Admissible Non-Injective Transitions as the Primitive of Physical Description

This repository contains the source of the Foundation Cosmochrony paper
Admissible Non-Injective Transitions as the Primitive of Physical Description.

This work establishes the axiomatic foundation of the Cosmochrony framework.

It introduces a minimal set of structural principles from which key features of physics — including irreversibility, temporal order, quantum coherence, and emergent symmetry — are derived without postulating spacetime, Hilbert space, or dynamics.

Quick Summary

Foundation-1.2 shows that the structure of physical reality can be derived from a single primitive:

Admissible transitions between observable states under a non-injective projection.

From this primitive, the paper proves that:

• Irreversibility and temporal order arise from non-injectivity alone (A2)
• Proto-states (unresolved physical states) are real and structurally required (A3)
• Phase coherence is preserved because admissibility forbids premature selection
• Quantum discreteness emerges from projection locking (A4)
• Heisenberg structure is forced by non-factorisability of admissible fibres
• The observable space is identified with the Weil representation (V_\rho)

No quantum postulates are assumed.

Context

This paper provides the axiomatic backbone of the Cosmochrony programme.

It underpins:

• the non-injectivity no-go theorem
• the spectral admissibility programme (O-series)
• the quantum reconstruction papers

It replaces traditional starting points such as:

• background spacetime
• Hilbert space formalism
• dynamical evolution laws

with a purely relational and projective structure.

Core Structure

The Primitive

The only primitive object is:

[ (O_{n-1}, F_n) ]

where:

• (O_{n-1}): observable state
• (F_n): admissible successor directions

There is:

• no underlying substrate
• no hidden variables
• no external configuration space

The Four Axioms

The framework is governed by four independent axioms:

• A1 — Local projective admissibility
  Admissible successor directions exist and are structurally constrained

• A2 — Structural non-injectivity
  Distinct admissible directions may project to the same observable state

• A3 — Non-premature selection
  No selection among admissible directions occurs before saturation

• A4 — Projection locking
  Resolution occurs only when continuous constraints meet discrete structure

Main Results

7. Projective Incompleteness

• Every admissible projection (\Pi_n) has a non-trivial kernel (Corollary to Theorem 5.4)
• No observable description is ever complete
• Direct consequence of ([X, \sigma(X)] = Z \neq 0) (Theorem 5.4)

8. Threefold Role of (\Pi_n) (Remark 3.5)

(\Pi_n) simultaneously plays three roles, each irreducible to the others:

1. Admissibility filter: selects observable directions from structurally compatible continuations; the discarded kernel is real and guaranteed by projective incompleteness
2. Generator of temporal order: the oriented sequence (\Pi_0, \Pi_1, \ldots) carries the partial order of Theorem on temporal ordering; time is the topology of this graph, not an external parameter
3. Revelation, not creation: (\Pi_n) resolves finer spectral content already present in (\Omega_n^{(c)}); increasing (n) reveals structure, it does not create it

These three roles are algebraically inseparable: the non-commutativity ([X, \sigma(X)] \neq 0) is the common algebraic source of all three simultaneously.

1. Irreversibility and Temporal Order

• Non-injectivity ⇒ information loss
• Information loss ⇒ irreversibility
• Irreversibility ⇒ temporal ordering

Time is not fundamental but emerges from projection structure.

2. Proto-State and Wave Structure

• The proto-state is a real unresolved configuration

• It exists between two observable states

• It is the physical origin of:

  • wave-like behaviour
  • superposition
  • coherence

3. Phase Coherence

• Coherence is not postulated
• It is forced by admissibility (A3)

Destroying coherence would correspond to premature selection → forbidden

4. Emergence of Heisenberg Structure

• Admissibility forbids factorisation of the fibre
• This forces a non-trivial commutator structure

Combined with minimality and parity:

[ \text{Symmetry group} \simeq \mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}) ]

• Observable space:

[ F_n \simeq V_\rho ]

5. Discrete Quantum Transitions

• Continuous Born–Infeld constraint + discrete shell structure
• ⇒ Projection locking
• ⇒ intrinsic discreteness of transitions

No quantisation postulate is needed.

6. Entanglement and Bell

• Entanglement = shared ancestral fibre

• Bell factorisation fails because:

  • projection is non-injective
  • information is already lost upstream

No nonlocal dynamics is required.

What This Paper Establishes

This paper provides:

• a minimal and complete axiomatic system

• a derivation (not postulation) of core physical structures

• a unified origin for:

  • time
  • quantum coherence
  • discreteness
  • symmetry

It serves as the foundation layer for all subsequent Cosmochrony results.

Effective Co-Metric Completion (v1.10; status revised in v1.16)

The paper integrates the downstream result that the effective co-metric is fully explicit. Under the Q5a–Q5b geometric chain, with Q8 (Casimir rigidity: $A_z=2$), Q10 (spectral universality: $A_H=2$), and Q11 (temporal Casimir rigidity: $A_\tau=2$), the effective co-metric is:

$$g^{\mu\nu} = \mathrm{diag}(-2,,2,,2,,2).$$

Status revision (v1.16). Q5a version 3.0 withdraws the Mosco derivation of the spatial limit operator: the canonical filtration is exactly a growing toric Fourier window, the published admissibility form converges to the zero form on it, and no common scalar normalisation produces a non-trivial toric differential operator. The spatial limit operator is now the explicit, unestablished hypothesis [H-L] (Q5b 2.0); the co-metric completion above is conditional on [H-L] ($D_{\mathrm{hom}}=4$ remains structural), and Q5 is open.

Relation to Established Frameworks (v1.10)

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 and what it presupposes that the present framework derives:

• Hamilton–Jacobi dynamics: the eikonal equation $g^{\mu\nu}\partial_\mu S,\partial_\nu S = 0$ appears as an effective description of projected dynamics, valid once the effective metric has been reconstructed (Q5b, Q6b). It is downstream of the admissibility layer, not a primitive.
• Symplectic geometry: the phase space $T^*M$ is not a primitive — $M$ emerges from the Carnot–Carathéodory geometry of Q5b. Symplectic structure is induced by the principal symbol of the admissibility operator.
• WKB approximation: the WKB phase $S$ 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 flow.

What This Paper Does Not Do

• Does not derive full quantum field theory
• Does not construct spacetime geometry explicitly (handled by Q5a–Q6b)
• Does not address all gauge sectors beyond SU(2)

These are handled in companion papers and the O-series.

Keywords

Non-injective projection; admissibility; emergence; quantum foundations;
phase coherence; Born–Infeld constraint; Heisenberg group; Weil representation;
temporal ordering; irreversibility; projection entropy; proto-state

Position in the Programme

Foundation-1.2 is the entry point of the framework: