The Gauge-Theoretic Origin of Fermi Statistics in Discrete Lattice Models

A strictly local one-dimensional lattice-register analysis shows how fermionic exchange statistics arise as a gauge constraint on discrete encodings of indistinguishable occupancy information, rather than as an independent postulate. The mechanism starts from a hard-core on-site Hilbert space and imposes operational locality axioms on the physically measurable even observable algebra—on-site number locality, strictly nearest-neighbor hopping locality, and compositional commutativity on disjoint regions—then demands that single-excitation (odd) representatives realize fermionic exchange through the canonical anticommutation structure. A sharp local-commutant lemma identifies on-site parity as the unique intermediate-site “sign resource” compatible with number-commuting locality while still flipping odd operators, forcing any consistent encoding to carry a parity-string tail. This yields a necessity-and-uniqueness-up-to-gauge result: the only way to implement fermions on a tensor-product 1D register without injecting nonlocal artifacts into physical observables is to pay the Jordan–Wigner positional-bookkeeping cost, with the apparent ordering dependence promoted to a gauge choice acting by conjugation on odd representatives while leaving the even observable algebra invariant. The paper further links exclusion to an information-theoretic capacity constraint induced by hard-core truncation and highlights an open-system consequence: under local noise and loss channels, occupancy-limited (hard-core/fermionic) dynamics restrict accessible local error weight and alter entropy-production pathways relative to unconstrained bosonic loss, making the encoding constraint operationally testable in discrete lattice settings.