Mass as pairing: an explicit Majorana mass on half of a generation does not seesaw in symmetric mass generation

Abstract

A fermion mass pairs a field with a partner of conjugate charge: elementary (Higgs-Dirac), self-conjugate (seesaw) or a three-fermion composite (symmetric mass generation, SMG). We give half of a Spin(10)-like generation an explicit Majorana mass in a four-dimensional SMG model and ask what the other half does in a phase with no condensate. With the source on, every mass-type term of the unsourced half averages to zero under the symmetry group of the sourced action, so a light mass term, and a seesaw's light-heavy mixing, requires a spontaneously broken symmetry; exact on the lattice and a general proposition. The source is colour-democratic (a Delta_R-type mass on every flavour's right-handed taste doublet); the single-nu_R split lies outside the positivity class used but inside the symmetry statement. Measured up to 8^4: (i) in the symmetric-gap phase, with the source screened to at most 1.1% of free, the unsourced half's single-fermion readout stays at 0.3% of free while its pair susceptibility rises from 0.2% to 1.2-1.7% at h=2; on like mode sets it falls with the volume, on (4^4,8^4) no inequivalent light mass channel grows faster than free, and symmetry breaking is not resolved at L<=8; (ii) at the critical point the explicit mass collapses the critical sigma channel and moves the light half toward free as a saturating crossover, not a power law; direction tests (MIXED by their pre-registered letter) are consistent with a critical coupling rising with the mass; y_c(h) is not located. Pole versus zero of the light propagator at p->0 is not measured. The outcomes are readings after pre-registered tests that mostly returned no verdict. Also: no first-order signature of the merged transition at L<=8, a Z_4-odd response handed to a lattice composite, and flavour democracy of the positivity class. Every claim has an evidence class and a runnable check.