A Prediction for DESI Full-Shape: Increasing Tomographic $\Omega_m(z)$ Trend
To confirm $\Lambda$CDM deviations are due to missing physics (not systematics), one should demonstrate that the model fitting parameters exhibit qualitatively similar redshift drift across independent observables. This is the only way one guarantees new physics. Here, we show that a recent Dark Energy Spectroscopic Instrument (DESI) DR2 Full-Shape (FS) modelling Lyman-$\alpha$ constraint at $z_{\rm eff} = 2.33$ combined with earlier DR1 FS modelling constraints with $0.295 \leq z_{\rm eff} \leq 1.491$ leads to a straight line $\Omega_m(z) = m z + c$ with slope $m = 0.022 \pm 0.012$, $1.8 \sigma$ removed from constant $\Omega_m$. Akaike Information Criterion and Bayesian evidence confirm that constant $\Omega_m$ and increasing $\Omega_m(z)$ are statistically indistinguishable. Through the $Om(z)$ diagnostic, we review how increasing and decreasing $\Omega_m(z)$ trends map to phantom and quintessence dark energy (DE) regimes, respectively. While FS modelling constraints map to phantom DE, the decreasing and increasing $\Omega_m(z)$ trends in DESI BAO and DESI with external data make a phantom crossing inevitable. Since dynamical DE is but one interpretation for $\Omega_m(z)$ trends, it is imperative that different datasets converge on their $\Omega_m(z)$ trends before one jumps to physical conclusions. We forecast how DESI FS modelling $\Omega_m$ constraints will improve up to the final data release and explore the implications for model selection.