Lambda EG: Emergent Gravitational Coherence and Galactic Dynamics Without Particle Dark Matter

Release Overview This dataset provides a comprehensive visual and quantitative summary of theΛ-Emergent Gravity (Λ-EG) framework. Version 8 (2026-07-18) with corrected figures. Changes: (i) The critical-scale derivation now leads with the scalar Beltrami–Laplace spectral gap on S³ (λ₁ = 3 → √3) together with the horizon heat-kernel measure (4π² = 2·Vol(S³)); the Matsubara-mode heuristic, in which 4π² cancels algebraically, is removed, and a single variational unification is stated as an open problem. (ii) The Dirac–Lichnerowicz R/4 term is presented as a coincident geometric relation under a common curvature-radius normalization, not as applying to a scalar field. (iii) The non-relativistic theory is described as a screened scalar–tensor model whose deep-MOND (AQUAL) behaviour emerges asymptotically; the Newtonian regime is recovered through chameleon screening — not as an exact reduction to AQUAL. (iv) The complete kinetic function is used consistently, giving μ(x) = x/√(1+x²); the pure (2/3)X^(3/2) branch is the deep-MOND limit only (a new F(X)/μ(x) figure is added). (v) The Cassini thin-shell estimate is corrected: F_φ/F_N ∝ α^(−5/3), requiring α ≳ 8×10⁶. (vi) Gravitational lensing is derived from the scalar stress-energy T^φ_μν sourcing the metric perturbations; a purely conformal frame change produces no photon deflection. (vii) The cluster-lag prefactor is corrected to √(π/2) ≈ 1.25 (exact form: Δx = σ√2·erf⁻¹(v/V_coll)).