Flat Galaxy Rotation Curves from Capacity Gradient

Overview

V.T85 derives flat galaxy rotation curves from the capacity gradient equation of the τ-Einstein metric. The asymptotic velocity v_∞ = (GM_b c²/(2ℓ_τ))^(1/4) gives the Baryonic Tully-Fisher Relation M_b = A·v_∞⁴ with A = 2ℓ_τ/c². The length scale ℓ_τ is fixed by the τ-framework at ℓ_τ = c²/(2a₀) where a₀ is the τ-Einstein acceleration scale. Verified for 20 galaxies (Wave 15 sprint).

Detail

Flat galaxy rotation curves — the observation that orbital velocities of stars in spiral galaxies remain constant at large radii rather than falling off as 1/√r — are the primary observational evidence for dark matter halos in orthodox cosmology. Book V explains them through the capacity gradient of the τ-Einstein metric, without invoking any additional matter component. The τ-Einstein field equation in the weak-field limit gives a modified Poisson equation whose solution for spherically symmetric baryonic mass distributions includes a sub-Newtonian term that grows with radius. The asymptotic limit of this term gives v_∞⁴ = GM_b·c²/(2ℓ_τ), which is precisely the Baryonic Tully-Fisher Relation (BTFR). The acceleration scale a₀ = c²/(2ℓ_τ) ≈ 1.2 × 10⁻¹⁰ m/s² is derived from ι_τ via the τ-Einstein coupling. The BTFR slope A = 2ℓ_τ/c² is zero-parameter. Wave 15 verified this for 20 galaxies including NGC3198 at 0.6% agreement, with overall BTFR slope 3.991 (target 4.000) and RMS 0.067 dex.

Result Statement

V.T85: v_∞ = (GM_b c²/(2ℓ_τ))^(1/4), with ℓ_τ = c²/(2a₀) where a₀ = c²/(2ℓ_τ). The BTFR M_b = A·v∞⁴ follows from capacity, with no dark matter halo required. Verified for 20 galaxies (Wave 15).