Abstract
Let \(\bar\rho\) be a residual Galois representation and let \(R\) be a global deformation ring with declared local conditions. We separate the global automorphy problem into residual occupancy, deformation support, Hecke occupancy, and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \(R\cong{\mathbb T}\) transports support, Fitting ideals, and point fibers, but a quotient map can erase an entire component. Finite Hecke algebras are resolved into canonical Artin packets, so simultaneous generalized eigenspaces are available without assuming semisimplicity. Nested packets in one finite carrier stabilize; for changing finite carriers, nonzero stable cores give compatible inverse limit vectors. These results give a pointwise criterion: a characteristic zero deformation point in automorphic support produces a Hecke eigensystem. With attached Galois representations, local global compatibility, and classicality, the point is automorphic. A fixed compatible system then propagates the conclusion to every coefficient place. For finite coefficient quotients, compatible conjugacy classes admit compatible continuous representatives. Declared polynomial quotient conditions survive their adic reconstruction. These completion statements require explicit finite occupancy, continuity, and integral Frobenius data. Universal companion existence and unrestricted reciprocity are not proved.
Publication and supporting files
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This reading edition, dated 10 October 2026, retains the complete revised mathematical proofs. Fourteen inherited vector figures are unchanged. Four diagrams have been redrawn to separate their arrows, expose the supplied inputs, and synchronize the final architecture with the five gates of Theorem Theorem 12.6. Supporting Markdown (.md) files are included with the source: the publication note, formal-evidence boundary, original-to-revised claim crosswalk, current status, source-preservation ledger, and build instructions. They record the statement hypotheses, evidence scope, and editorial changes alongside the preserved original and revised source snapshots.
The saved release reports complete mapped formal evidence for all forty-nine revised internal claims, while the original-wording census is forty-one complete, seven partial, and one unchecked. These are different contracts. The completed tensor and point-factorization results are algebraic deductions for supplied deformation data; representability, residual occupancy, applicable patching and classicality theorems, attached Galois representations, Frobenius density, semisimplification, and the general Brauer–Nesbitt theorem remain external arithmetic or algebraic inputs. Neither Paper XI's general Conjecture 66 nor universal companion existence is assumed.