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

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 parameter L 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, with Newton's constant set by the cell's length, and the largest ring ties the cosmological constant to L. The measured G and Λ fix the two constants, the cell at √(π ln 2) Planck lengths and L = 4.4 × 10⁶¹; nothing else is fitted. Newton's law, read as an equal share of energy per string, holds for an isolated mass as long as the mass removes more entropy than a sphere's strings hold inside it. Below the acceleration c²/πR_Λ, R_Λ the horizon radius, it removes less, and the entropy the strings still hold responds as an elastic medium, which 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. The observed 1.2 × 10⁻¹⁰ respects it, at 85%; the transition between the two regimes is not derived. Only a quantum computer can measure L on its own: we predict that quantum mechanics fails for a random state of more than 204 qubits, the same ceiling for every technology, testable within the decade. The count, the chain of relations under the stated inputs, and every quoted number are machine-checked in Lean 4.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 v14: the five readings of the postulates in v13 are now derived; Postulate 2 says that two cells not opposite lie on exactly one largest ring; a register is shown to exist for every L with L/2 − 1 a prime power; and the Lean 4 check (release v14) has 189 theorems: https://github.com/AlphaLatitude/RegisterGravity/tree/v14 v15: the strings of a sphere are counted at one cell, without placing the rings in space; the qubit-ceiling test is one-sided, and the paper states what the register predicts for cluster states; a new section shows that the strings' heat is ln 2 of their energy, compared with today's measured Ω_Λ. Lean formalization RegisterGravity v15, 191 theorems: https://github.com/AlphaLatitude/RegisterGravity/tree/v15 v4 (2 Sep 2026) remains the priority record.