Redshift factor and the first law for non-singular black cores: exterior invariance of the horizon test and a regular interior representation

Zimmerman, Lewis and Pfeiffer (Phys. Rev. Lett. 117, 191101, 2016) extracted the redshift factor z of binary black holes from numerical simulations through the normalized surface gravity of the apparent horizons, and found that the first law of binary mechanics, M − 2ΩJ = Σ z_i m_i, holds to a remarkable degree. This note asks what changes if each hole is a non-singular black core of Relativity of Matter (RM): a regular hemisphere of S³ with lapse N = A cos(ρ/R_S), joined at its throat to the Schwarzschild slice. The exterior surface gravity is exactly the Schwarzschild value 1/(4m) for every interior lapse amplitude A. The Born–Infeld corrections are of relative order (3/2)(ℓ⋆/R_S)⁴, below 10⁻¹⁵⁶ for stellar masses. Three falsifiable null predictions follow: (N1) the horizon-based extraction and the first-law test are unchanged; (N2) the post-ISCO downturn of Δz and ΔQ, which tracks the nonadiabaticity Ω̇/2Ω² (0.027 at the ISCO, 0.052 at mΩ = 0.1) and exceeds the adiabatic radial-velocity correction by more than an order of magnitude, is an exterior effect that a regular interior can neither produce nor remove; (N3) the core spin susceptibility, I_c = ½χ r_c³ ≈ 2×10⁻⁷⁹ R_S³, makes the core contribution to the spin term (Ω_A − Ω)δS_A of the first law negligible. The regular core also adds interior structure absent in general relativity: a Tolman–Komar representation of the horizon term, m = −2Am + (1+2A)m; a central worldline whose clock rate is z_c = A z in the matched-asymptotic frame of the binary; a smooth lapse vanishing at the throat and an exact local frame-dragging condition B⁴ω′ + 3r_c³ω = 0 at the core wall. Statements are labelled FORMAL on the static witness, and CONDITIONAL on the nonlinear equivalence of the RM exterior dynamics with general relativity, which remains open. Binary core solutions, the dynamical throat and Zone II rotation are listed as open problems.