Tomographic ISW Cross-Correlation as a Probe of Time-Varying Gravitational Coupling

We present a tomographic prediction for the Integrated Sachs–Wolfe (ISW) effect in Energy-Flow Cosmology (EFC), a framework in which the effective gravitational coupling evolves with cosmic entropy through a transition function μ(a). Unlike ΛCDM, where the ISW signal arises solely from late-time decay of gravitational potentials due to dark energy, EFC introduces an additional time-localized contribution proportional to dμ/ d ln a. This generates a distinctive partial cancellation in the ISW kernel near the transition epoch zt≃0.82z_t \simeq 0.82zt​≃0.82. Using linear theory with an Eisenstein–Hu transfer function and DESI-motivated tracer windows, we compute the ISW–galaxy cross-power spectrum CℓTgC_\ell^{Tg}CℓTg​ over multipoles ℓ = 2–60. We introduce a cancellation index C\mathcal{C}C that quantifies interference between the standard and transition-driven ISW contributions. In EFC, C\mathcal{C}C increases monotonically with tracer overlap with ztz_tzt​, reaching values C≳0.8\mathcal{C} \gtrsim 0.8C≳0.8 for bins centered near the transition. This produces a redshift-dependent amplitude suppression of the ISW signal, while leaving the ℓ-shape largely unchanged. Bias-calibrated amplitude fits show that current ISW measurements, including DESI Legacy Survey × Planck and ATLAS LRG × Planck, are consistent with EFC predictions within 1–2σ due to large uncertainties at z≳0.6z \gtrsim 0.6z≳0.6. The decisive test is a DESI tomographic cross-correlation at z∼ztz \sim z_tz∼zt​, where EFC predicts strong kernel cancellation and AISW