Preprints on the granularity of Hilbert space and its consequences for gravity, the cosmological constant, and the dimension of space. The citable versions are on Zenodo; this page mirrors them.
ORCID 0009-0005-4569-2228
Version v7 · 5 September 2026
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 within a factor of 1.4 on a sixty-two-digit number. This is the old coincidence between Milgrom's acceleration and the Hubble scale, and here it is 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.
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A. Korytko, Gravitation from Hilbert-Space Granularity: Pinning the Parameter L of Rational Quantum Mechanics Across Scales, Zenodo (2026). doi:10.5281/zenodo.22255461
Version v1 · 10 September 2026
Quantum field theory with a Planckian cutoff predicts a vacuum energy density of 4.6×10¹¹³ J m⁻³, 123 orders of magnitude above what the observed cosmological constant allows. This paper resolves the discrepancy in two steps, within a framework in which every quantum state carries the same finite budget of L bits and gravity follows from that limit. First, the zero-point energy does not gravitate: a stress tensor proportional to the metric has no flux through any null surface, so it never enters the thermodynamic derivation of Einstein's equations, and Λ survives there as an integration constant that the horizon fixes. Second, the error in the standard estimate is in the state count. The energy Λ carries inside the horizon is the horizon entropy times the horizon temperature, which gives ρΛ = 3πρPlanck/2L² with L = 6.4×10⁶¹ the number of Planck lengths around the horizon: one factor of L because the count is of a boundary and not of a volume, and one because the energy scale of a pixel is EP/L and not EP. The acceleration scale of galactic rotation curves, a₀ = (1.20 ± 0.24)×10⁻¹⁰ m s⁻², fixes L independently at (4.6 ± 0.9)×10⁶¹ and gives ρΛ = (1.0 ± 0.4)×10⁻⁹ J m⁻³, against the observed 5.2×10⁻¹⁰; in the measured variable a₀ the two agree at 1.4σ. The same L caps a state spread across its Hilbert space at 205 qubits, a third determination with no astronomy in it.
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A. Korytko, The Vacuum Catastrophe and the Granularity of Hilbert Space, Zenodo (2026). doi:10.5281/zenodo.22683543
Version v2 · 8 September 2026
Palmer's Rational Quantum Mechanics writes a qubit as L bits: the count m of +1 bits gives the Born probability m/L and a cyclic shift the phase. Previously we made L universal, identified one phase step with one Planck length on the horizon, and derived Newton's constant and Einstein's equations at L ≈ 6×10⁶¹. The two axioms become one: a dial of L notches on the horizon, each a Planck length; every qubit is a pattern on it; one notch is a Planck-length translation, under which a wave of κ wavelengths gains κ phase steps, Palmer's phase step being the slowest wave's. From it: light moves one notch per tick; velocity is the excess of right- over left-movers, v/c = 2m/L − 1; in matter a right-mover becomes a left-mover every quarter Compton period, its Compton clock being Palmer's hidden global phase; the circumference is the longest wavelength, the previous paper's second axiom; and Born's rule as a frequency bounds a state spanning its Hilbert space at 2N ≤ L, Nmax = 205, for any physical realization. Free dynamics is exact mode by mode in the plane of any elementary vertex, whose momenta the horizon's pixels label; whole-cell position holds along one direction only; several particles in several planes need three dimensions, not constructed here. The predictions are inherited: the qubit ceiling, now a random circuit and its inverse failing to return past 205 logical qubits, and constant Λ; it entails no energy-dependent light speed and no intrinsic matter-wave decoherence, unlike collapse models.
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A. Korytko, One Register: A Common Origin for the Granularity of Hilbert Space and of Space, Zenodo (2026). doi:10.5281/zenodo.22669416
Version v8 · 10 September 2026
Why do we live in a three-dimensional world? It has been asked since antiquity. We now know the answer; it follows from two postulates. The first is that quantum mechanics is granular: every quantum state is a record of L bits, L = 6.4×10⁶¹ a universal constant, on a ring of L Planck cells, and each record's phase is hidden. The second is that the momenta an elementary interaction can carry are labeled by the cells of a sphere in n-dimensional space built on such rings, with n left open, integer or not. From the first, identical cells make momentum conserved at a vertex, and the hidden phase, respected cell by cell, makes the ring's shift carry a link phase: a gauge field, coupled to the current at a vertex with three legs. A vertex with k legs has momenta spanning k−1 dimensions; Palmer's rule that every probability is a multiple of 1/L gives such a momentum Lk−1 labels and the sphere Ln−1 cells; equating them gives n = k, so every elementary vertex has the same number of legs, and the three-leg one makes n = 3. Ref. [2]'s Planck pixels, defined by the horizon entropy, agree to a factor π; as an equation in real n their single root is 3.008, and 2 or 4 would need either count wrong by a factor 3×10³⁰. The sphere is one, so the space is one. An elementary vertex with four legs, or an energy-dependent speed of light, would falsify it.
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A. Korytko, The Dimension of Space from the Granularity of Hilbert Space, Zenodo (2026). doi:10.5281/zenodo.22678537