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Published July 10, 2026
| Version v17
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The Emergence of Time and Space
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Fogeborg, Robert<br>(Researcher)
Description
This paper derives the known architecture of particle physics from one rule: two states, one transition, no null. The rule is a type with exactly two values, a single operation that flips between them, and the theorem that no state equals its opposite. That is the entire premise. From this rule, checked by the Lean 4 proof assistant in 205 machine-verified theorems, the following are forced: • Three spatial dimensions — forced because fewer axes cannot enclose a loop.
• One time dimension — three temporal axes unified into one shared arrow<br>• Spin-½ — an odd-length loop returns inverted after one turn, needing 720°<br>• The Pauli exclusion principle — exchange antisymmetry forces identical fermions apart<br>• Three quark colors — the structural mirror of space’s three dimensions<br>• Eight gluons — 3²−1 from the swap-minus-singlet construction<br>• Three weak bosons — 2²−1 from the same construction on the contrast pair<br>• One photon — the unified phase needs no internal swaps<br>• Twelve fermion flavors — |Role|×|Axis3|×|Bool|<br>• Fractional charges ⅓ and ⅔ — the unique integer partition of 3² = 9<br>• Charge quantization — closure forces whole-unit neutral composites<br>• An inverse-square law — flux in three dimensions spreads as 1/r²<br>• A universal speed limit — the floor below which no transition can be subdivided<br>• Time dilation — motion and aging share one capped budget<br>• Inverse fine-structure constant α−1 = 137 — exactly from 4·8³+8²+8 = 2120, minus the 6D tensor deficit 6⁴+2 = 1298, divided by 6 axes: (2120−1298)/6 = 822/6 = 137 (FORCED)<br>• Total synchronized state boundary 822 = 137×6 across the six Z6 lattice axes (FORCED)<br>• Hexagonal normalization 822/36 = 137/6 = 22.833… — per-axis per-direction state density (FORCED)<br>• Proton/electron mass ratio integer 1836 — from structural composition 2·822+137+36+19 of Z6 lattice constants (SUGGESTIVE)<br>• Mass ratio coefficient 6 = 3 quarks × 2 spin, exponent 5 = 6 ambient axes − 1 anchor (FORCED)<br>• A dark-sector candidate at phase 3 — the unique nonzero phase invisible to all gauge symmetries<br>• Exact matter/antimatter balance forbidden — would require the null state the seed excludes<br>• Parity violation forced in dynamics — negation cannot commute with the tick in ℤ₆<br>• Neutrino: unique unpinned generation, unique self-conjugate flavor<br>• CHSH classical bound |CHSH| ≤ 2 proven — naive deterministic reading cannot exceed it<br>• Graviton count 0 — swap-minus-singlet on contrast-free substrate yields zero carriers<br>• Mass-exponent sector mapping — 8=colorCount, 4=weakTotal, 3=spatialDim<br>• Gauge dimension match SU(3)×SU(2)×U(1): 8+3+1 = 12<br>• Residual curvature axis = 1 (6 − 3 space − 1 time − 1 anchor)<br>• Kochen-Specker structural root: -(x+1) ≠ -x+1 in ℤ₆ — contextuality from tick-noncommutativity<br>• ΠZ → π as the unreachable continuum limit of a discrete angular invariant
The Standard Model does not derive these numbers; it selects its gauge group and parameters to fit them. Here every number is a theorem, not an input.
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EuroSciVoc
Quantum physics
Mathematical physics
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Physics
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10.5281/zenodo.21294371
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