Sub-Planckian Discrete Spacetime Calibrated by Dirac's Large-Number Ratio: Effective Field Theory Bounds, Calibration Identities, and a Ten-Condition Falsifiability Roadmap

We evaluate the continuum limit of a discrete spacetime substrate with fundamental spacing fixed by Dirac’s large-number ratio Q ≈ 4.17 × 10^42 at the sub-Planckian scale δ_p ≡ ℓ_P / Q ≈ 3.88 × 10^-78 m. Treating the substrate as a hypercubic lattice, spatial parity symmetry eliminates odd-order finite-difference operators, excluding dimension-5 Lorentz-violating corrections. The leading residual effect is a dimension-6 operator at O(δ_p^2) with a directional angular prefactor g(θ, ϕ), yielding a Q^-2 ~ 10^-84 suppression relative to Planck-scale cutoff models that comfortably satisfies ultra-high-energy cosmic-ray bounds and predicts zero vacuum birefringence. In the electromagnetic sector, inserting δ_p yields the scale-free identity α = Z_trans / Z_rot, recovering the fine-structure constant as a calibration input rather than a lattice prediction. In the gravitational sector, the same substitution enforces the infrared calibration identity G_N / G_raw = Q^2. These are consistency checks, not independent derivations. To formalize future tests, we construct a ten-condition falsifiability roadmap (BC1-BC10) defining explicit mathematical failure criteria. Of seven conditions evaluated, isotropy (BC2), substrate stability (BC8), and parity protection (BC9) pass necessary-condition checks, generic to any discrete lattice. Kinematic covariance (BC3) is satisfied at the toy-model level pending action-level convergence. As Q-specific results, bulk volume counting (BC1), Newtonian coupling strength (BC4), and fine-structure constant matching (BC5) identify specific gaps where microscopic derivations remain incomplete. The framework stands as a consistent effective theory pending resolution of the remaining open conditions.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-06
DOI
https://doi.org/10.5281/zenodo.22646572
Primary Topic
Noncommutative and Quantum Gravity Theories
Type
preprint
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preprint

Sub-Planckian Discrete Spacetime Calibrated by Dirac's Large-Number Ratio: Effective Field Theory Bounds, Calibration Identities, and a Ten-Condition Falsifiability Roadmap

Peter Luh
Zenodo (CERN European Organization for Nuclear Research)
Noncommutative and Quantum Gravity Theories
preprint

Sub-Planckian Discrete Spacetime Calibrated by Dirac's Large-Number Ratio: Effective Field Theory Bounds, Calibration Identities, and a Ten-Condition Falsifiability Roadmap

Peter Luh
preprint en

Abstract

We evaluate the continuum limit of a discrete spacetime substrate with fundamental spacing fixed by Dirac’s large-number ratio Q ≈ 4.17 × 10^42 at the sub-Planckian scale δ_p ≡ ℓ_P / Q ≈ 3.88 × 10^-78 m. Treating the substrate as a hypercubic lattice, spatial parity symmetry eliminates odd-order finite-difference operators, excluding dimension-5 Lorentz-violating corrections. The leading residual effect is a dimension-6 operator at O(δ_p^2) with a directional angular prefactor g(θ, ϕ), yielding a Q^-2 ~ 10^-84 suppression relative to Planck-scale cutoff models that comfortably satisfies ultra-high-energy cosmic-ray bounds and predicts zero vacuum birefringence. In the electromagnetic sector, inserting δ_p yields the scale-free identity α = Z_trans / Z_rot, recovering the fine-structure constant as a calibration input rather than a lattice prediction. In the gravitational sector, the same substitution enforces the infrared calibration identity G_N / G_raw = Q^2. These are consistency checks, not independent derivations. To formalize future tests, we construct a ten-condition falsifiability roadmap (BC1-BC10) defining explicit mathematical failure criteria. Of seven conditions evaluated, isotropy (BC2), substrate stability (BC8), and parity protection (BC9) pass necessary-condition checks, generic to any discrete lattice. Kinematic covariance (BC3) is satisfied at the toy-model level pending action-level convergence. As Q-specific results, bulk volume counting (BC1), Newtonian coupling strength (BC4), and fine-structure constant matching (BC5) identify specific gaps where microscopic derivations remain incomplete. The framework stands as a consistent effective theory pending resolution of the remaining open conditions.

Zenodo (CERN European Organization for Nuclear Research)
Phi Beta Kappa Society (US)
Noncommutative and Quantum Gravity Theories
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