Reduction of the lattice chiral gauge measure problem to symmetric mass generation for the 16 of Spin(10): a conditional closure with machine-verified hypotheses

This deposit contains an eight-page note, with complete reproduction scripts, establishing a conditional closure of the lattice chiral gauge problem for the matter content of the Spectral Quantum Gravity (SQG) framework: n_g copies of the 16 of Spin(10). Main result: a reduction theorem with a four-step proof. Given (i) perturbative anomaly freedom, (ii) global anomaly freedom, and (iii) G-symmetric gappability of the mirror sector (symmetric mass generation, SMG), the substrate-computed fermion determinant defines a gauge-invariant, local measure for the chiral wall theory; the entire lattice measure problem for this content reduces to assumption (iii). Verified hypotheses and supporting computations: (1) The cubic anomaly symbol of the 16 vanishes exactly over all 45^3 generator triples (machine verification; SU(3) fundamental as nonzero control). (2) 2D U(1) overlap measure phase: quadratic anomaly scaling; the anomaly-free multiplets 3450 and 1^8-2 cancel; the residual is the q^4 irrelevant operator. (3) 4D U(1) overlap measure phase: cubic anomaly scaling (flat to 0.3% over q=1..6 at L=5); the multiplet 3^3+4^3+5^3=6^3 cancels; the residual is identified as the q^5 irrelevant operator to 0.2% and falls by a factor 18 from L=4 to L=5. (4) On-site no-go: the Fock space of a single 16 contains no Spin(10)-invariant state at any partial filling; exact sector formula min C2 = (3/4) N (16-N). (5) Exact multiplicity-free decomposition of all exterior powers of the 16 (Freudenthal recursion in exact arithmetic); operator identity sum_a (J^a)^2 = (3/4) N (16-N) on-site; the quartic Casimir is likewise scalar on Lambda^4 (Casimir rigidity to degree 4). (6) The half-filled two-site cluster contains exactly 27 invariant singlets; the invariant exchange operator splits the two singlets of the (4,12) sector by 51.9 with a gap of 16 to the non-singlet spectrum, yielding an explicit fully invariant interaction with a unique gapped singlet ground state — the strong-coupling anchor of assumption (iii). (7) Theorem 2 (strong-coupling gap stability): the dimerised SMG Hamiltonian retains a unique, gapped, Spin(10)-invariant ground state in an open region |t| < t0*g, via the Yarotsky and Nachtergaele-Sims-Young stability frameworks; no symmetry breaking occurs anywhere in the region. Honest scope: assumption (iii) itself beyond this strong-coupling region — the intermediate-coupling regime and the continuum limit — remains open; the note states throughout what is established and what is not. An open question is posed to the community: does the invariant cluster interaction admit a sign-free representation suitable for determinantal quantum Monte Carlo? All numbers are reproducible from the nine included Python scripts (numpy/scipy; deterministic linear algebra and exact integer/fraction arithmetic; no stochastic estimates). Companion note to: Spectral Quantum Gravity (v437), Zenodo, [DOI v437]. Version 1.1.0 (26 July 2026): adds the positivity/ extension — exact singlet spectrum via a hopping su(2), a sign-problem-freeness theorem for the cluster interaction on bipartite lattices (even n_g), and a validated BSS-QMC demonstration with measured tables (dimer and 4-site chain)