Foundational Constants in the S3-Field Framework

A self-contained program note that organizes how the familiar constants (c), (G), and (\alpha) should arise in a single-field (S^{3}) (SU(2)) framework. The paper distinguishes definitions, derived relations, closure assumptions, and open matching steps; gives an operational definition of the emergent metric from the principal symbol/characteristics of the field equations; and states the conditions required for a universal causal cone. It derives the trace–projection factor (c_{\mathrm{grav}}=1-1/D) (equal to (3/4) in (D=4)) controlling the kinetic screening coefficient in the calm gravitational mode, clarifies the role of the coarse-grained scale (\mu^{2}), and separates it from any optional identification with the potential curvature scale. For electromagnetism it records the Hopf/fiber construction, the effective Maxwell normalization and coupling parameters, and the specific Wilsonian matching tasks needed to obtain (\alpha). The result is a precise ledger of what is fixed by the (S^{3})-field dynamics and what must still be computed to obtain numerical predictions.