The Standard Model Symmetry Atlas and Exhaustion Closure

SYM-11 of the Standard Model Symmetry Arc Target-Indexed Normal Forms, Physical Role, and Same-Scope Exhaustion This paper presents the capstone classification and exhaustion theorem for the Standard Model Symmetry Arc. It treats “symmetry” not as a single group label, but as a typed physical statement whose meaning depends on its acting object, physical quotient, anomaly status, vacuum realization, global form, operator locus, and renormalization-group scope. The manuscript integrates the preceding SYM modules covering gauge architecture, spacetime symmetry, anomalies, discrete symmetry, flavor, chiral and custodial structure, accidental charges, electroweak vacuum fixation, global form, generalized symmetry, and scale/RG transport. For every closure-eligible Standard Model target, the construction: freezes the target, record language, source hypotheses, role architecture, and support criteria; routes every same-target physical or status difference through nine formation loci and 54 source-theorem subfamilies; constructs target-qualified normal-form class spaces before reading atlas population; forms physical projections independently of role burdens, residues, atlas rows, and compatibility tests; transports exact physical-signature equality to independently generated source branches; derives source-licensed carrier support and proves relative removal minimality; partitions all proposed rivals into lawful equivalence, same-target obstruction, target update, or genuine same-target counterexample. The principal theorem states that, for any closure-eligible target DD, every complete admissible atlas satisfying the same target, record language, candidate protocol, source hypotheses, role architecture, and support criteria is isomorphic to the canonical atlas under lawful identity-preserving redescription: AD′/≡D≅AD/≡D,\mathcal{A}'_D/\equiv_D \cong \mathcal{A}_D/\equiv_D, and the generated genuine same-scope survivor branch is empty: Surv⁡Dsym(CD∖AD)=∅.\operatorname{Surv}^{\mathrm{sym}}_D \left(\mathcal{C}_D\setminus\mathcal{A}_D\right) = \varnothing. The isomorphism preserves target-qualified physical and status identity, role incidence, support ancestry, and source/evidence status. It does not require preservation of names, notation, row order, or other descriptive skin. The capstone addresses four independent closure burdens: candidate completeness; redescription-stable physical identity; compatible role occupation and source-licensed support; genuine-rival elimination. The result establishes that every closure-eligible target fiber has a unique atlas up to lawful redescription and no unclassified same-scope symmetry-relevant survivor. The theorem is target-indexed and relative to the declared target family, source hypotheses, record language, role architecture, support criteria, and comparison class. The manuscript explicitly distinguishes inventory, normalization, and exhaustion. A populated atlas is not assumed to be complete merely because it contains many rows; a normalized atlas is not assumed to be unique without a lawful equivalence theorem; and an empty rival ledger is not treated as exhaustion without a candidate-generation and rival-disposition theorem. The publication manuscript includes the complete proof at journal granularity, together with source-first derivations, theorem-identity crosswalks, target-indexed normal forms, support and provenance structures, compatibility transport, and reproducibility boundaries. Companion audit and atlas artifacts expose the finite implementation surfaces but are not used as substitutes for the semantic proof. The result does not assert uniqueness of the Standard Model under arbitrary beyond-Standard-Model or ultraviolet premises, select a unique global form, determine numerical parameters, or close targets whose assumptions differ from the declared target family. Principal contributions A target-indexed capstone atlas for Standard Model symmetry. A formal distinction between inventory, normalization, and exhaustion. Nine formation loci and 54 source-theorem subfamilies for candidate completeness. Lawful redescription and extensional quotient construction. Independent physical projection and source-branch formation. Full-domain role-occupancy and compatibility transport. Source-licensed carrier-support derivation and minimality. Disjoint rival classification and genuine-counterexample elimination. Fiberwise atlas uniqueness under fixed target data. Same-scope exhaustion with an empty generated genuine-survivor branch. Cross-module integration of SYM-0 through SYM-10. Explicit theorem, source-authority, and reproducibility boundaries.