Remark. The counting array of a precision operator [fdip-002O]
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  • October 25, 2026

Let \(Q:X_V\to X_V\) be a positive precision parameter. Choose an indicator enumeration and its counting chart \(C:X_V\to \mathbb R^n\). Write \(A=CQC^{-1}\) for the resulting array. Then \[h_V(x,Qy)=(Cx)^TA(Cy).\] Thus \(Q\) is self-adjoint exactly when \(A^T=A\), and is strictly positive exactly when \(z^TAz>0\) for every nonzero \(z\). The action is \(S_Q(x)=\tfrac 12(Cx)^TA(Cx)\).

Proof. 

  • October 25, 2026

The chart sends the indicator basis to the standard orthonormal basis, so the counting pairing becomes the ordinary dot product. Substitution yields the displayed identity. Testing it on every pair of basis vectors proves the symmetry equivalence; bijectivity of \(C\) proves the positivity equivalence. Using this same basis for source and target gives \(\det A=\det Q\); a simultaneous re-enumeration conjugates \(A\) by a permutation matrix and does not change the determinant.

In the ensuing Gaussian formulas, the customary matrix symbol \(Q\) denotes this counting array of the endomorphism. An arbitrary nonorthonormal chart need not make its array symmetric: its Gram matrix must also enter the pairing. This calculation is a coordinate recovery of Definition [fdip-0004], not a new normalization result. The half-action convention agrees with [wingate2025lattice, Section 4, Equation (4.5), p. 3].