Vacuum Information Density as the Fundamental Geometric Scalar: UIDT v3.5.5: A Proposed Theoretical Framework for the Yang–Mills Mass Gap and Gamma-Scaling Unification (incl. Pillar I-III)

The Unified Information-Density Theory (UIDT) presents a constructive proposed theoretical framework that explores geometric connections between Quantum Field Theory and General Relativity via information-geometric methods. Version 3.9 consolidates the Four-Pillar Architecture by combining a rigorously verified QFT core with a covariant scalar-field extension, a lattice-torsion model, and a photonic analog platform. Canonical parameters are obtained self-consistently within the UIDT framework using the Extended Functional Renormalization Group (FRG) and a Banach fixed-point construction for the Yang–Mills sector. The analysis yields a numerically stable vacuum solution characterized by the spectral gap Δ = 1.710 ± 0.015 GeV, coupling ratio κ = 0.500 ± 0.008, the self-coupling λS := 5κ²/3 (exact RG fixed-point definition), and the phenomenological invariant γ = 16.339. The kinetic vacuum expectation value is v = 47.7 MeV [A]. These parameters exhibit numerical closure with residuals below 10⁻⁴⁰ in the constructive core and are consistent with continuum-extrapolated lattice-QCD results where applicable. The invariant γ = 16.339 is phenomenologically calibrated from the kinetic vacuum expectation value (Category A⁻) rather than derived from renormalization-group first principles in UIDT v3.9. A recent algebraic analysis identifies a candidate expression γbare = (2Nc+1)²/Nc = 49/3 ≈ 16.333 from SU(3) Casimir structure, matching the canonical value to 0.037%. The physical dressing shift δγ ≈ 0.006 from the bare to the calibrated value remains under investigation via momentum-dependent FRG vertex flows (Limitation L4, Category D). Version 3.9 completes the synthesis of the Four-Pillar Architecture: the QFT foundation (Pillar I), lattice topology and torsion binding energy (Pillar II), spectral expansion and thermodynamic noise thresholds (Pillar III), and macroscopic isomorphism with nonlocal optical media (Pillar IV). Within UIDT, the lattice torsion binding energy ET = 2.44 MeV is defined as an entropic scale that parametrizes a torsion-related binding contribution in the hadronic sector and is explicitly classified as a quantitative Category D prediction. Within this structure, the framework implements a multi-stage suppression mechanism for the effective vacuum-energy density, combining the non-perturbative spectral gap, the invariant scaling factor γ, and an empirically defined 99-step renormalization-group cascade (N99). The N99 cascade represents a leading-order phenomenological scaling rule; a next-to-leading-order correction yields N ≈ 94.05, and both are documented in the canonical audit with their respective precision bounds. Thermodynamic censorship is formalized through a characteristic noise threshold near ∼17 MeV (the Wolpert limit), which generates harmonic resonance patterns within UIDT. Within UIDT, these structures are used only as an interpretive framework for reported X17-scale anomalies and the BESIII X2370 signal; they are classified as Category D predictions and make no claim of having identified the physical origin of any specific resonance. Cosmological calibration enters through observational constraints from DESI and related datasets, which motivate a holographic length scale λ ≈ 0.66 nm and an effective dark-energy equation of state w(z) with w0 ≈ −0.99. In this context, H0 ≈ 70.4 ± 0.16 km s⁻¹ Mpc⁻¹ is used as a DESI-calibrated, Category C reference value rather than as a strict prediction of UIDT, keeping cosmological statements explicitly in the calibrated-model regime. All results are explicitly classified by evidential strength: Category A (mathematically robust constructive proofs within the UIDT axioms), Category B (lattice-QCD alignment and numerical corroboration), and Category C/D (cosmological calibration and experimentally unverified predictions).