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.
Version 2.1.0 corrects the status of structural non-injectivity and temporal order. A2 remains a local axiom in its generic form unless the applicability conditions of the ENI theorem are established for the local transition map. Immediate information loss is proved only on non-injective steps, permanent non-recovery remains open, and a strict temporal order requires the unestablished acyclicity hypothesis [H-acyc]. The carrier-selection correction introduced in version 2.0.0 remains in force.
It introduces a minimal set of structural principles from which key features of physics — including immediate information loss, a conditional temporal order, quantum coherence, and emergent symmetry — are derived without postulating spacetime, Hilbert space, or dynamics.
The paper studies how much structure follows from a single primitive:
Admissible transitions between observable states under a non-injective projection.
From this primitive, the paper proves that:
- Loss in the immediate image arises from non-injectivity alone (A2), along the transitions A2 asserts to be generically non-injective
- Temporal order is conditional: non-invertible steps admit cycles, so orienting the chain needs an acyclicity hypothesis [H-acyc] that is not derived
- 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)
- Under an explicit representation dictionary, non-factorisability implies irreducibility and non-commutation
- These conditions do not select a Heisenberg group: an explicit (\mathfrak S_3) countermodel has a non-central commutator
- The Heisenberg carrier (\mathcal H_c) used by the O-series is a supplied, conditional realisation, distinct from its associated Weil module (V_\rho)
No quantum postulates are assumed.
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.
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 framework is governed by four axioms with distinct roles (no independence theorem is claimed):
-
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
- Every admissible projection (\Pi_n) that is non-injective has a non-trivial kernel, which A2 asserts to be the generic case
- No description resting on a non-injective step is complete; A2 being generic, an exceptional injective transition is not excluded
- Direct consequence of structural non-injectivity (A2): distinct admissible configurations share an observable image, so their difference is a non-zero kernel element. No commutator or carrier-selection theorem is involved.
(\Pi_n) simultaneously plays three roles, each irreducible to the others:
- Admissibility filter: selects observable directions from structurally compatible continuations; where the step is non-injective, the discarded kernel is real and guaranteed by projective incompleteness
- Generator of temporal order: the oriented sequence (\Pi_0, \Pi_1, \ldots) carries the partial order of the temporal-ordering theorem under the acyclicity hypothesis [H-acyc]; time would then be the topology of this graph, not an external parameter
- 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 inseparable, but their common source is the axiomatic layer, not an algebraic commutator: the constraints A1--A3 and Born--Infeld parity that force the admissibility filter also generate the temporal order under [H-acyc] and, for the non-injective transitions A2 asserts to be generic, guarantee the incompleteness of the resolution. They do not by themselves select the carrier group.
- Non-injectivity ⇒ information loss in the immediate image
- No inverse ⇒ the step is not reversible
- Steps compose into an order only under [H-acyc]: two different non-injective maps can carry (O_0 \to O_1) and (O_1 \to O_0), giving a cycle and no strict partial order
Time is not fundamental. It would emerge from projection structure once [H-acyc] is supplied; the immediate loss at each step is proved, the orientation of the whole chain is not.
Permanent non-recovery is not established: projection entropy is non-negative step by step, which is not a monotonicity statement along the chain. It would require a temporal sufficiency property that the framework does not yet supply.
-
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
- Coherence is not postulated
- It is forced by admissibility (A3)
Destroying coherence would correspond to premature selection → forbidden
- With a representation-theoretic dictionary, admissibility forbids invariant factorisation of the fibre
- In dimension greater than one, irreducibility forces the generating pair to not commute
But minimal generation and parity do not make the commutator central:
[ \mathfrak S_3,\quad X=(12),\quad Y=(23),\quad [X,Y]=(132)\notin Z(\mathfrak S_3) ]
Thus the finite Heisenberg group and its carrier must be supplied by an additional typed hypothesis. Stone–von Neumann then fixes the Heisenberg carrier (\mathcal H_c) up to unitary equivalence; it does not identify that action with the distinct Weil action on (V_\rho).
- Continuous Born–Infeld constraint + discrete shell structure
- ⇒ Projection locking
- ⇒ intrinsic discreteness of transitions
No quantisation postulate is needed.
-
Entanglement = shared ancestral fibre
-
Bell factorisation fails because:
- projection is non-injective
- information is already lost upstream
No nonlocal dynamics is required.
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.
The paper integrates the downstream result that the effective co-metric is fully
explicit. Under the Q5a–Q5b geometric chain, with Q8 (Casimir rigidity:
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] (
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.
- 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)
- Does not derive the finite Heisenberg carrier from A1–A4
These are handled in companion papers and the O-series.
Non-injective projection; admissibility; emergence; quantum foundations;
phase coherence; Born–Infeld constraint; Heisenberg group; Weil representation;
temporal ordering; irreversibility; projection entropy; proto-state
Foundation-1.2 is the entry point of the framework: