Put gauge transport on oriented lattice links

Use a four-dimensional Euclidean hypercubic lattice with spacing a, coordinates x=an, and natural units ℏ=c=1. At each positively labeled link assign Uμ(x) in SU(N). Adopt the transformation Uμ(x)→Ω(x)Uμ(x)Ω†(x+aμhat), so Uμ multiplying a neighboring matter field transports its components back to the local frame at x. Reverse orientation is represented by the Hermitian conjugate at the corresponding endpoint. Choosing the opposite convention is possible, but every product must then be changed consistently.

Near a smooth continuum configuration write Uμ≈exp[iag₀Aμ], with Hermitian Aμ=AμaTa and generators normalized by tr(TaTb)=δab/2 in the fundamental representation. The link is dimensionless, while A has inverse-length units and g₀ is dimensionless in four dimensions. This exponential is a smooth-field expansion, not a globally unique logarithm of every sampled link. Large lattice fluctuations remain valid group configurations even when no small-link expansion describes them well.

Multiply four links around an elementary square, reversing the two return links. Internal endpoint transformations cancel pairwise, leaving conjugation by Ω at the chosen base site. Taking the trace eliminates that final conjugation. A traced open path does not generally have the same invariance, because its endpoint transformations are unrelated. Trace cyclicity also makes the chosen starting corner immaterial, while reversing the loop complex conjugates its trace.

The Wilson gauge action sums one contribution for each unoriented elementary square, labeled by μ<ν. Include the real trace and divide by N, so the identity plaquette contributes zero after subtracting from one. This is a dimensionless Euclidean action entering exp(−SW). It is distinct from a real-time phase weight, and its parameter β is a bare gauge coupling convention rather than an inverse thermodynamic temperature.

For smooth fields the plaquette is the identity plus i g₀ a²Fμν and higher powers of spacing. Expanding its real trace yields the continuum Yang–Mills action with the declared generator normalization. Irrelevant discretization terms vanish in a suitable continuum limit, but taking that limit also requires tuning bare parameters while physical scales are held fixed. A finite lattice is a regulator, not by itself a proof that a desired four-dimensional continuum quantum theory exists.

The modular lab illustrates closed index bookkeeping only. Reset and set modulus to 4 and additive step to 1, then Step four times; predict residue 0 and orbit period 4. Reset and set modulus to 4 and step to 2, then Step twice; predict residue 0 and period 2. Returning an index to its starting value is a simple loop analogy. The instrument contains no group matrices, gauge transformation, Wilson action, or gauge-field simulation.