Admissible Non-Injective Transitions as the Primitive of Physical Description

Results and status

• Information loss and temporal order (A1+A2): a non-injective transition loses information in its immediate image and admits no inverse (Proposition 4.1). Neither permanent non-recovery nor the arrow of time follows from this: temporal order holds only under the acyclicity hypothesis [H-acyc] (Theorem 4.4).
• Proto-state (defined; indeterminacy derived from A1, A3): the proto-state is defined as the unresolved configuration shared between two successive states; A1 and A3 make it structurally indeterminate among its admissible directions. Reading it as a genuine intermediate physical state, not an epistemic placeholder, is the paper's interpretation.
• Phase coherence (not derived): A3 forbids premature selection among open directions, so it preserves their multiplicity, but it does not by itself forbid an erasure of their relative phases that selects no direction. Relative phases need the supplied complex carrier, and their coherence follows from A3 only under an additional reading that is not derived.
• Irreducibility and non-trivial commutator (conditional): given a representation dictionary, A1 and A3 make the admissible generators act irreducibly only under an additional sector reading of A3, not derived, on which a decomposition of the fibre stable under those generators counts as premature selection (Lemma 5.6). Under that condition, a pair generating the acting group cannot commute on a fibre of dimension greater than one.
• Carrier selection: A1–A3 and BI parity do not imply that the commutator is central, so they do not select \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\).
• Representation data (conditional): once the finite Heisenberg group and a non-trivial central character are supplied, Stone–von Neumann identifies the Heisenberg/Schrödinger representation $\pi_c$ on $\mathcal H_c$. The associated Weil action is distinct further symplectic data. The antecedent and that action are not derived.
• Discrete quantum transitions (postulated by A4): the discreteness of resolution is stated by A4, not derived. On the supplied Heisenberg shell realisation, a realised Born–Infeld saturation depth is an integer shell depth (O22); the existence of such a depth is not established (Theorem 5.12).
• Projective incompleteness (Corollary 4.9): under A2 alone, a non-injective realisation map \(\Pi_n\) has a non-trivial fibre, so its observable description is incomplete. This corollary does not guarantee the initial transition: A1 posits admissible directions, but the existence of a successor satisfying all four axioms is a distinct condition (Remark 4.11).
• Threefold role of \(\Pi_n\) (Remark 3.1): \(\Pi_n\) simultaneously acts as (i) admissibility filter, (ii) generator of temporal order, conditionally on [H-acyc], and (iii) revelation operator (not creation), an interpretive reading: A1–A4 do not decide whether the distinctions resolved are revealed, transferred or produced. These roles do not by themselves select the carrier group.
• Projective Observability Principle (Remark 2.6): identifying the physical content with the invariants of \(\Pi_n\) is a principle adopted by the paper, not a consequence of A1–A4.
• Effective co-metric: no coefficient value is established, and \(g^{\mu\nu} = \mathrm{diag}(-2,\,2,\,2,\,2)\) is not derived. In Q10 the \(\mathfrak{su}(2)\) Casimir value 2 determines no coefficient and the uniformity hypothesis [U] is not established; Q11 shows that spatial \(\mathrm{SU}(2)\) invariance leaves \(A_\tau/A_H\) unconstrained. The co-metric is conditional on the unestablished spatial limit hypothesis [H-L] and on the open lifting hypothesis [H-lift] of Q5b. The homogeneous dimension \(D_{\mathrm{hom}}=4\) is a property of the supplied Heisenberg carrier, holds on the finite group only for balls of radius \(n \ll \sqrt{q}\), and is not identified with the \(3+1\) structure of spacetime.
• Complex amplitudes (conditional): a classical probability mixture does not represent the phase relations of the proto-state, but this does not exclude real amplitudes; real and quaternionic formulations exist, and whether they are distinguishable depends on the rule for composing subsystems. The complex structure enters with the supplied carriers, the Heisenberg–Schrödinger carrier and the spinor carrier of O23; no selection of complex amplitudes from the axioms is established. Phase invariance, positivity and normalisability do not single out the squared norm, so the Born weight is an additional rule.
• Gravity and gauge dynamics (conditional): on a supplied metric, the Einstein tensor is the dominant local two-derivative term of the renormalized response when the matched Einstein coefficient is non-zero and \(R L_4^2 \ll 1\), a sufficient condition that does not bound the non-local terms (Gravity); the source-free Yang–Mills equations hold at a supplied base metric for a supplied compact gauge group (Q12); the joint Einstein–Yang–Mills system, with its cosmological term, is conditional on the matched coefficients and Euclidean (Q13). The couplings are matching data, and no hierarchy between them is derived.
• Gauge structure: the \(\mathrm{SU}(2)\) gauge structure is conditional on the spinor carrier supplied by O23, and no identification of the Standard Model gauge group \(\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)\) is established; O31 version 2.0 is a withdrawal notice.

Relation to established frameworks

A further 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, takes conditionally, or supplies. In the present paper the emergence is partial, on supplied carriers, and the geometric continuum is conditional on [H-L].

• 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), conditionally on [H-L] and the open lifting hypothesis [H-lift]. It is downstream of the admissibility layer, not a primitive.
• Symplectic geometry: the phase space \(T^*M\) is not primitive — \(M\) would emerge from the Carnot–Carathéodory geometry of Q5b, conditionally on [H-L], and the cotangent structure would be induced by the principal symbol of the admissibility operator, under the further open hypothesis [H-lift]. Symplectic geometry describes the effective dynamics on \(T^*M\) once the configuration space is constructed.
• WKB approximation: the WKB phase is an effective quantity defined on the supplied carrier, not derived from the axioms; 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. The companion papers evaluate the spectral entropy functional on supplied geometric and gauge data, not on a constructed limit operator, and the production of field content by the limit is not established.

Significance: keeping identifications explicit

In the Cosmochrony white paper, the relational substrate \(\chi\), the relaxation mechanism, and the Heisenberg group structure are postulated as starting points. Foundation reduces part of that axiomatic footprint, but not the carrier: a finite \(\mathfrak{S}_3\) countermodel satisfies the extracted carrier contract without a central commutator, so A1–A3 and BI parity do not select the Heisenberg group.

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

Calculations internal to the supplied carrier are unaffected. The missing bridge is identified, which prevents those calculations from being mistaken for a derivation of their input.

Status and open directions

Not proved: the algebraic identification of the admissible fibre as \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) and its representation data. Stone–von Neumann applies only after the group and central character are supplied, and does not itself construct the associated Weil action.

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 3.3 proves that the published form has a zero-form Mosco limit 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 of TopologicalInvariants 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 (including the carrier obstruction) → noscale

The Q-series then studies quantum-mechanical structure, an \(\mathrm{SU}(2)\) sector and Lorentzian spacetime on top of the Foundation axioms combined with O-series spectral data, on supplied carriers and under explicit hypotheses; the \(\mathrm{SU}(2)\) sector rests on the spinor carrier supplied conditionally by O23, which the axioms do not select.

References

Jérôme Beau. Admissible Non-Injective Transitions as the Primitive of Physical Description, 2026. doi:10.5281/zenodo.19616329

Carrier-selection obstruction and countermodel: doi:10.5281/zenodo.19635395.