Emergence–Edge Cosmology in Pattern Field Theory

A curvature-replication model of large-scale structure, Trishift dynamics, and collapse-based formation.

Abstract

Emergence–edge cosmology in Pattern Field Theory (PFT) constructs a complete cosmological framework based on the Allen Orbital Lattice (AOL) as a discrete curvature carrier. Large-scale structure grows by curvature replication at the cosmological edge, and observable shifts accumulate through a three-component Trishift field rather than a recession-based redshift. Internal instability regions generate collapse events with two endpoints: PCS-L (finite prestellar collapse) and PCS-H (horizon-anchored black-hole states). These collapse channels seed structure, regulate curvature flow, and provide a load-balancing mechanism across the manifold. The model delivers a falsifiable alternative to the distance–redshift relation of ΛCDM via a replication integral and Trishift slope condition, and defines an observational program for testing PFT cosmology as an independent framework.

1. Foundations of Pattern Field Theory Cosmology

1.1 Overview

PFT cosmology begins from a discrete, resonance-governed substrate rather than a prior smooth metric. The basic objects are:

  • AOL (Allen Orbital Lattice) – a prime-indexed hexagonal lattice carrying curvature, phase, and coherence weights.
  • Differentiat – the continuity-governor that enforces identity-preserving change and assigns window budgets to cascades.
  • Event Cascades – sequences of Phase Alignment Lock (PAL) driven transitions that move curvature and structure through the lattice.
  • Equilibrion – the identity-stable state between cascades.

Cosmology in this setting follows the same principles as micro-scale dynamics: continuity, discrete cascades, and curvature-driven structure.

1.2 AOL, PAL, CCE

The Allen Orbital Lattice is the discrete carrier on which curvature resides. Each site carries phase, coherence weight, and adjacency rules, with prime indexing controlling allowed closure patterns. Phase Alignment Lock (PAL) synchronises regions for joint cascades, while the Curvature Conservation Envelope (CCE) ensures that total curvature is tracked and redistributed in controlled ways. The cosmological model builds on these three ingredients:

  • AOL as geometry carrier.
  • PAL as ignition and phase synchronisation for large-scale cascades.
  • CCE as the global book-keeping of curvature flow.

1.3 Null

Null is the pre-geometric state with no coordinates, no metric, and no temporal progression. It contains tendency to structure but no expressed geometry. There is no vacuum, no background, and no space in the usual sense. Null is the precondition out of which the first geometric event emerges.

1.4 From Null to Geometry

The first geometric object in PFT cosmology is a π-loop: a closed curvature loop whose closure ratio equals π. One π-loop is the first dimension-bearing event. Its existence supports:

  • Geometric persistence.
  • Measurable curvature.
  • A basis for substrate formation.

Once a π-loop exists, copies can tile into a 2D substrate. From there, curvature stacking and resonance stability allow a lift into a 3D manifold carried by the AOL.

1.5 Departure from Metric Expansion

PFT cosmology does not rely on an expanding metric, scale factor, or cosmological constant. Distances increase because curvature replicates at the edge of the occupied region, not because a pre-existing metric stretches. Observable frequency shifts accumulate from Trishift dynamics along propagation paths. The cosmological engine is emergence and replication from Null through π-loop closure, substrate formation, and 3D lift, followed by ongoing curvature replication at the edge.

2. Emergence–Edge Architecture

2.1 Core Sequence

The emergence–edge architecture is a fixed sequence:

  1. Null – pre-geometric state with dormant fractal tendency.
  2. π-loop – first curvature loop; origin of geometry.
  3. 2D substrate – tiled π-loops and early lattice formation.
  4. 3D lift – dimensional extension to a volumetric AOL carrier.
  5. Fractal stabilisation – refinement into a stable, self-similar carrier lattice.
  6. Curvature replication – controlled copying of curvature rings outbound.
  7. Edge conversion – conversion of replicated curvature into structure at the cosmological edge.

This architecture replaces the role of inflation. Early rapid replication phases fill out a finite, structured, nearly isotropic carrier without requiring an inflationary metric stage.

2.2 Emergence Front and Edge

The manifold grows by curvature replication. The emergence front is the active boundary where new curvature shells appear on the AOL. Behind the front, structure forms as cascades traverse the lattice, producing galaxies, voids, and filaments. The cosmological edge is the farthest radius where curvature has replicated and converted into expressible geometry.

2.3 Baseline Symmetry

The emergence–edge process yields a baseline statistical isotropy, because replication proceeds ring by ring and defects are local. Large-scale homogeneity does not come from stretching a pre-existing uniform fluid, but from the regularity of replication and the averaging of many local cascades on the AOL.

3. Trishift Cosmology

3.1 Replication Integral

The key scalar driving observable shifts is the curvature replication integral along a radial path of length d: 

χ(d) = ∫₀ᵈ R(s) η(s) ds

Here R(s) is the replication rate and η(s) is an efficiency field that accounts for conversion, dissipation, and local conditions. χ(d) is the accumulated curvature replication along the path.

3.2 Trishift Field

The Trishift field T(d) is a three-component observable shift: 

T(d) = (T₁(d), T₂(d), T₃(d))
  • T₁(d) – spectral component (frequency/wavelength shift).
  • T₂(d) – temporal component (clock drift / timing shift).
  • T₃(d) – geometric component (apparent angular and size distortions).

Each component is derived from χ(d) through channel-specific response functions. The spectral slope condition 

dT₁ / dd = H_pft · η(d)

acts as a falsifiable alternative to the distance–redshift relation of ΛCDM. H_pft is an effective PFT Hubble-like parameter derived from replication, not from metric expansion.

3.3 Trishift Flow

Trishift flow along a given line of sight follows the local replication history, medium properties, and geometry. Different channels (CMB, supernovae, lensing) probe different combinations of T₁, T₂, and T₃, yielding a multi-channel constraint set on χ(d) and η(d).

4. Observational Mapping

4.1 CMB Mapping

In PFT cosmology, the cosmic microwave background (CMB) encodes the history of curvature replication and Trishift accumulation rather than a relic of a hot, dense early phase of an expanding fluid. Anisotropies correspond to variations in replication history and local efficiency η(d). The multipole spectrum arises from AOL-scale interference, defect structure, and cascades.

4.2 Large-Scale Structure

Galaxy clustering, filaments, and voids emerge from curvature replication and event cascades on the AOL. Underdensity seeds collapse zones, while over-loaded regions anchor PCS-H endpoints. Observed two-point and higher-order statistics map to the distribution of collapse channels and replication histories in the lattice.

4.3 Multi-channel Pipeline

A full PFT observational program uses:

  • CMB temperature and polarization maps.
  • Galaxy redshift surveys and weak-lensing maps.
  • Standard candle / standard ruler measurements.
  • Time-delay and clock comparison data where available.

Each channel constrains T(d) in different regimes. Joint inversion recovers χ(d) and η(d) under PFT assumptions, allowing direct comparison with ΛCDM fits.

5. Prestellar Zones and Collapse Dynamics

5.1 Prestellar Zones

Prestellar zones are underdensity regions in the curvature replication field where internal tension and mismatch between replication and local load generate instability. These regions are not empty; they store curvature imbalance that must resolve through cascades.

5.2 Collapse Trigger Condition

Collapse is triggered when local parameters cross a PFT collapse threshold that combines:

  • Replication imbalance.
  • Curvature mismatch vs neighbouring regions.
  • PAL compatibility conditions for large-scale cascades.

Once triggered, a curvature imbalance collapses via an Event Cascade channel. Two canonical endpoints emerge: PCS-L and PCS-H.

5.3 Collapse Evolution

During collapse, curvature flows inward through AOL paths selected by PAL, subject to continuity constraints enforced by Differentiat and CCE. The evolution is discrete: a sequence of curvature moves on the lattice, not a smooth fluid flow. Energy and curvature are redistributed while identity continuity of the manifold is preserved.

5.4 PCS-L and PCS-H Endpoints

  • PCS-L (Prestellar Collapse State – Local) Finite collapse that stabilises at a non-horizon endpoint, seeding star-forming regions, galaxy interiors, and bound structures that do not anchor horizons.
  • PCS-H (Prestellar Collapse State – Horizon) Collapse that crosses a horizon-forming threshold, generating black-hole anchored structures. PCS-H acts as a sink and regulator for curvature flow, stabilising the global carrier.

The bifurcation between PCS-L and PCS-H is determined by local replication gradients, PAL structure, and global curvature balance.

6. Cosmological Predictions and Tests

6.1 Replacement Structure for ΛCDM Elements

PFT cosmology provides explicit replacements for key ΛCDM elements:

  • Metric expansion → emergence and curvature replication at the edge.
  • Redshift from recession → Trishift T(d) from χ(d).
  • Inflation → early replication-driven filling of AOL rings and defect smoothing.
  • Dark energy → long-range behaviour of replication and η(d), with no free cosmological constant required.

6.2 Trishift Slope Test

The condition 

dT₁ / dd = H_pft · η(d)

yields a family of distance–shift curves that can be confronted with supernova data, BAO measurements, and other standard-ruler observations. Deviations from the ΛCDM distance–redshift relation correspond to specific features in η(d), rather than an unexplained dark component.

6.3 Collapse Statistics

The distribution of PCS-L and PCS-H endpoints predicts patterns in:

  • Black hole mass functions.
  • Void sizes and underdensity statistics.
  • Cluster richness and environment dependence.

These statistics provide a way to test whether curvature replication and collapse dynamics on a lattice produce the same large-scale structure that surveys observe.

7. Conceptual Summary

Emergence–edge cosmology in Pattern Field Theory builds the cosmos from:

  1. Null and the first π-loop closure.
  2. Substrate formation and AOL lift to 3D.
  3. Fractal lattice stabilisation.
  4. Curvature replication at the edge.
  5. Event Cascades that route curvature into structure.
  6. Trishift accumulation along observational paths.
  7. Prestellar collapse into PCS-L and PCS-H endpoints.

Rather than stretching a pre-existing metric, the universe grows as a structured carrier that replicates curvature and routes it through cascades. Observables record how often and where that replication occurred. This yields a concrete, testable alternative to ΛCDM, grounded in the same discrete dynamics that Pattern Field Theory applies at every scale.

External PDF

The full technical paper, including all equations, TikZ figures, and extended glossary, is available as a PDF:

Emergence–Edge Cosmology in Pattern Field Theory (full PDF on Academia)

How to Cite This Article

APA

Allen, J. J. S. (2025). Emergence–Edge Cosmology in Pattern Field Theory. Pattern Field Theory. https://www.patternfieldtheory.com/articles/emergence-edge-cosmology/

MLA

Allen, James Johan Sebastian. "Emergence–Edge Cosmology in Pattern Field Theory." Pattern Field Theory, 2025, https://www.patternfieldtheory.com/articles/emergence-edge-cosmology/.

Chicago

Allen, James Johan Sebastian. "Emergence–Edge Cosmology in Pattern Field Theory." Pattern Field Theory. December 8, 2025. https://www.patternfieldtheory.com/articles/emergence-edge-cosmology/.

BibTeX

@article{allen2025pft,
  author  = {James Johan Sebastian Allen},
  title   = {Emergence–Edge Cosmology in Pattern Field Theory},
  journal = {Pattern Field Theory},
  year    = {2025},
  url     = {https://www.patternfieldtheory.com/articles/emergence-edge-cosmology/}
}