Conditional-Expectation Coarse-Graining and Anchored Twist–Calm Coupling in a Single-Field S3 Framework

This paper reformulates the calm-sector coarse-graining and the twist–calm coupling extraction in the single-field (S^{3}) framework using only classical measure theory. The commutative von Neumann algebra (\mathcal A=L^\infty(\Omega,\mu)) is used as a precise language for observables, with calm coarse-graining defined canonically as conditional expectation onto a calm subalgebra generated by the coarse calm datum (g=C_\ell[\nabla\alpha]) and its anchored Hodge decomposition (g=\nabla\tau+w), (\tau(\infty)=0). A localized anchored insertion (\tau_{\rm ext}=\tau_0\chi) ((\chi\to0) at infinity) defines a bulk-cancelled interaction-energy observable with a well-posed infinite-volume limit, and the twist–calm coupling (a) is defined invariantly as the anchored, sector-fixed derivative of this interaction energy at (\tau_0=0) (with an equivalent centered-slope estimator). A symmetry obstruction is stated—symmetry-equivalent insertions force (a=0)—clarifying why anchoring is required. The resulting weak-field closure map is summarized, yielding (K_0\nabla^2\tau=a c^2\rho) and (G=a^2c^4/(4\pi K_0)), with the Newtonian (1/r) regime characterized by an infrared masslessness condition on the calm quadratic kernel.