Non-Factorisability and the Emergence of Heisenberg Structure from Admissibility Constraints

Core contributions

• Non-factorisability theorem: proof that the admissibility constraints on the projection $\Pi$ cannot be expressed as a product of independent constraints on subsystems, establishing an irreducible global character.
• Forced non-Abelian structure: demonstration that non-factorisability forces a non-Abelian group structure on the relational graph of the projection.
• Identification as Heis₃: when the admissibility dimension is three, the non-Abelian structure is uniquely identified as the Heisenberg group $\mathrm{Heis}_3$, accommodating position ($X$), momentum ($P$), and central phase ($Z$) degrees of freedom.
• Structural origin of $[X, P] = i\hbar$: the Heisenberg commutation relation follows as a structural consequence of the non-factorisability, not as an axiom of quantum mechanics.
• Foundational role for the Q-series: the emergence of Heis₃ from admissibility provides the structural basis for the Q-series papers on quantum emergence, BFS geometry (Q5a), and Mosco convergence.

Non-factorisability as a structural principle

In the Cosmochrony framework, the projection $\Pi$ from the substrate $\chi$ to the observable space $\mathcal{O}$ is non-injective. The admissibility constraints are the conditions that a relational configuration must satisfy to be consistent with a valid projection.

The factorisability question is: can these constraints be decomposed into independent conditions on separate subsystems? A factorisable set of constraints would be compatible with an Abelian group structure — the relational degrees of freedom would be independent.

This paper shows that the admissibility constraints are non-factorisable: the constraints on position-like and momentum-like degrees of freedom are coupled through the phase. This coupling is precisely what generates non-Abelian structure, and the minimal non-Abelian structure consistent with a three-dimensional admissibility is $\mathrm{Heis}_3$.

The result has a concrete consequence: the uncertainty principle is not a postulate of quantum mechanics imported into the framework but a structural theorem about what kinds of relational configurations can be projected. The non-commutativity of $X$ and $P$ reflects the non-factorisability of the admissibility constraints.

Relation to the Cosmochrony programme

This paper is a foundational contribution to the quantum emergence programme within Cosmochrony. It provides the structural justification for the use of the Heisenberg group throughout the spectral admissibility series (O-series) and the quantum mechanics emergence papers (Q-series).

The Q5a Mosco convergence theorem (and its completion by H2) describes how the discrete Weil representation on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ converges to the Schrödinger representation of the continuous $\mathrm{Heis}_3$. The present paper explains why $\mathrm{Heis}_3$ is the relevant group in the first place: it is the minimal non-Abelian structure forced by the admissibility constraints.

Together with the spectral admissibility series, this paper forms part of the structural argument for why quantum mechanics — including the Heisenberg algebra, the Schrödinger equation, and the Born rule — emerges from the Cosmochrony framework without being postulated.

References

Jérôme Beau. Non-Factorisability and the Emergence of Heisenberg Structure from Admissibility Constraints. Preprint. doi:10.5281/zenodo.19635395