Finite generation of permutation twisted cohomology

We settle Miller's Noetherianity conjecture for the full canonical effective permutation twisted cohomology ring of every finite p-group in defining characteristic. More generally, for every finite group and every finite family of actual finite-dimensional real representations, we prove that the effective section ring over any commutative Noetherian base is finite as a module over an explicit finitely generated central commutative algebra. The finite-module theorem also holds after subgroup restriction, on coordinate faces, and with bounded permutation coefficient complexes; over fields it extends to bounded complexes of finite-dimensional modules. Over a field of positive characteristic, the section ring of the doubled regular representation gives a Noetherian bigraded model whose fully homogeneous spectrum contains the Balmer spectrum of permutation complexes as the complement of a single closed point. Suitable homogeneous localizations, followed by passage to twist degree zero, give affine Dirac charts. This provides an explicit Noetherian model for the geometric conclusion announced by Gómez and Miller. For unit coefficients over nonzero bases, we also determine the minimum number of fully homogeneous coefficient generators of nonzero twist needed for module-finiteness. The proof combines geometric forerunners with a signed cone and induction on subgroups and coordinate faces; a uniform integral torsion bound and universal coefficients give the extension to arbitrary Noetherian bases. Version 2 revises the abstract and introduction, adds an introductory cyclic example, reorganizes the coefficient-bound discussion, updates references, and includes an AI declaration. The mathematical results and proofs are unchanged from version 1. 47 pages. The original version remains available at doi:10.5281/zenodo.23697586. File: manuscript.pdf; 606,692 bytes.SHA-256: 21f2e427c9ae38320d6f4c0c4d385dfe58c601931f8fa5dc022583e6a8346caf.