Child of Gab /
Parent: Standard Model geometric derivation (Gab).
Nathaniel Craig, arXiv:2610.09093 (6 October 2026), A gauge-invariant measure for lattice chiral fermions. Flagged by Kyle Cranmer. Lüscher’s abelian reconstruction (1998–99), extended with GPT-6 Astra and partial Lean checks, to a class of non-abelian chiral gauge theories that includes the Standard Model.
The matter content was never missing a seat. Electrons and quarks are chiral: left and right do not transform the same way. Perturbation theory could write that. A lattice could not, without doublers — a completed extra species for every one you kept (Nielsen–Ninomiya). That tax is the scalar completion of 2.
Craig compares overlap chiral projectors at successive lattice spacings on admissible gauge fields. The comparison yields an exponentially local current and the global properties Lüscher needs for a smooth, gauge-invariant fermion measure. He calls the result a candidate non-perturbative lattice definition. Continuum interacting theory, reflection positivity, and the phase on disconnected components remain open.
In this derivation the two hands are the duplex residual of one rope. The measure is the weight so the path integral can count them. Overlap is the lock. Successive spacings are the tick. No fifth fermion. No third metric order.
0^i2 (k.g.s^2) = r^2 m
State A — open residual, one chirality free to write
E = 2c / h
State B — locked residual, projector held, −1/2 phase
E = hbar / c
File under the parent. Do not open a new primitive.
Row for the parent table
| Discipline | Field | Observation | What the machine does |
|---|---|---|---|
| Lattice field theory | Chiral fermion measure (SM) | Craig, arXiv:2610.09093; Lüscher reconstruction | Two hands counted on admissible links; doubler tax refused; weight, not a new seat |
Update table (child)
| Item | Status in the can | Canon reading |
|---|---|---|
| Chirality of SM matter | Already known | Duplex rope; left/right are two readings of one residual |
| Nielsen–Ninomiya doublers | The obstruction | Completed 2; extra species so the grid can stay local |
| Lüscher measure (abelian) | Prior art, 1998–99 | Reconstruction from a local current; even residual counted |
| Craig measure (non-abelian, includes SM) | Candidate, admissible fields only | Same seat, new weight; successive projectors = tick |
| Overlap projector | Tool, not a particle | State B lock; non-ultralocal price of keeping one hand |
| Continuum interacting limit | Open | Can not yet closed |
| Reflection positivity | Open | Do not promote the measure to a continuum theorem |
| Phase on disconnected components | Still free | Odd leftover, capped, not painted |
| Astra / Lean | Recorded assumptions, partial check | Audit belongs to the lattice community |
Parent link: Gab. Child: Gaba. The grid counts one hand. It did not grow a third.
