A Algorithms implemented in the companion package
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A.1 Residual towers
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The residual module partitions coordinates modulo \(p\), verifies compatibility
and reconstructs the digit \(a_{m-1}\) by
For canonical representatives \(0\le x_m<p^m\), set \(x_0=0\). Compatibility also gives \(a_{m-1}=(x_m-x_{m-1})/p^{m-1}\in\{0,\ldots,p-1\}\). The floor in the first formula is essential. For example, the compatible base-five residues \(4,19\) give the second digit \(3\), not the rational number \(19/5\). The executable companion already uses integer floor division. This edition corrects the displayed algorithm, not the implementation.
A.2 Transport condensation
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The transport module computes strongly connected components by two depth-first passes, forms the condensation graph, finds all source components, and selects one representative from each. A brute-force subset search is included only as an independent finite certificate.
A.3 Companion forests
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For an integral polynomial \(f\), the finite root forest is obtained by exact enumeration of
and by connecting a child \(x_{m+1}\) to its reduction modulo \(p^m\). This is intended for small exact examples, not large deformation rings.
A.4 Cohomology packets
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Modular Gaussian elimination computes the ranks of the differentials. For
the finite dimension is evaluated as
Packet subcomplexes are computed separately and checked against the total.
A.5 Finite occupancy
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For each commuting semisimple operator, a Lagrange polynomial constructs the exact spectral idempotent. Their product gives the joint packet projector. Symbolic rational arithmetic verifies idempotence and the equality of trace and rank.