Combined SN+BAO+Compressed-CMB Constraints on Minimal Late-Time G-Field Cosmology: BAO–CMB Trade-off

<div> The December 2025 SN+BAO-only analysis of minimal G-field extensions of Lambda-CDM identified a promising background-only late-time branch: mildly negative Planck-mass running improved the fit to Type Ia supernova (SN) and baryon acoustic oscillation (BAO) distances and raised the inferred Hubble constant to H0 ~ 70-71 km/s/Mpc. In this work, we test whether that branch survives a more conservative combined analysis that adds compressed cosmic microwave background (CMB) distance information. </div> <div> <br> </div> <div> To do so, we study the compensated Density-Triggered Relaxation family (DTR/DTR4). This construction anchors M_*^2 to unity both at early times and today, and enforces zero integrated running (integral alpha_M d ln a = 0), thereby eliminating net early-time and present-day normalization leakage. Once the SN+BAO+compressed-CMB likelihood is imposed, the same-family null reference already shifts to a low-H0 baseline, H0 ~ 66.66 km/s/Mpc, and the free-DTR optimum moves only to a near-null deformation, alpha_M,amp ~ -0.005, with total Delta-chi-squared ~ -0.036 relative to the DTR null case. </div> <div> <br> </div> <div> A component-level decomposition shows that this near-null convergence is driven by a BAO-CMB trade-off: for alpha = 0 -&gt; -0.005, Delta-chi-squared_SN = -0.687, Delta-chi-squared_BAO = +1.474, and Delta-chi-squared_CMB = -0.823. The fixed-threshold DTR4 benchmark remains statistically acceptable, only Delta-chi-squared ~ +0.43 worse than the free-DTR best point, but it likewise does not restore a meaningful high-H0 combined branch. We therefore present DTR/DTR4 not as a resolution of the Hubble tension, but as a restricted late-time null benchmark: under the assumptions of fixed early physics, a single compensated late-time transition, and compressed CMB distance information, the background-only high-H0 branch does not survive the combined test. We also report a concrete out-of-sample standard-siren prediction for this near-null baseline. </div>