Spin(10) chiral gauge theory on the lattice: the construction, its single assumption, and a conditional nonperturbative definition

Abstract

This deposit contains one paper and the deterministic scriptsreproducing every computed statement in it.The paper defines a chiral Spin(10) gauge theory on the lattice infull — gauge group, field content, lattice and measure, a singleG-invariant gapping interaction, and the order of limits — and provesan existence statement conditional on one explicitly statedassumption: that the mirror layer is G-symmetrically gappable in thethermodynamic limit at intermediate coupling. Everything else isunconditional.Established without assumption: the anomaly conditions that makesymmetric gapping possible at all (the cubic coefficient vanishesidentically in the 16, Witten’s anomaly is absent, and the Z16Dai-Freed anomaly vanishes for every generation number, so no anomalydistinguishes odd from even); a CRT theorem forcing the mirror sectorand fixing its unique invariant coupling, with the Kramers signTheta^2 = (-1)^F; the closed-form strong-coupling singlet spectrum ofthe gapping interaction, organising the 27 half-filled dimer singletsas 17+9+1, with unique gapped anchors; determinantal positivity ateven flavour number, proved from a single local anti-selfadjointnesscondition and verified in four dimensions at three volumes, where thecondition holds exactly and the hypotheses remain sharp; a positivitytheorem for Kramers antiunitaries together with an exact CRTrealisation, whose hypothesis holds automatically for a vectorlikeinteraction, and a no-go excluding every site-local antiunitary oncethe interaction acts on the mirror alone, as symmetric mass generationrequires; an exactly solved interface theory with the contaminationlaw cont = C t/Delta_c and the falsifiable prediction C(16) =0.478836; evidence that the mirror gaps G-symmetrically throughdimerization, extended beyond one dimension, where the only newfeature is a first-order rearrangement of the dimer pattern along aline t_w = 0.5-0.8 g with the global Casimir vanishing exactly on bothsides, and where the minimum gap rises again from two to threedimensions at fixed volume; and domain-wall results in which theresidual mass falls exponentially in the fifth extent at exactly thetransfer-matrix rate and survives mirror gapping up to a constantfactor, together with a staggered formulation that satisfies thepositivity condition exactly.Granting the assumption, the conditional theorem states what theconstruction delivers at even flavour number: a well-definednon-negative measure, a symmetrically gapped mirror whosecontamination of the light sector is quantitatively bounded, and alight spectrum retaining exactly the intended Weyl content. Each 16then contains one Standard-Model generation together with aright-handed neutrino, with hypercharges quantised by the embedding.The paper is explicit about what is not established: the assumptionitself in four dimensions at intermediate coupling, which is thecentral open problem of the symmetric-mass-generation programmegenerally; odd flavour number, hence three physical generations, forwhich the site-local route is closed by the no-go proved here; thecontinuum limit; the assembly of chiral kinetics with the positivitystructure; and all quantitative phenomenology downstream of thedefinition.Every one of the nineteen formal statements carries a proof or anexplicit citation. Claims are graded [proved], [computed], [imported],[argued] or [open]. Scripts are deterministic (fixed seeds; nostochastic step enters any exact statement) and require only numpy andscipy.