The Lattice-Fermion Doubling Gate
Date: 17 October 2026 Branches: B03, B05, B11 and B13 Claim status: exact naive zero count and Wilson lifting check; no chiral Standard-Model lattice constructed
1. Naive lattice Dirac symbol
For a symmetric lattice derivative,
the momentum symbol is
The massless Dirac operator vanishes whenever
for every spatial or Euclidean momentum component.
2. Doubling count
In \(d\) lattice dimensions there are
corners with zero naive Dirac symbol.
The intended low-momentum fermion at
is accompanied by doublers at Brillouin-zone corners.
For four Euclidean dimensions, the naive operator has sixteen zeros.
3. Wilson lifting
The Wilson term contributes
It vanishes at \(k=0\), while a corner with \(n\) components equal to \(\pi/a\) receives
Thus the doublers become heavy and only the physical corner remains massless in the free check.
The price is explicit breaking of the naive chiral symmetry at finite spacing.
4. Why this matters here
The Furey and Standard-Model branches require chiral matter. A discrete substrate cannot be assessed using scalar dispersion alone.
It must specify one of the known types of resolution or a genuinely new one:
- Wilson fermions with controlled chiral restoration;
- staggered fermions;
- domain-wall fermions;
- overlap/Ginsparg–Wilson structure;
- a quantum cellular automaton whose net chirality and anomaly flow are explicit.
Merely placing Weyl labels on cells does not evade doubling.
5. Anomaly gate
Even if unwanted poles are lifted, the lattice theory must reproduce:
- perturbative gauge-anomaly cancellation;
- global anomalies;
- baryon/lepton anomaly structure;
- the correct number of chiral generations;
- locality and unitarity.
The previously established anomaly-free finite TROs are representation data. They do not by themselves prove that a local lattice regulator realizes those representations without mirrors.
6. Relation to \(Spin(2,3)\)
\(Sp(4,\mathbb R)\cong Spin_0(2,3)\) organizes the proposed response carrier. It does not alter the Brillouin-zone topology of a chosen microscopic lattice derivative.
Any claim that the Jordan structure removes doublers must exhibit the modified lattice Dirac operator and count its zeros and index explicitly.
7. Stop-everything criteria
Advance: construct a local gauge-covariant lattice Dirac operator for the selected matter TRO and verify its index, anomaly and continuum spectrum.
Mirror failure: the regulator necessarily restores the mirror copy excluded by the chiral selector.
Nonlocal escape: doublers disappear only through nonlocality that spoils the proposed causal cellular interpretation.
Anomaly failure: the continuum limit does not reproduce the established one-generation anomaly cancellation.
8. Certificate
The fermion-doubling certificate counts all \(2^d\) naive zeros for \(d=1,2,3,4\) and verifies that the Wilson mass lifts every nonzero Brillouin corner in the free model.