Find the anomaly in the fermion measure

Expand a Euclidean fermion in orthonormal modes as $\psi=\sum_n a_n\phi_n$, with independent Grassmann coefficients. A linear change of coefficients $a'=Ca$ transforms their integration measure by $(\det C)^{-1}$, rather than by the ordinary bosonic determinant. This inverse is already visible for one variable because its defining integral must remain normalized. The conjugate fermion has an independent set of coefficients in the Euclidean integral, so its axial transformation contributes a second determinant.

For a local axial rotation, the matrix C contains overlaps of the modes with the rotation parameter and chirality operator. Taking its logarithm produces a formal trace of $\alpha(x)\gamma_5$. That trace involves infinitely many modes and is undefined without a prescription. Declaring it zero by pairing symbols before regulating is not a derivation of symmetry. The basis and regulator are chosen using the gauge-covariant Dirac operator so that the vector gauge symmetry remains the exact symmetry of the quantum definition.

Let H be a self-adjoint Euclidean Dirac operator obtained by multiplying the anti-Hermitian massless operator by i. On a closed compact manifold with smooth background, the heat kernel of $H^2$ gives a well-defined regulated trace. Insert $e^{-H^2/\Lambda^2}$ and take the large-cutoff limit only after evaluating the local trace. The regulator is gauge covariant, so its background dependence respects the vector gauge symmetry. That choice reproduces the same symmetry tradeoff as the vector-preserving triangle calculation.

At short distance, the heat-kernel expansion organizes terms by powers of the field strength and cutoff. The chirality trace kills many terms, while the term with enough gamma matrices remains finite after momentum integration. In four dimensions its gauge part is proportional to the topological density quadratic in curvature. The resulting axial divergence is twice the regulated index density. Curved backgrounds add gravitational terms; the flat electromagnetic result is not a complete formula for an arbitrary curved spin manifold.

The nonzero spectrum of $H^2$ has paired opposite-chirality subspaces because H maps one to the other. Their contributions to the integrated regulated chirality trace cancel, leaving the difference of chiral zero-mode counts. This does not imply that the local regulated density vanishes point by point. Local heat-kernel information and an integrated spectral identity answer different questions. A finite matrix example can illustrate inverse Jacobians and cancellation, but cannot establish the continuum local anomaly coefficient by itself.

The answer also depends on what symmetries the regulator is asked to preserve. A gauge-noninvariant subtraction can move a term into a different current definition without removing the obstruction to preserving all required symmetries. Boundaries introduce additional spectral data and boundary conditions; the closed-manifold identity cannot simply be reused after discarding surface terms. Finally, zero modes can make an unsourced massless determinant vanish. The Jacobian then still controls Ward identities and zero-mode-saturating correlation functions, rather than being interpreted as a measurable phase multiplying a nonzero vacuum amplitude in every background.