Non-Injectivity and Strict Information Loss in Projective Effective Descriptions

This repository contains the source of the paper Non-Injectivity and Strict Information Loss in Projective Effective Descriptions: A Structural No-Go Theorem.

Main result

Let $\Omega$ be a fine-grained configuration space, $\mathcal{O}$ a space of observable states, and $\Pi : \Omega \to \mathcal{O}$ a surjective projection whose observable structure is entirely induced by $\Pi$. If $\mathcal{O}$ is genuinely distinct from $\Omega$, then $\Pi$ cannot be injective. An injective projection would preserve every distinction and yield only a structural relabelling $\mathcal{O} \cong \Omega$, not genuine emergence.

Under the paper's hypotheses, genuine emergence therefore entails descriptive redundancy and strictly positive projection entropy, $S_\Pi > 0$. The argument assumes no metric, Hilbert space, Lagrangian, or dynamics.

Notation boundary

The symbol $\Omega$ belongs to the paper's framework-independent theorem and denotes its fine-grained configuration space. It is not a synonym for $\chi$, which names the proposed static relational substrate in the preliminary Cosmochrony white-paper vocabulary. This paper neither identifies $\Omega$ with $\chi$ nor constructs a Cosmochrony-specific $\Omega$; identifying a concrete fundamental space and projection for standard physics remains an open programme.

Further consequences

• Non-injectivity, descriptive redundancy, and positive projection entropy are co-extensive under the stated assumptions.
• Compression is recursive and exactly additive across towers of emergent levels.
• Any non-trivial gauge structure requires non-trivial fibres.
• When colour-neutral states genuinely emerge from coloured degrees of freedom, free colour is structurally unobservable through a non-injective colour projection.
• The theorem admits categorical and information-theoretic formulations.

Video

Editable sources for One Reality, Multiple Descriptions are in video/. The published video is available on YouTube and Instagram.

Repository contents

.
├── tex/       # LaTeX sources
├── out/       # Compiled paper output
└── video/     # Editable animation and narration sources

Citation

J. Beau, Non-Injectivity and Strict Information Loss in Projective Effective Descriptions: A Structural No-Go Theorem, 2026. https://doi.org/10.5281/zenodo.19391746

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All claims, interpretations, and final formulations remain the author's responsibility.