagonal block exactly at an even number, and the pattern is independent of d -- the coordination enters only through the scalar f(k), whose value moves with d while the block structure of every term does not. This is not a tautology of writing a 2x2: switching on an intra-sublattice term destroys the parity, and that negative world is exhibited. The mass enters as the weight of a segment and, in the velocity basis, as the weight of a reversal, measured as a change of basis with a sigma_x-type mass as the control in which the roles do not swap. The zero set f = 0 has dimension d-2 generically; that was first printed by other authors (2024) and our independent derivation agrees with it, so it is carried as context and not as a claim. What generalizes to higher dimensions is the chain, not the counting of paths: the structural obstruction to the latter is known, is stated in the text rather than in a footnote, and is not disputed. Statements at d = 2, 3 are exact and d >= 4 is extrapolation where so marked. No physical reading of any object is made or implied.