pauli_exclusion
plain-language theorem explainer
The declaration proves that any antisymmetric function from a state space to itself vanishes on the diagonal. Quantum field theorists deriving Fermi-Dirac statistics from symmetry would cite it when reducing the exclusion principle to the algebraic identity x = -x. The proof is a short tactic sequence that substitutes the antisymmetry hypothesis at equal arguments, forms the sum 2x = 0 via add_neg_cancel, and applies mul_eq_zero together with two_ne_zero to conclude x = 0.
Claim. Let $S$ be any type and let $ψ : S → S → ℂ$ satisfy $ψ(a,b) = -ψ(b,a)$ for all $a,b ∈ S$. Then $ψ(a,a) = 0$ for every $a ∈ S$.
background
Recognition Science obtains the spin-statistics connection from phase accumulation over the eight-tick cycle. A 2π rotation traverses one full cycle; half-integer spin requires two cycles to recover the identity, producing a minus sign under exchange. The module QFT.SpinStatistics therefore works inside the discrete-time ledger whose atomic unit is the Tick structure. Upstream, the phase definition supplies the explicit values kπ/4 for k in Fin 8, while mul_eq_zero from IntegersFromLogic encodes the ring property that a product vanishes only if a factor vanishes. The IntegrationGap constant A = 1 anchors the counting of active edges per tick.
proof idea
The tactic proof introduces the state space, the wavefunction, the antisymmetry hypothesis and the diagonal argument. It obtains ψ a a = -ψ a a directly from antisymmetry, rewrites the sum ψ a a + ψ a a = 0 by substitution and add_neg_cancel, multiplies by 2 to reach 2ψ a a = 0, and invokes mul_eq_zero together with two_ne_zero to force the diagonal value to zero.
why it matters
The result supplies the algebraic core of the Pauli exclusion principle inside the Recognition Science derivation of spin-statistics from the eight-tick octave. It feeds the parent theorem pauli_exclusion in Foundation.SpinStatistics, which closes the link between phase rotation and fermionic antisymmetry. The declaration therefore realizes the module goal of obtaining Fermi-Dirac statistics from the discrete 8-tick structure without further dynamical input.
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