2026/03/08 by Eric McLean · 1 voice
Physics and Astronomy · Mathematics · #Quantum and Classical Electrodynamics #Relativity and Gravitational Theory #Advanced Mathematical Theories
paper · doi:10.5281/zenodo.18912388
openalex publication_date 2026/03/08 · openalex created_date 2026/03/09 · openalex updated_date 2026/07/01
Quantum Mechanics from the 600-Cell: Measurement, Collapse, and the Born Rule Solved Pentagon Physics Programme Paper 36 This paper argues that the measurement problem in quantum mechanics is not a dynamical problem but a geometric one, and solves it accordingly. The physical state space is the 600-cell, a four-dimensional regular polytope with 120 vertices whose coordinates all lie in ℚ(√5). The 600-cell is not postulated. It is derived from the self-referential axiom σ = 1/(1+σ) via E₈, as established in earlier papers in this series. A 3D observer is by definition a hyperplane in ℝ⁴. Measurement is the 4D state touching that hyperplane. Measurement is therefore projection — not by assumption or analogy, but by definition. Given a 4D state space and a 3D observer, no other operation is geometrically available. From this, six results follow: The Born rule is not a postulate. It is the projection identity |⟨m|ψ⟩|² = 1 − v₄², automatic from the Pythagorean theorem once the state space is the 600-cell. The squaring is forced by Gleason's theorem: the quadratic form is the unique consistent probability measure on subspaces of dimension ≥ 3. The mean Born probability is exactly 3/4. By the spherical symmetry of S³, each coordinate has expected square 1/4. The three visible coordinates account for 3/4 of the state. The hidden fraction is exactly 1/4. This is the same 3/4 : 1/4 partition that appears in the Spiral Engine as the hydrogen-to-helium transmission ratio — the same geometric fact at two levels of the sublimation hierarchy. Hilbert space is the coordinate system. The 600-cell is the geometry. ℂ² is ℝ⁴ with complex bookkeeping. Every approach that worked within ℂⁿ without identifying the 600-cell was handling the right evidence inside the wrong frame. Superposition, collapse, wave-particle duality, and entanglement all dissolve. Superposition is what a 3D observer describes when the fourth coordinate is inaccessible. Collapse is not a process — it is a definite 4D vertex seen from ℝ³. Entanglement is shared structure in the fourth coordinate, locally present in the geometry, non-local only from inside three dimensions. Quantum mechanics is entirely a fourth-dimension phenomenon seen from inside three. The path integral is finite. The Dirac–Frenkel action restricted to the 600-cell is a sum over 720 edges, all weights in ℚ(√5). It converges without regularisation. QFT divergences are artefacts of a continuous state space that was never the correct object. The VAC/σ² cell in the Constants Table is identified. The coherence floor φ⁶ is derived from the 600-cell's angular structure: six steps of π/5 close the signed inner-product cycle. The 600-cell is the second geometric mode. Five falsification conditions are given. The sharpest: if any experiment specifically engineered to prepare and probe states with full icosahedral symmetry in four dimensions yields measurement probabilities outside 0, (5−√5)/8, 3/4, (5+√5)/8, 1, the framework fails. No such experiment has been performed. No laboratory has been looking for it. Part of the Pentagon Physics programme deriving all Standard Model constants from the single axiom σ = 1/(1+σ) with zero free parameters. Keywords: quantum foundations, Born rule, measurement problem, 600-cell, polytope, quantum geometry, E₈, icosahedral symmetry, Pentagon Physics, projection, fourth dimension