THE HERNANDEZ HYPOTHESIS A Unified Phenomenological Model of Density-Dependent Vacuum Dynamics V4

Version 4 is a fundamental reformulation of the Hernandez Hypothesis.Versions 1–3 implemented the framework as an exponential suppression ofquantum-vacuum recombination by local matter density, with galacticdynamics supplied by an ultralight scalar field (a fuzzy-dark-mattervariant). Version 4 replaces that mathematics entirely with a singlealgebraic modified-gravity law, g_obs = g_bar + C · f(g_bar/g_crit), f(x) = √x / (1 + √x), with C = 6.5×10⁻¹¹ m/s² and g_crit = 5.0×10⁻¹¹ m/s². The physicalmotivation — that vacuum behavior depends on the local gravitationalenvironment — is retained; the mathematical implementation, thedark-matter mechanism, and the dark-energy conclusion are new. The framework is AQUAL-class (its character function is algebraically aFermi-Dirac distribution) and its deep-regime limit reduces to MOND withimplicit scale a₀ = C²/g_crit = 8.45×10⁻¹¹ m/s². It is tested acrossseveral domains against public data, with successes and failures reportedsymmetrically: • Outperforms MOND-simple on low-density gas-dominated dwarfs (Sample B; 17/19 galaxies at matched parameter freedom) and on six ultra-diffuse galaxies (6/6 head-to-head).• Predicts the Baryonic Tully-Fisher scatter (0.067 dex) and slope (3.89) consistent with observation, plus a novel residual-correlation signature.• Satisfies LIGO/Virgo constraints structurally (~20 orders of magnitude).• Implicit a₀ matches the cosmological de Sitter acceleration c²√(Λ/3)/(2π) within ~2–3%.• Underperforms MOND on Milky Way classical dwarf spheroidals (7/8) and at cluster cores; the cosmological sector predicts the wrong sign for the dark-energy equation of state (a candidate manifold-tension fix is offered but requires fitted parameters); solar-system consistency requires a screening parameter that is not derivable from the framework. This version also corrects two specific v3 claims: the dark-energy sector’sphantom prediction (w < −1) is the wrong sign relative to DESI DR2, and thepaper cited by v3 as independent confirmation (Moffat 2025) in fact reportsw₀ > −1, opposite to the v3 prediction. Both are documented in Appendix A,“Changes from Version 3.” The deposit includes the full paper, the analysis code, the curated galaxysamples, and a methods-and-assumptions ledger, so that every numericalclaim can be reproduced against the public SPARC database. The work ispresented as a documented phenomenological exploration — an idea articulatedin reproducible, falsifiable form — not as a validated theory.