Douglas E. Allen

A FINITE UNIVERSE

The sky is a message

A cosmology exploring a finite, deformable universe as an alternative to the expanding model.

Citation
Allen, D. (2026). “A Finite Universe”. Zenodo. doi:10.5281/zenodo.20492474.
https://doi.org/10.5281/zenodo.20492474

What you will find here

This project consists of four connected resources: a technical web book, a plain-language companion guide, an interactive visual model, and a dedicated AI chatbot. Together, they explore the possibility that the universe is finite and geometrically deformable, rather than infinite and governed by a single universal expansion scale.

The web book develops this model step by step. It begins with the fundamental logic of measurement and redshift before expanding into complex observations—from the cosmic microwave background and early galaxies to the behavior of the universe's outer boundaries. Rather than just presenting equations, the book connects these diverse observations into a single, shared framework.

Whether you are a researcher, a student, or a curious observer, these tools are designed to let you move from intuitive ideas to technical proofs at your own pace.

Finite deformable universe schematic
Conceptual schematic. This diagram illustrates the finite deformable universe model. A light path travels through a finite domain, accumulating a "redshift response" as it moves inward, then transitioning through a boundary-layer projection. This shows how local structure and far-field projections are linked parts of the same geometric picture.
Open full-size image

The key symbols used throughout the model are summarized in the Measurement Registers page.

Companion Resources

Use these tools to explore the model through plain-language explanations, interactive visuals, and AI-assisted queries.

Companion Guide

The Sky Is a Message

A plain-language guide that explains the core physical ideas before you dive into the technical calculations.

Read the Companion Guide
Visual Companion App

A Finite Universe — Visual Companion

An interactive app to compare how the standard expansion reading and the finite-geometry reading interpret the same data.

Open the Visual Companion App
Reader support

Finite Deformable Geometry AI

An AI assistant trained on this web book to answer questions about the equations, figures, and general concepts in plain language.

Open FDG AI
Front Matter

A Note from the Author

Motivation, scope, and the role of AI in this project.

A Note from the Author
Chapter 1

Redshift as Path Response

Foundations of how light responds to the geometry of the path.

Read Chapter 1
Chapter 2

Observational Registers

Establishing the scales and measurements used to read the sky.

Read Chapter 2
Chapter 3

A Finite Deformable Universe

The core theory of the deformable cosmic model.

Read Chapter 3
Chapter 4

A Finite Deformable Boundary Cosmology

Applying the model to the edges and boundaries of the observable universe.

Read Chapter 4
Appendix

Quantitative Appendix

Technical definitions, units, derivations, and diagnostic equations for the redshift-distance framework.

Open Appendix