The Dimension of Space from the Granularity of Hilbert Space

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22814261
Primary Topic
Quantum Mechanics and Applications
Type
preprint
Controls
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preprint

The Dimension of Space from the Granularity of Hilbert Space

Andrew Korytko
Zenodo (CERN European Organization for Nuclear Research)
Quantum Mechanics and Applications
preprint

The Dimension of Space from the Granularity of Hilbert Space

Andrew Korytko
preprint en

Abstract

Why is space three-dimensional? We now know: because of the two postulates of quantum granularity. The first: every qubit is a string of L bits on a ring of L Planck cells, L = 6.4×10⁶¹ universal, with a phase on every link. The second: a direction is the state of one qubit. A qubit's state is a bounded count and a cyclic turn, which close to a sphere of L² cells when no cell is distinguished. The directions of n-dimensional space have Lⁿ⁻¹ cells, one rational probability per angle. The two sets are one, so n = 3. A momentum is a direction on one ring and a magnitude, L values each, so a vertex's momenta have L² labels with no n among them. Finite L and the gauge clause fill the sphere with those labels one to a cell, so every elementary vertex has three legs, a prediction the register's own interaction meets. Momentum is conserved, a four-leg event is two three-leg events, everything that interacts with light shares one sphere, and it is the cosmological horizon. Could n be a non-integer? The grid exists only for integer n, but the horizon's Planck pixels count for real n, with one root, three within the pixel error, 0.008 at the physical L and zero as L → ∞; 2 or 4 would need a count wrong by 3×10³⁰. A four-leg elementary vertex whose coupling is not the square of a cubic one, or an energy-dependent light speed, would falsify it. v14: Postulate 2 is now "a direction is the state of one qubit," and n = 3 is counted from a qubit's two digits; every elementary vertex having three legs is a prediction, with four-leg events as two of them, and the register's own interaction is the check. Postulate 1 carries the phase on every link and the hidden phase of a state. The Planck pixel enters only to continue the count to real n, where its one root tends to 3. Consequences for the register updated to its current version; language revised throughout. v5 (9 Sep) remains the priority record.v12: Postulate 2 weakened by one word: its addresses are directions of n-dimensional space, not the directions. That every direction is an address is now derived, from the gauge clause and the one address space. Two words of Postulate 2 remain postulated, one and directions, correcting v11's "only one." Citation removed from the abstract. Wolfram's hypergraph-rewriting models added to the introduction as a third prior approach, one that leaves dimension open. v11: Postulate 2 restated as an address space; the sphere, every great circle a ring, and one momentum per address (n ≥ 3 from finite L, n ≤ 3 from the gauge clause) are derived. The three-leg vertex is exact: a crossing shifts a link's flux by one unit, so a four-leg event is two crossings. The L²/2 accounting is withdrawn; the root is 3.008 and π is the whole residual. Minor corrections and unused material removed.v8: The two-body premise is removed; conservation, the three-leg vertex, and n = k are derived from the postulates through the hidden phase, which forces a gauge coupling. References to Wilson, Peierls, and 't Hooft added; concept DOI on the title page.

Zenodo (CERN European Organization for Nuclear Research)
Quantum Mechanics and Applications
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