A tilt statistic for structure-coupled dark energy: a systematics-immune discriminant in fsigma8

Redshift-space distortion measurements constrain the product f*sigma8, whose overall amplitude is degenerate with sigma8 and with any multiplicative calibration of the analysis pipeline. Models in which dark energy couples to structure formation generically predict not a uniform offset in f*sigma8 but a tilt across redshift, with the deviation changing sign at intermediate redshift. This paper argues that the tilt, not the amplitude, is the appropriate discriminant, and quantifies the trade. Profiling out a free overall amplitude removes the degeneracy at a cost. For a representative structure-coupled prediction across the six DESI redshift bins, a 3-sigma detection requires a per-bin precision of 1.28% on the shape against 1.67% on the raw amplitude, a tightening by a factor 1.30. In exchange the test is immune to any multiplicative systematic. The paper shows further that this premium shrinks as the shared systematic grows: for a common mode with correlation 0.8 the cost factor falls to 1.02, so the protection is nearly free precisely where it is most needed. The shape signal is concentrated at the ends of the redshift range, the two lowest and two highest bins carrying 96% of it. Once the measured errors are folded in, however, the bins that matter most in practice are LRG1 and ELG2 rather than the extremes, and a jackknife shows that no single measurement dominates: removing any one bin costs at most 19% of the present significance. Against DESI DR1 full-shape errors the present significance of the shape test is 0.32 sigma, so an improvement by a factor 9.4 is required for a 3-sigma detection, of which the bispectrum supplies a documented 1.10. Growth-rate measurements outside the DESI range, from peculiar-velocity surveys below z=0.1 and from the Lyman-alpha forest above z=2, change the required precision by only a few per cent at their present errors. Six predicted values are given for one worked example. The paper discusses what the present observational situation does and does not support, and states explicitly the limits of the analysis, including one internal inconsistency in the worked example that is not resolved here.