Color Realization from a Premetric Response Atlas

Invariant Networks, Global Response, and Physical Continuation This paper develops the BCRS-2C color realization as a downstream specialization of the premetric bosonic response architecture established in BCRS-I. It constructs a source-relative atlas connecting primitive color transport to invariant observables, global-form response, screening, Yang–Mills sectors, metric source response, and physically continued neutral boundaries. The construction begins with one color-transport carrier and its jointly attached matter and Higgs source. On the (SU(3)) adjoint realization, the eight real coordinates form one irreducible invariant support. The coordinate dimension is therefore distinguished from the number of primitive carriers, invariant supports, or asymptotic particles. The paper derives the canonical local color law from the source structure. Connections, curvature, covariant matter action, invariant pairings, and non-Abelian self-interactions are retained as one typed response system. Compact-group averaging and witnessed tensor contractions then generate neutral responses and invariant observables from admitted source operations. A central result is the reconstruction of color transport from complete finite networks. Boundary-respecting forest reductions, separating invariant probes, full-path transport, and smooth extensions are developed on their stated domains. On explicitly witnessed tensor branches, elementary pairings and volume tensors generate the complete invariant-polynomial observation algebra, so observational completeness follows from source operations rather than from an unstructured list of assumed observables. The global-form analysis reconstructs electric descent and screening on the received Standard Model representation family. The paper also treats the associated magnetic lattice, mutually local dyonic branches, coupled phase data, theta-period structure, and determining joint characters. These results preserve the distinction between algebraically available branches and physically realized sectors: screening data do not by themselves establish a universal confinement law or the physical existence of every formal line branch. The physical continuation layer connects actually represented color sectors to neutral boundaries through a two-level configuration-to-boundary interface. Its maps retain preparation data, amplitudes, boundary conditions, mixed inputs, and continuation domains. Nonzero-amplitude criteria and matched continuation are treated as source-relative properties rather than inferred from color singletness alone. On matched Yang–Mills branches, the paper incorporates dynamical GNS comparison, spectral transport, local-vacuum neutrality, and source-qualified subgap rigidity. These results remain attached to their exact local, completion, matter-coupling, and boundary hypotheses. They are not promoted into an unrestricted statement about all of QCD or the full electroweak–color theory. The metric realization adds complete source probes, local-to-global response gluing, a unique distributional color source, and a balanced matter–Higgs–color exchange identity on the canonical local branch. This demonstrates how the color response contributes to geometric variation and energy–momentum exchange without treating gravity as a color carrier or claiming a complete theory of gravitational dynamics. The complete native response algebra then yields: an exact observational quotient; whole-interface role uniqueness over each complete realization datum; source-preserving reconstruction of admitted response interfaces; an exact criterion for lossless deletion of information; explicit separation of carrier identity, invariant response, physical attachment, and asymptotic continuation. The resulting atlas shows how one color carrier can support distinct but related field, network, line, phase, sector, metric, and physical-boundary responses. These responses are unified through their common source attachments while remaining distinct in type and realization authority. Scope and realization boundary The paper establishes a target-relative color realization on its received and independently specified source branches. It does not assert unrestricted physical confinement, all-sector asymptotic completeness, a universal hadron spectrum, a free asymptotic quark or gluon, a mass formula for every composite state, or a physical continuation for every algebraically defined color configuration. Structural neutralization, charge screening, metric closure, Yang–Mills spectral rigidity, and physical confinement remain separate claims with separate source requirements. Where those requirements are not supplied, the corresponding conclusion remains unresolved rather than being converted into either a positive result or a structural obstruction. Role in the BCRS programme BCRS-2C provides the color-sector realization of the premetric bosonic response atlas. It carries BCRS-I’s distinction between carrier, response, and projection into non-Abelian transport, global form, screening, Yang–Mills realization, metric sourcing, and neutral physical continuation. The companion BCRS electroweak/Higgs paper develops the corresponding electroweak branch, while the BCRS atlas paper integrates the realized sectors into the broader carrier–response–projection architecture. Keywords color realization; premetric gauge theory; Standard Model color; non-Abelian transport; (SU(3)) adjoint representation; invariant theory; global gauge form; electric screening; magnetic duality; dyonic response; Yang–Mills theory; metric source response; confinement interface; observational quotient; AASC