The Denomination Desert: Gravitational Confinement, the Hubble Tension, and Cross-Domain Structure in the 840-State Manifold

Abstract A logarithmic dimension function D(x) =-ln x/ ln φ, where φ = (1 + √ 5)/2 is the golden ratio, arises as the unique solution to the chiral fold identity D(z) + D(1/z) = 0 when the base is fixed by the Lucas recurrence Ln = Ln-1 + Ln-2 with seeds (3, 4). The twelve Lucas numbers so generated sum to 840 = lcm(1,. .. , 8), whose algebraic closure GF(841) * of order 840 contains ten distinguished residues, the Brahim numbers, with center K = 107 and mirror constant S = 2K = 214. In this framework, the ratio of local to early-universe Hubble constant measurements decomposed as φ 1/6 corrected by a series of golden-ratio powers whose exponents emerged from a greedy algorithm with gaps equal to Lucas numbers. At third order, the expansion predicted H late 0 = 73.4996 km s-1 Mpc-1 ; the H0 Distance Network Collaboration subsequently reported 73.50 ± 0.81 km s-1 Mpc-1 , a match at 5 parts per million. A geometric correction ladder of the form Dn = nK + 2 n-3 /π, verified at 2.3 ppm for both the electron-to-Planck mass ratio (n = 1, corresponding to me/M Pl = 4.185 × 10-23) and the gravitational fine structure constant (n = 2, corresponding to αG = 1.752 × 10-45), yielded the gravitational confinement mass M char = M 3 Pl /m 2 e = 6.25 × 10 6 M⊙, consistent with the peak of the JWST supermassive black hole population at redshifts z = 4-10. Between the ninth and tenth Brahim numbers (B9 = 172 and B10 = 187), a structural void of width L5 + L3 = 11 + 4 = 15 in the dimension coordinate spanned the interval 183.55, 184.99, which mapped to the neutron star-to-black hole mass gap at 2.5-5.0 M⊙. Three gravitational wave detections by the LIGO-Virgo network fell inside this void. Forty-three equations were proposed and tested against six independent observational datasets spanning gravitational wave astronomy, X-ray astrophysics, exoplanet populations, circumgalactic medium physics, and fast radio burst phenomenology. Four Monte Carlo null tests dismissed six equations, confirmed twenty-four, and revealed deeper invariants behind three others. Five falsifiable predictions were registered for verification by DESI, LIGO O5, and JWST. No parameters were fit to the data. Keywords: Golden ratio, dimension function, Hubble tension, mass gap, fast radio bursts, gravitational confinement, little red dots, circumgalactic medium, Lucas numbers, Brahim numbers Declaration of Interest The autor confirms that no personal or financial relationship with other people or organizations have influenced the study Funder Statement Elias Oulad Brahim, Individual Researcher Suggested Citation Oulad Brahim, Elias, The Denomination Desert: Gravitational Confinement, the Hubble Tension, and Cross-Domain Structure in the 840-State Manifold (April 17, 2026). Available at SSRN: https://ssrn.com/abstract=6596698