Gravitation from Hilbert-Space Granularity: The Sphere, the String, and the Ring

Abstract

For over a century, physics has sought a theory that unites general relativity and quantum mechanics. We argue that the answer has been staring us in the face: gravitation is what a granular quantum mechanics looks like at every scale. In Tim Palmer's Rational Quantum Mechanics, Hilbert space is granular, with a finite parameter L that he attributes to gravity. We reverse the claim. L comes first, one universal integer, and we start from two postulates: every qubit is a string of bits on a ring of cells, with no ring of more than L cells, and any two cells not opposite lie on one ring of L cells. The postulates give L²/4 largest rings, and on the cosmological horizon each holds one bit of entropy. Jacobson's argument then gives Einstein's equations with G = c³ℓ_c²/(πℏ ln 2), so the measured G fixes the cell length ℓ_c at √(π ln 2) Planck lengths. Newton's law, read as an equal share of energy per string, holds until a mass removes less entropy than a sphere's strings hold inside it, at the Newtonian acceleration c²/πR_Λ; below it, galactic rotation curves flatten without dark matter, with v⁴ = a_M G M and a_M = πc²/12R_Λ = 1.4 × 10⁻¹⁰ m s⁻², an upper bound, against 1.2 observed. The largest ring fixes the cosmological constant, GΛ = 12πc³/(ℏL² ln 2). The cosmological and galactic scales, each with the measured G, give L = 4.4 × 10⁶¹ and L ≤ 5.2 × 10⁶¹, within 18%. Only a quantum computer measures L on its own, and that test will come within the decade. We predict a ceiling of 204 qubits in a random state, the same for every technology. Every formula and number the paper derives is verified in Lean 4 from the postulates, thermodynamics, and elasticity.v12, changes since v4: Newton's constant now derived from Jacobson's argument, not Verlinde's entropic force; Einstein's equations arrive in Sec. III with Λ free. Sec. V: the two pins test an order-one coefficient, not sixty-two digits; a0's explanation credited to emergent gravity. Qubit ceiling from 2^N≤L: Nmax=205, was 212; test is a random circuit and its inverse. Prediction 4 carries the DESI DR2 significances and notes w=−1 is derived, not assumed. Prediction 2 names the surviving results; the shift map keeps only its clock; deep regime flagged as Verlinde 2017's; a0 at 1.4σ; three new references. v9: Cites Palmer 2016 and Hance–Palmer–Rarity 2025 (their Sec. V) on gravity's role and IST's Λ = 0 dark sector; notes one a0 across five dex of galaxy mass, so the 1.4 is a coefficient, not a mass effect. No other changes. v11: Major revision: gravity mathematically derived from two postulates using only thermodynamics and elasticity. v12: Space's three dimensions and the horizon's identification with the sphere of the largest rings now derived in Secs. II and IV D, with the symmetry of space named as the one premise; qubit ceiling 204 with its block argument; a_M as an upper bound (85% reached); Lean package extended to 168 theorems (tag v11.2, Software Heritage archived); references and language revised throughout. v4 (2 Sep 2026) remains the priority record.