The Symmetry Status

SYM 0 of Symmetry ARC Constraint, Redescription, Physical Action, and Standing Kernel Necessity and Independent Role Occupation: A Fixed-Domain Normal Form for Standard Model Symmetry Claims Symmetry claims in quantum field theory can perform fundamentally different functions. They may express constraints on admissible dynamics, equivalences between representatives, actions on physical states or operators, theorem consequences, anomaly or matching obstructions, parameter-limit regularities, target updates, or transports on theory space. Treating all of these functions as a single exact/broken or fundamental/derivative label obscures the distinctions required for rigorous classification. This paper develops a fixed-domain symmetry-status calculus that types these functions independently and constructs a unique role-and-status normal form for a controlled Standard Model benchmark universe. The framework begins from the Admissibility–Standing–Reference–Irreversibility kernel. It treats determinate symmetry claims as already requiring target identity, admissible comparison, standing-bearing use, lawful transformation, and irreversible act-time fixation. Conventional quantum-field-theoretic structures then provide the sector-specific realization within that standing-bearing domain. Each atomic symmetry claim is classified by: claim kind: action, structure, diagnostic, or transport; physical disposition; classical and quantum status; anomaly and obstruction status; vacuum and boundary status; global-form and topological status; operator and causal-ancestry status; scale, effective-theory, and renormalization-group status; target-update and redescription status. The role-occupation construction is deliberately independent. Target obligations generate role positions without naming their occupants, while physical and dependency facts generate candidate residues before role lookup. Every role-position/residue pair is evaluated, complete matchings are tested by family-specific joint compatibility, and compatible assignments are quotiented by the full automorphism group preserving the theorem-bearing invariant tuple. At the declared SYM-0 benchmark scope, the paper analyzes forty-six atomic records organized into sixteen maximum-stress benchmark families. The independent construction yields sixty-eight role positions, sixty-eight presentation-quotiented physical residues, ninety eligible incidence edges, twenty-nine complete preassignments, seven substantive compatibility rejections, twenty-two compatible assignments, and exactly one lawful assignment orbit per benchmark family. The principal theorem states that, for every role-realization-complete certificate and fixed benchmark target, ∣Assign⁡(C,T)/Γlawful(C,T)∣=1.\left|\operatorname{Assign}(C,T)/\Gamma_{\mathrm{lawful}}(C,T)\right|=1. The public symmetry-status record is the canonical presentation of that unique orbit. Subsequent fixed-domain gate and admissible-interior results exclude a residual second same-scope selector or governance-non-equivalent regime. The paper includes a controlled atlas, evidence and source-traceability structures, reproducibility architecture, independence audits, adversarial mutations, fail-closed dispositions, and an explicit theorem boundary. The result is complete for the declared SYM-0 calibration universe. It does not claim full candidate exhaustion for every symmetry-relevant structure of the Standard Model, numerical closure of couplings or masses, or arbitrary ultraviolet uniqueness; those remain reserved for the later SYM-11 programme. Principal contributions A typed calculus separating constraint, redescription, physical action, diagnostic, and transport. A fixed-domain normal form for Standard Model symmetry claims. Independent formation of target-derived role positions and physical residues. Complete eligibility and joint-compatibility analysis. Quotienting by lawful identity-preserving automorphisms. A singleton role-occupation theorem for every declared benchmark family. A no-second-same-scope-selector result. A reproducible, fail-closed audit and evidence architecture. An explicit separation between the completed SYM-0 benchmark theorem and the broader SYM-11 exhaustion problem.