Gravitation from Hilbert-Space Granularity: Pinning the Parameter L of Rational Quantum Mechanics Across Scales

Abstract

Ever since general relativity and quantum mechanics were discovered over a century ago, physics has been on the quest for a theory that unites them. We argue that the answer has been staring us in the face: gravitation is a consequence of discrete quantum mechanics. Tim Palmer's Rational Quantum Mechanics claims that a quantum state cannot carry infinite information. Hilbert space is granular, with a finite parameter L, and Palmer attributes that granularity to gravity. We reverse the claim. L comes first, and gravity is what a granular Hilbert space looks like at every scale. From one universal integer L and one length, the cosmological horizon radius, we derive the Planck length and Newton's constant. Einstein's equations follow; with the same L, galactic rotation curves flatten without dark matter, and Newton's constant and the cosmological constant cease to be independent: their product is fixed by L. We test the reversal by pinning L on every scale where it appears on its own rather than through the Planck length. The galactic scale gives 4.6 × 10⁶¹; the cosmological scale gives 6.4 × 10⁶¹. They agree to a factor of 1.4: the old coincidence between Milgrom's acceleration and the Hubble scale, here one L seen on two scales. The third place where L appears on its own is the quantum computer, which will report within the decade. We predict what it will find: a ceiling near 200 entangled qubits, the same for every technology.v8, 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. v4 (2 Sep 2026) remains the priority record.