A Direct Empirical Test of the Curvature Texture Index in Quantum Fractal Geometry: Evidence from Supermassive Black Hole Scaling Relations

This paper presents the first empirical test of the Curvature Texture Index (CTI) in Quantum Fractal Geometry using an implementation faithful to its formal definition. A prior preprint (DOI: 10. 5281/zenodo. 19523163, withdrawn) employed a proxy statistic (CTIᵥ1) that is shown here to be a first-order morphological statistic rather than an implementation of the geometric concept. The proper CTI is implemented using the QFG projection operator directly: density profiles are convolved with a Gaussian kernel at 35 logarithmically spaced resolution scales, the spherical Laplacian is computed at each scale, and CTI is obtained by integrating the absolute rate of change of the spatially-averaged squared Laplacian across scales. Applied to N = 70 galaxies with published black hole masses from McConnell & Ma (2013) and Kormendy & Ho (2013), with density profiles reconstructed via the deprojected Sérsic approximation, the proper CTI yields a statistically significant negative coefficient in the MBH scaling relation (δ = −0. 357 ± 0. 131, t = −2. 72, p = 0. 0083, 95% bootstrap CI −0. 621, −0. 093, ΔAIC = −5. 4). Galaxies with higher resolution-dependent curvature texture host smaller black holes at fixed velocity dispersion and stellar mass, consistent with the QFG prediction. Cross-validated R² does not improve at N = 70 and independent replication on a larger sample is required. This constitutes the first direct empirical test of the geometric hypothesis as formally defined in QFG. Addendum (April 2026). A morphology-independence test has been conducted to establish whether CTI carries information beyond Sérsic structural parameters. After orthogonalising CTI against stellar velocity dispersion, stellar mass, and Sérsic index n simultaneously (Model E: σ + M★ + n + CTI), the CTI coefficient strengthens rather than weakens (δ = −0. 702, p < 0. 0001, 95% CI −1. 110, −0. 370). The morphology-independent component CTI⊥ — from which all shared variance with σ, M★, and n has been removed — remains highly significant (δ = −0. 158, p < 0. 0001, residual correlation r = −0. 465). The morphology-independent component of CTI (CTI⊥) captures a statistically significant, irreducible contribution to SMBH scaling relations. Supporting figure: CTIᵢndependenceₜest. pdf. CTI empirical paper (19562898) Note (April 2026). Follow-up independence testing using multi-component profiles and TNG100-1 matched simulations shows that the CTI signal is absorbed by morphological structure when profiles are sufficiently well-characterised. QFG has not been falsified by these results. However, the specific CTI implementation tested does not currently demonstrate independent physical signal beyond morphology. The framework therefore remains theoretically viable but empirically unconfirmed, and requires improved observables or first-principles derivations before it can be considered predictive.