Two constraints on an emergent gauge sector: a centre condition on the gauge-group extent, and the spectral axis as the renormalization-group scale

Companion note to the Spectral Quantum Gravity manuscript (v436). Two short, self-contained results are recorded. (1) An odd-centre exclusion: if the deconfined Z2 topological order of the framework — fermion parity — is to survive any extension of the emergent gauge group, it must embed centrally and act as (−1)^F; every extension whose centre has odd order is thereby excluded (Georgi–Glashow SU(5), E6, trinification SU(3)^3), while Spin(10) — on whose spinor 16 the unique central order-two element acts exactly as fermion parity — and Pati–Salam remain open. (2) An RG-scale identity: at one loop, the layer density of the parton-induced gauge coefficient along the spectral axis equals the four-dimensional running coupling at the layer’s mass scale; the spectral axis realises the renormalization-group scale geometrically, and the layered kinetic term is ghost-free at quadratic order with no new light states. Neither result solves the long-standing problem of constructing a chiral gauge theory non-perturbatively; both are constraints that sharpen where that problem sits. All claims carry explicit epistemic tags; independent verification is expressly invited.