What question did this study set out to answer?

Investigate whether the three spatial dimensions derived from discrete Heisenberg group geometry can be rigorously identified with the three-dimensional neutral spin sector in Cosmochrony.

August 16, 2026Open Access

Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate between Heff ≃ C³ and the Horizontal Geometry of Heis3(R)

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Investigate whether the three spatial dimensions derived from discrete Heisenberg group geometry can be rigorously identified with the three-dimensional neutral spin sector in Cosmochrony.
Applied Schur's lemma and representation theory to analyze equivariant maps between the semisimple algebra su(2) and the nilpotent Heisenberg algebra heis3.
Formulated the Schrödinger Quadratic Form Lemma to evaluate Casimir operator pullbacks under Carnot and spin-weight gradings.
Performed numerical checks across prime checkpoints q in {61, 101, 151, 211} to calculate cross-term vanishing and power-law scaling of the spatial isotropy gap.
Proved that the irreducibility of the spin-1 module Sym²(V_rho) forces any equivariant map to the spatial sector to be unique up to a positive scalar.
Established that the chirped discrete Fourier transform commutes with L_Weil for all (q, c), structurally eliminating cross terms in the effective symbol.
Demonstrated numerical convergence of the isotropy gap via power-law fit |A_H - A_z| ≈ 0.364 q^(-0.52) (R² = 0.98), confirming consistent O(q^(-1/2)) decay toward asymptotic isotropy A_H = A_z = 2.

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Implication

Theoretical analysis demonstrates a unique equivariant bridge between spin-1 representations and Heisenberg geometry, indicating asymptotic spatial isotropy across discrete scales.

Cite This Study

Jérôme Beau (2026) studied this question.

synapsesocial.com/papers/6a817a80f2fb91fc834ae68f https://doi.org/10.5281/zenodo.21947516

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Implication

Theoretical analysis demonstrates a unique equivariant bridge between spin-1 representations and Heisenberg geometry, indicating asymptotic spatial isotropy across discrete scales.

Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate between Heff ≃ C³ and the Horizontal Geometry of Heis3(R)

Key Points

Discussion

Implication

Cite This Study

Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate between Heff ≃ C³ and the Horizontal Geometry of Heis3(R)

Key Points

Discussion

Implication

Cite This Study

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