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 v13 · 30 September 2026
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/πRΛ, RΛ 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 aM = πc²/12RΛ = 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.
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A. Korytko, Gravitation from Hilbert-Space Granularity: The Sphere, the String, and the Ring, Zenodo (2026). doi:10.5281/zenodo.22255461
Version v2 · 15 September 2026
Quantum field theory with a Planckian cutoff predicts a vacuum energy density of order the Planck density, 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, whatever its magnitude, and Λ survives as an integration constant that the horizon fixes. Second, the standard estimate's error 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 a pixel's energy scale 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 residual being the order-unity coefficient the framework has yet to derive. The same L caps a state spread across its Hilbert space at 205 qubits, a ceiling whose measurement would be 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 v16 · 17 September 2026
In the paper on the origin of gravitation, we postulated separately that Hilbert space and space are granular with one universal granularity L, extending Palmer's Rational Quantum Mechanics. We derived Newton's constant and Einstein's equations, and pinned L at two scales, galactic and cosmological. Here the two axioms become one postulate: a qubit is a record of marks, one at each cell of a ring of L Planck cells, and a state of N qubits is N records on the same cells, read jointly. A record's fraction m/L of +1 marks is a Born probability and its cyclic position a phase, one step of phase being one cell, a full turn one circuit. The ring is the register of the title. Placing the ring on a great circle of the cosmological horizon, as before, gives L = 2πRΛ/ℓP ≈ 6×10⁶¹. Momentum is quantized at ℏ/RΛ; a state spread over all 2ᴺ outcomes of N qubits needs cells for each, 2ᴺ ≤ L, at most ⌊log₂ L⌋ = 205, whatever realizes them. Along a direction, light moves one cell per tick, velocity is v/c = 2m/L − 1, and matter at rest reverses every quarter Compton period, its clock Palmer's hidden global phase; several particles in several planes need three dimensions, a count derived elsewhere. Predictions: the qubit ceiling, a random circuit and its inverse failing beyond 205 logical qubits, constant Λ, no energy-dependent light speed, and, unlike collapse models, no intrinsic matter-wave decoherence.
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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 v16 · 19 September 2026
We derive the three dimensions of space from two postulates. The first is the register: every qubit is a string of L bits around a ring of L cells of one Planck length, with L = 6.4×1061 universal. Its bits are the outcomes of one measurement and its position is the phase, and each cell counts phase from its own zero mark, on which no law depends. The second is that there is a sphere in d-dimensional space, d unknown, every great circle of which is such a ring, and its cells label the momenta an elementary interaction can carry, one cell per momentum. An elementary interaction is an event at which its legs exchange momentum. Momentum is conserved, so the momenta of an interaction with k legs span k−1 dimensions, and each of them is one of Lk−1. The sphere has Ld−1 cells, so d = k for each interaction, and d need not be universal. In the register a moving string can pass one step of phase to the offset of two neighboring cells. That event has three legs, so d = 3. The offsets are light. An observation is an exchange of momentum, and a string's momentum changes in no other way, except when two strings meet at one cell, which for uncorrelated strings happens once in L pairs. Every observation is therefore made with this event. A four-leg interaction, rare or common, would be seen only through it, and its four momenta span three dimensions, so it would appear three-dimensional. With L from the cosmological constant the sphere is the horizon of an observer at the interaction. A count relaxed to real d finds no other solution.
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A. Korytko, The Dimension of Space from the Granularity of Hilbert Space, Zenodo (2026). doi:10.5281/zenodo.22678537