Fix ordering before integrating fermionic variables

For generators θi, exchange of two distinct generators changes the sign, and each generator squares to zero. Every function of finitely many such generators is a finite polynomial. For one generator define the integral of one to be zero and the integral of θ to be one. This rule behaves like differentiation and extracts the linear coefficient; it is not integration over real values of θ. A positive-density interpretation would therefore be inappropriate from the beginning.

For several generators, declare the orientation by stating which ordered top monomial integrates to plus one. Swapping two integration differentials changes that orientation’s sign. Swapping two variables inside the integrand also changes its sign. These two operations can compensate when performed consistently, but changing only one silently changes an answer. Left and right derivatives likewise differ on odd quantities; a product rule must retain the parity of the factor moved past the derivative.

Treat ψ and its barred partner as independent Grassmann generators rather than ordinary complex conjugate integration coordinates. For N pairs define the oriented measure by assigning integral plus one to ψ₁barψ₁ through ψNbarψN in that order. With exponent minus barψ Aψ and ℏ=1, its finite polynomial expansion has a top-degree coefficient equal to det A. One complex pair makes the sign transparent: minus barψ aψ equals plus aψbarψ.

For real Majorana-like generators with an antisymmetric quadratic matrix, the Gaussian instead gives a Pfaffian, with an orientation-dependent sign and its square equal to the determinant. Thus one should not replace every fermionic integral by the same determinant power. In a field regulator the determinant may be negative or complex, and a zero eigenvalue can make the partition function vanish unless insertions saturate the missing generators. None of this defines an ordinary positive bosonic sampling weight automatically.

For an invertible fermion matrix, normalized two-generator insertions produce inverse-matrix entries with signs determined by their order. In the one-pair example below, ψbarψ has normalized expectation plus one over a, while barψψ has minus one over a. A reference using another orientation or source-derivative order can display different intermediate signs while giving the same consistently defined physical correlator. Always compare the entire convention rather than a detached determinant formula.

The modular lab illustrates only permutation parity. Reset and set modulus to 2 and additive step to 1, then Step once; predict residue 1 and period 2. Reset and set modulus to 2 and step to 2, then Step once; predict residue 0 and period 1. A single swap reverses parity while two swaps preserve it. The instrument computes additive residue orbits, not Grassmann multiplication, Berezin integrals, fermion spectra, or a physical exchange trajectory.