Finite-group comparison with precision loss (Block 17).
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Let \(R\) be a commutative domain, \(\Gamma\) a finite group, and \(\rho,\tau\) actual integral matrix representations. Suppose \(a=|\Gamma|\ne0\) in \(R\) and \(a\mid\pi^e\). An equivariant matrix modulo \((\pi)^{e+k}\) lifts to an integral equivariant matrix with the prescribed reduction modulo \((\pi)^k\). Indeed, lift its entries to \(A\) and form
The finite equations imply \(S-aA=\pi^{e+k}E\) with \(E\) integral. Choose \(b\) with \(\pi^e=ab\) and put \(P=A+b\pi^kE\). Then \(aP=S\), so cancellation of \(a\ne0\) proves equivariance. Over a local ring, if \(\pi\) is a nonunit, \(k>0\) and the supplied comparison is invertible, \(P\) is an integral unit. Neither completeness nor a finite residue field is needed for this step.
For a DVR, \(e=v_\pi(|\Gamma|)\) is derived from the actual group order. The single precision \(e+1\), independent of rank, detects integral isomorphism. It makes reduction injective on integral isomorphism classes, and finite coefficient quotients give finitely many integral classes in fixed rank. This is a formalization of classical finite-group rigidity (Maranda, 1953, Theorem 2), with an explicit proof of the comparison's precision loss.
For complete local coefficients with finite quotients, downward-compatible allowed representations occupied at every precision, and the required finite-group type at depth \(e+1\), reconstruction followed by this comparison produces the required integral representation. Continuity is proved for a finite discrete source and for pullback through a fixed continuous map to that source. Occupancy at only one precision is insufficient. The result lifts comparisons between actual integral representations; it does not lift arbitrary quotient representations or supply the missing arithmetic packet interpretation.
For \(C_2\) over \(\mathbb Z_2\), take
The matrix \(A\) intertwines modulo \(4\) and its average is \(P=\left(\begin{smallmatrix}1&-1\\0&1\end{smallmatrix}\right)\). No exact intertwiner can retain \(A\) modulo \(4\), although \(P\) retains it modulo \(2\). The actual trivial and sign representations also agree modulo \(2\) but are not integrally isomorphic. These checked examples show why residual-only detection and full preservation of the input precision cannot be asserted here.