Admissible Non-Injective Transitions as the Primitive of Physical Description

Results and corrected status

• 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 route: within the paper's irreducibility and minimal-generation formalisation, the admissible pair cannot commute.
• Carrier selection (corrected): A1–A3 and BI parity do not imply that the commutator is central, so they do not select \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\).
• Weil representation (conditional): the Stone–von Neumann endpoint remains valid once the finite Heisenberg group and a non-trivial central character are supplied. The antecedent is not derived.
• 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 determined, conditionally (v1.16): the results table records the downstream result that the effective co-metric is \(g^{\mu\nu} = \mathrm{diag}(-2,\,2,\,2,\,2)\), obtained jointly by Q5b, Q6b, Q8 (Casimir rigidity), Q10 (spectral universality), and Q11 (temporal Casimir rigidity) — conditional on the unestablished spatial limit hypothesis [H-L] since the Q5a 3.0 withdrawal; the homogeneous dimension \(D_{\mathrm{hom}}=4\) remains structural.

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 (conditional on [H-L]).
• Functional renormalisation group: the conjectured Mosco limit of the admissibility Dirichlet forms (hypothesis [H-L]) would play a structurally analogous role to the Wetterich effective average action — both describe an infrared fixed point of a renormalisation flow.

Significance: keeping identifications explicit

In the original Cosmochrony formulation, the relational substrate \(\chi\), the relaxation mechanism, and the Heisenberg group structure were postulated as starting points. Foundation attempted to reduce that axiomatic footprint; the carrier-selection part of that attempt does not survive the finite countermodel.

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

• temporal-ordering and other independent consequences keep their stated hypotheses;
• the Heisenberg/Weil carrier is a supplied realization, not a derived theorem;
• the Heisenberg uncertainty relation is conditional on that supplied carrier.

This correction does not invalidate calculations internal to the supplied carrier. It identifies the missing bridge and prevents those calculations from being mistaken for a derivation of their input.

Status and open directions

Corrected: the algebraic identification of the admissible fibre as \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) with Weil representation \(V_\rho\) is not proved.

Open: carrier selection comes first. The continuum limit and passage from the discrete group \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) to the continuous Heisenberg group \(\mathrm{Heis}_3(\mathbb{R})\) — Q5a version 3.0 withdraws the former Mosco route and proves the obstruction; the spatial limit is the open hypothesis [H-L] (Q5 open).

Open directions include:

• Carrier selection: add an independently motivated hypothesis that excludes the \(S_3\) countermodel, or declare the Heisenberg/Weil carrier as a model input.
• 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 → carrier obstruction → 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

Status correction and countermodel: doi:10.5281/zenodo.21710123.