Non-Local Tensional Geometry as an Effective Embedding Model for Dark-Sector Phenomenology Creator: David Garcia Arrate

This work formulates Hyperdimensional Tensional Field Theory (HTF) as an effective embedding framework for dark-sector phenomenology. The central claim is not that dark matter is an unseen material fluid, particle population, or additional mass component. Rather, HTF interprets the dark sector as the three-dimensional gravitational reconstruction of positive tensional energy stored in a four-dimensional geometric substrate. Observable space is modeled as a three-dimensional section Σ3τ embedded in a four-dimensional substrate C4. Ordinary matter is the only material source and is treated as confined to the observable history of these sections. Matter deforms the substrate in the normal direction; the substrate responds through a tensorial internal stress state σij with finite relaxation. Observers confined to Σ3τ do notmeasure the normal restoring response directly. They reconstruct the positive stored tensional energy as additional attractive curvature in an effective spacetime description.The present formulation strengthens the original phenomenological proposal by introducing:(i) a dynamic embedding map, (ii) a tensorial viscoelastic constitutive law for internal tension,(iii) a normal-restoration transition variable χ whose weak-field projection appears as g/aHTF,(iv) a configurational non-additive projection rule, (v) a minimal lensing closure ΨT = ΦT,(vi) a saturating basal-tension model for dark-energy-like behavior, and (vii) a controlled source functional that distinguishes persistent coherent matter from diffuse radiation. In the galactic weak-field limit the model recovers a MOND-like projected equation with transition function µ(x) = x/√1+x2 and hence a baryonic Tully-Fisher scaling v4f = GMbaHTF, with configurational environmental corrections. Controlled numerical consistency checks are also presented: idealized rotation-curve solutions, baryonic Tully-Fisher scaling, non-additive configurational tensional energy, a viscoelastic dynamic-lag illustration, basal-tension cosmology, and a simplified linear-growth illustration.The framework is not presented as a validated replacement for ΛCDM. Instead, it is a mathematically structured theoretical program whose claims are constrained by explicit equations and falsification routes. The present manuscript reports the maximum validation level completed in this work: full-SPARC rotation-curve inference, reduced late-time BAO/SNe diagnostics, an official Planck PLC/clik likelihood evaluation, and a Phase-A official Planck scan of the CMB-safe basal branch. A full posterior or evidence-level sampling of the entire QHTF parameter space is explicitly identified as the computational boundary of the present study rather than claimed as completed.