Causal Pregeometry I: Dual-Layer Foundations and Split–Contraction Calculus

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-16
DOI
https://doi.org/10.5281/zenodo.22787709
Primary Topic
Quantum many-body systems
Type
preprint
Controls
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preprint

Causal Pregeometry I: Dual-Layer Foundations and Split–Contraction Calculus

A-Z
Zenodo (CERN European Organization for Nuclear Research)
Quantum many-body systems
preprint

Causal Pregeometry I: Dual-Layer Foundations and Split–Contraction Calculus

A-Z
preprint en

Abstract

Without presupposing external coordinates, continuous space, or physical time, this paper formulates a dual-layer structure separating a "committed causal history" and "current local adjacencies." Thereby, it establishes exact graph calculus and an algebraic central identity regarding Unresolved Causal Dependency States (UCDS), providing a microscopic mathematical foundation for discussing the emergence of macroscopic physical laws. [Key Highlights] Definition of the event–state dual-layer structure: It mathematically separates the "causal history," which is append-only upon updates and accumulates, from the "currently active routing state," which allows local rewiring, thereby eliminating the conflation of causal relations. Proof of the central identity: It proves the central identity ($D_{CC} = a + b - G_J = r_{\rm new}$), showing that the complement-aware causal compatibility (CCA) debt coincides exactly with the counterfactual unresolved load that would remain if the target pair were edge-contracted. Establishment of dimensionless graph calculus: Independent of physical time and spatial dimensionality, it yields exact finite-change laws for the numbers of vertices and edges, local degrees, the first Betti number ($\Delta\beta_1^J=-m_S$), and the mean UCDS burden of the causal network. [Abstract] Without presupposing external coordinates, continuous space, or physical time, how can a single discrete framework consistently represent both a committed causal history and current local adjacencies without conflating the two? Here, we formulate a committed causal history (which is append-only and accumulates events without deleting or rewriting existing vertices and edges) and a currently active routing state as two mathematically distinct layers. As our main result, we prove the central identity, $D_{CC}(x,y) = a + b - G_J(x,y) = r_{\textrm{new}}$, showing that the complement-aware causal compatibility (CCA) debt coincides exactly with the counterfactual unresolved load that would remain if the target pair were contracted. On this dual-layer structure, we further introduce dimensionless algebraic balance laws for unresolved causal dependency states (UCDS), where forks are defined as vertex splits and joins as edge contractions. This yields exact finite-change laws for the numbers of vertices and edges, local degrees, the first Betti number ($\Delta\beta_1^F=0, \Delta\beta_1^J=-m_S$), and the mean UCDS load. The scope of this paper is strictly limited to establishing these dimensionless discrete algebraic structures and exact identities. The derivation of physical time, physical distance, and spatial dimensionality, together with the investigation of macroscopic cosmological relaxation laws and kinetic selection, lies outside the claims of the present work and is deferred to subsequent studies.

Zenodo (CERN European Organization for Nuclear Research)
Peace, Justice and strong institutions
Quantum many-body systems
DOIOpen Access PDFOpenAlex
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