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

Abstract

We argue that 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, in two postulates. Every qubit is a string of bits on a ring of cells, no ring having more than L cells; and any two cells not opposite lie on one ring of L cells, any two such rings sharing a cell. The postulates give L²/4 largest rings, and an observer's cosmological horizon holds one bit of entropy for each. Jacobson's argument then gives Einstein's equations, and the measured G fixes the cell 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 acceleration c²/πR_Λ, R_Λ the horizon radius; below it, the entropy the strings still hold responds as an elastic medium, and that response bounds from above the acceleration scale of flat rotation curves without dark matter, at a_M = πc²/12R_Λ = 1.4×10⁻¹⁰ m s⁻², which the observed 1.2 respects; Verlinde's equality on the strain attains the bound. The largest ring ties the cosmological constant to Newton's constant, and the measured Λ and G give L = 4.4×10⁶¹, inside the galactic bound L ≤ 5.2×10⁶¹, at 85% of it. Only a quantum computer measures L on its own: we predict a ceiling of at most 204 qubits in a random state, the same for every technology, testable within the decade. Every count, relation, and number is machine-checked in Lean 4, with every input and reading named in the files.v13, 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) v13: The register is an incidence structure, not a drawing on a sphere, joined to the horizon's area by a reading; the count is of the empty de Sitter horizon, one per observer at rest, a mass shortening its throat circles, not the register's rings; the ceiling 2^N ≤ L holds for every state with all joint outcomes, is soft below 204, and is tested one way by the random circuit; galactic test one-sided; five readings, all inputs named; Lean v13, 168 theorems: https://github.com/AlphaLatitude/RegisterGravity/tree/v13 v4 (2 Sep 2026) remains the priority record.