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

Overview

This paper attempted to derive the Heisenberg group from admissibility constraints on the projection $\Pi$. Its non-factorisability setup remains a structural proposal, but the carrier-selection theorem is not established.

The key mechanism is non-factorisability. The admissibility constraints on $\Pi$ cannot be expressed as a product of independent constraints on subsystems — they are irreducibly global. This can motivate non-commutativity, but it does not imply a central commutator or a class-two nilpotent group.

The finite $S_3$ countermodel has a faithful irreducible unitary carrier and a minimal generating pair exchanged by an involution, yet its non-trivial commutator is not central. Therefore neither $\mathrm{Heis}_3$ nor $[X,P]=i\hbar$ follows from the published premises.

Status correction. The carrier-selection theorem is refuted. The final Stone–von Neumann step remains valid conditionally when a finite Heisenberg group and a non-trivial central character are supplied.

Audited argument

Non-factorisability setup: formalisation in which the admissibility constraints on the projection $\Pi$ cannot be expressed as a product of independent constraints on subsystems.
Non-commuting generators: a structural route to a non-trivial commutator within the paper's formalisation.
Failed centrality step: a non-trivial commutator is promoted to a central one without a valid implication. The $S_3$ countermodel refutes this step.
Reversed Schur step: irreducibility constrains central operators; it does not make an arbitrary commutator central or prove prime central order.
Conditional endpoint: after supplying a finite Heisenberg group and non-trivial central character, Stone–von Neumann theory identifies the irreducible carrier.

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 proposes that the admissibility constraints are non-factorisable: the constraints on position-like and momentum-like degrees of freedom are coupled through the phase. That proposal can motivate non-Abelian structure, but the minimality claim does not select $\mathrm{Heis}_3$.

Accordingly, the uncertainty relation is conditional on a supplied Heisenberg carrier. Non-commutativity alone does not determine its central constant or representation.

Relation to the Cosmochrony programme

This paper documents the attempted structural justification for the Heisenberg carrier. The O-series and Q-series calculations that use that carrier remain mathematically meaningful, but their carrier is a supplied realization.

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 does not explain why $\mathrm{Heis}_3$ is selected in the first place; that bridge is open.

The correction is local: it changes the status of the carrier-selection edge, not every theorem proved after the carrier has been supplied.

References

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

Countermodel and status correction: doi:10.5281/zenodo.21710123