Overview
Q7 addresses a central structural question of the Cosmochrony programme: the coincidence between two independently derived three-dimensional structures.
On the one hand, Q5b derives a four-dimensional Lorentzian geometry with three spatial directions from the Carnot structure of \[ \mathrm{Heis}_3(\mathbb{R}). \]
On the other hand, O23–O29 establish that the admissible projection space satisfies \[ H_{\mathrm{eff}} \simeq \mathbb{C}^3, \] corresponding to three stable admissible directions.
The central objective of Q7 is: to determine whether these two “3”s are the same structure, and if so, under which constraints.
Core contributions
- Algebraic obstruction: proves that $\mathfrak{su}(2)$ and $\mathfrak{heis}_3$ are non-isomorphic, forbidding any direct identification at the Lie algebra level.
- Representation-theoretic identification: establishes \[ H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho), \] the spin-1 irreducible representation.
- Rigidity of the bridge: by Schur’s lemma, any equivariant identification is unique up to scalar.
- Quadratic form constraint: the $\mathfrak{su}(2)$ Casimir induces a positive-definite quadratic form matching the spatial metric structure.
- Metaplectic symmetry result: proves that \[ [F_c, L_{\mathrm{Weil}}] = 0, \] implying structural absence of cross terms.
- Reduction to a single condition: the identification holds if and only if \[ A_H = A_z. \]
Interpretation
Q7 transforms the status of the spatial dimensionality problem within the programme.
- Before Q7: two independent “3”s coexist
- After Q7: their relation is fully constrained
The problem is no longer conceptual but spectral:
- either the two structures coincide via a unique bridge
- or they are definitively distinct
In both cases, the ambiguity is removed.
Relation to the Cosmochrony program
Q7 connects two major branches of the programme:
- O-series: admissible structure and quaternionic minimality
- Q-series: emergent geometry from spectral limits
It provides the missing bridge between representation theory and geometry, reducing their compatibility to a single testable condition.
Current outcome and open directions
Q7 establishes that the identification problem reduces to: \[ A_H = A_z. \]
Numerical evidence shows:
- vanishing cross terms (proved analytically)
- \(A_z \approx 2\), matching the Casimir
- \(|A_H - A_z|\) decreasing with \(q\)
Remaining directions include:
- Asymptotic verification: test convergence for larger primes.
- Analytical closure: compute the sub-principal symbol in the Z-direction.
- Geometric implications: determine whether spatial isotropy is exact or emergent.
References
Jérôme Beau. Three Admissible Directions and Three Spatial Dimensions: A Structural Bridge Candidate, 2026.