Anomalies as Structural Obstructions

SYM-3 of Symmetry ARC Gauge Inconsistency, Global Response, Inflow, Matching, and Normal-Form Closure in the Standard Model A Fixed-Target Anomaly Reference for the Standard Model Symmetry Arc Quantum field theory uses the word “anomaly” for several physically distinct phenomena: failure of a dynamical gauge symmetry at the quantum level, anomalous nonconservation of a meaningful global current, obstruction to gauging a global symmetry, topology-sensitive determinant phases, mixed-background noninvariance, bulk–boundary inflow, ultraviolet–infrared matching constraints, and renormalized trace identities. This volume develops a fixed-target classification of these phenomena within the Standard Model symmetry programme. It distinguishes their physical loci, causal ancestry, obstruction types, repair mechanisms, presentation freedoms, and target-update boundaries rather than collapsing them into a single exact/broken or conserved/nonconserved label. The analysis is formulated under the Admissibility–Standing–Reference–Irreversibility kernel. A determinate anomaly claim already requires fixed identity, lawful comparison and transformation, consequence-bearing use, and irreversible fixation. Conventional anomaly theory then supplies the target-specific currents, measures, determinant phases, descent classes, index data, global obstructions, inflow relations, matching conditions, and renormalized operator identities. The closed SYM-3 family contains ten reference-complete anomaly targets: the one-generation Standard Model local gauge-anomaly ledger; vector/axial descent with Bardeen structure; a symmetric vectorlike six-Weyl theory; the ordinary and newer SU(2)SU(2) global anomalies; the massless three-flavor QCD U(1)AU(1)_A anomaly; electroweak B+LB+L; a four-dimensional Weyl boundary with five-dimensional inflow; three-flavor QCD gauged-Wess–Zumino–Witten matching; the massless-QCD trace anomaly in a fixed MS‾\overline{\mathrm{MS}} operator target. Across these targets, the construction contains: 63 independently derived role positions; 590 role-blind physical anomaly candidates; 4,412 evaluated same-fiber pairs; 91,695 complete candidate maps; 91,597 physical-compatibility rejections; 98 surviving compatible objects; one lawful normal-form isomorphism class for each closed target. The volume establishes a common anomaly grammar distinguishing: gauge inconsistency from anomalous global response; local descent from global determinant and bordism obstruction; consistent, covariant, and Bardeen-shifted current representatives; counterterm redistribution from genuine physical cancellation; anomaly inflow from retrospective repair; ’t Hooft anomaly matching from uniqueness of infrared realization; trace anomalies from gauge anomalies; closed anomaly targets from generalized, higher-group, non-invertible, and global-form frontier interfaces. The resulting normal-form groupoid retains nontrivial presentation morphisms, automorphisms, and stabilizers within a unique isomorphism class. Public anomaly coordinates and repair claims are admitted only when they preserve the fixed target and its invariant physical content. The volume also provides explicit interfaces for later symmetry modules. Target-relative anomaly records are transported to SYM-4 through SYM-10 under target-preserving rules. Global-form, line-operator, monopole, higher-group, and non-invertible anomaly inventories remain explicit SYM-9 frontier interfaces, while whole-arc candidate exhaustion remains reserved for SYM-11. The result does not claim that anomaly cancellation uniquely derives the Standard Model, fixes numerical couplings, or determines a unique infrared phase. It is complete only for the declared target and constructor grammars, and the executable audit layer does not rederive imported continuum results such as determinant-line theory, bordism classification, or renormalized quantum-field-theoretic theorems. Principal contributions A fixed-target anomaly-status calculus for the Standard Model symmetry arc. A typed separation of gauge, global, mixed, gravitational, topological, inflow, matching, and trace anomalies. A distinction between physical obstruction and presentation-dependent current or counterterm choices. A local anomaly-polynomial, Wess–Zumino, and relative BRST descent framework. A global-anomaly treatment using determinant lines, eta invariants, bordism, and faithful global form. A mixed-anomaly and discrete-anomaly classification. An explicit ’t Hooft anomaly and infrared-matching boundary. A bulk–boundary inflow and repair-relation analysis. A generalized, higher-group, and non-invertible anomaly frontier interface. A trace-anomaly and renormalization-group transport handoff. Exact Standard Model perturbative and global anomaly ledgers. A reproducible finite normal-form and compatibility audit architecture.