Admissible Non-Injective Transitions as the Primitive of Physical Description

Core derivations from axioms A1–A4

• Arrow of time (A1+A2): irreversibility follows structurally from the combination of local admissibility and non-injectivity — the projection cannot be inverted, so the effective description acquires a direction.
• Proto-state as physically real (A3): the unresolved configuration retaining phase coherence is not an epistemic placeholder but a genuine intermediate physical state.
• Non-trivial commutator (A1+A3): a non-trivial commutator between conjugate generators is forced by A3 and the minimality of A1, yielding \([X,P] \neq 0\) without postulating it.
• Heisenberg group (A1–A3 + BI parity): the admissible fibre symmetry group is identified as \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) for a prime \(q\).
• Weil representation (Theorem 5.6): the identification \(F_n \cong V_\rho\) — the fibre as the Weil representation — is derived as a theorem, not postulated. The detailed proof is provided in the companion paper HeisenbergStructure.
• Discrete quantum transitions (A4): the discrete, locally finite character of admissible transitions follows directly from the fourth axiom.
• Projective incompleteness (Corollary to Theorem 5.4): every admissible projection \(\Pi_n\) has a non-trivial kernel — no observable description is ever complete. This follows directly from \([X, \sigma(X)] = Z \neq 0\).
• Threefold role of \(\Pi_n\) (Remark 3.5): \(\Pi_n\) simultaneously acts as (i) admissibility filter, (ii) generator of temporal order, and (iii) revelation operator (not creation). These three roles are algebraically inseparable and all sourced in the non-commutativity of the fibre.
• Effective co-metric fully determined (v1.13): the results table now records the downstream result that the effective co-metric is \(g^{\mu\nu} = \mathrm{diag}(-2,\,2,\,2,\,2)\), established jointly by Q5b, Q6b, Q8 (Casimir rigidity), Q10 (spectral universality), and Q11 (temporal Casimir rigidity).

Relation to established frameworks (v1.13)

A new section situates the four axioms A1–A4 relative to established formalisms — not by reduction, but by structural translation: identifying what each framework corresponds to in the emergent hierarchy and what it presupposes that the present framework derives.

• Hamilton–Jacobi dynamics: the eikonal equation \(g^{\mu\nu}\partial_\mu S\,\partial_\nu S = 0\) is an effective description of projected dynamics, valid once \(g^{\mu\nu}\) has been reconstructed from the principal symbol (Q5b, Q6b). It is downstream of the admissibility layer, not a primitive.
• Symplectic geometry: the phase space \(T^*M\) is not primitive — \(M\) emerges from the Carnot–Carathéodory geometry of Q5b, and the cotangent structure is induced by the principal symbol of the admissibility operator. Symplectic geometry describes the effective dynamics on \(T^*M\) after the configuration space is constructed.
• WKB approximation: the WKB phase is a derived quantity encoding the projective compression of the admissible fibre; geometric optics is the ray approximation of admissible propagation in the continuum limit (Q5a).
• Functional renormalisation group: the Mosco limit of the admissibility Dirichlet forms (Q5a) plays a structurally analogous role to the Wetterich effective average action — both describe an infrared fixed point of a renormalisation flow.

Significance: from postulates to theorems

In the original Cosmochrony formulation, the relational substrate \(\chi\), the relaxation mechanism, and the Heisenberg group structure were postulated as starting points. The Foundation paper reverses this: it shows that all of these emerge from a smaller set of structural commitments.

The four axioms A1–A4 are not physical laws — they are minimal commitments about what a description of physical transitions must satisfy. From these commitments alone:

• the substrate \(\chi\) and relaxation emerge as consequences;
• the Heisenberg group structure is not assumed but derived;
• quantum uncertainty \([X,P] = i\hbar\) is not a postulate but a structural theorem.

This refoundation significantly narrows the axiomatic footprint of the programme and strengthens the claim that the quantum-mechanical formalism is structurally unavoidable rather than chosen.

Status and open directions

Proved: algebraic identification of the admissible fibre as \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) with Weil representation \(V_\rho\) (Theorem 5.6).

Structurally motivated: continuum limit and the passage from the discrete group \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) to the continuous Heisenberg group \(\mathrm{Heis}_3(\mathbb{R})\) — this is the subject of the Q5a programme.

Open directions include:

• Canonical model extension: \(\pi_2(\mathcal{C}_{\mathrm{eff}}) = 0\) beyond the canonical model (Remark 4.2) remains open.
• Matter sector: derivation of fermionic structure from A1–A4 without additional input.

Relation to the programme

The Foundation paper sits at the top of the logical dependency hierarchy. The recommended foundational reading path is:

ENI → Foundation → HeisenbergStructure → noscale

The Q-series then builds quantum mechanics, \(\mathrm{SU}(2)\) symmetry, and Lorentzian spacetime on top of the Foundation axioms combined with O-series spectral data.

References

Jérôme Beau. Admissible Non-Injective Transitions as the Primitive of Physical Description, 2026. doi:10.5281/zenodo.20258438