Significance
This paper gives a rigorous formal result in the study of emergent geometry from discrete relational systems. The SERD framework starts without a background metric: separation information is not assumed from an external space, but reconstructed from local rewrites and transported records on a typed substrate of point particles, space elements, and information gaps. The main theorem package focuses on a restricted frozen regime of the constant-point-particle metric kernel. After local updates stop, the remaining receiver-indexed records form a deterministic finite transport system. The paper proves finite settling, one-generation rebroadcast, and an exact terminal correction to each observer’s internal distance representation as a finite functional of the full frozen state. Under an explicit recovery-compatibility condition, observers recover and agree on the same frozen latent geometry. The result does not claim a complete physical theory of spacetime; its significance is narrower and precise: it provides a theorem-level mechanism by which observer-relative distance can be reconstructed from internal information flow in a background-free discrete model.
Abstract
The Space Element Reduction Duplication (SERD) framework is a discrete, background-free relational framework in which observer-indexed separation information is reconstructed from local updates and transported records on a typed latent substrate rather than imposed by an external manifold or metric. The parent framework is built from three primitive element types — point particles, space elements, and information gaps — together with a small family of strictly local update operations and a deterministic propagation phase for gap-based records.
This paper develops the formal middle layer of the current SERD programme. Its aim is not to establish complete theorem-level closure of the full SERD rule space, but to define a common parent framework, separate microscopic law from scheduler family, identify restricted kernels derived from that framework, and state the strongest currently available rigorous results within those kernels. The paper studies two implementation-faithful restricted kernels: a split-enabled structural kernel and a constant-point-particle metric kernel.
The strongest results are obtained for the frozen regime of the constant-point-particle kernel. After local updates have been switched off, the remaining receiver-indexed transport dynamics are formalized as a deterministic frozen queue system. In this regime the paper proves deterministic evolution after freeze, one-generation-only rebroadcast, finite-horizon settling of residual transported records, and, for observer-symmetric frozen states, an exact terminal frozen correction theorem expressing the eventual observer-side correction as a finite functional of the full frozen state. On that basis, it gives a conditional characterization of late-time observer recovery and observer agreement in terms of recovery compatibility of the full frozen state. A separate conditional bridge theorem identifies when a structurally quiescent late regime of the split-enabled structural kernel may be explicitly identified with a constant-point-particle frozen-state description, and hence brought under the same frozen-regime theorem package.
The paper does not claim a complete treatment of unrestricted active dynamics, a final continuum identification, or a theorem-level account of every numerical phenomenon discussed in companion computational work. Its contribution is narrower and more precise: it formalizes the parent framework, isolates its current restricted kernels, and proves a bounded frozen-regime theorem package together with a conditional bridge result from the split-enabled structural sector.
Key Findings
Formalizes the SERD framework as a discrete, background-free relational model based on point particles, space elements, and information gaps.
Separates the parent microscopic law from scheduler-level choices such as update probabilities, freeze times, and execution order.
Defines two restricted kernels: a split-enabled structural kernel and a constant-point-particle metric kernel.
Proves deterministic evolution after freeze in the constant-point-particle metric kernel.
Proves one-generation-only rebroadcast and finite-horizon settling of residual transported records.
Proves an exact terminal frozen correction theorem for observer-symmetric frozen states.
Characterizes late-time observer recovery and observer agreement through recovery compatibility of the full frozen state.
Establishes a conditional bridge from structurally quiescent Kernel A regimes into the frozen Kernel B theorem package.
Keeps physical interpretation bounded: the paper does not claim a final continuum theory, but provides a rigorous formal layer for future work on emergent relational geometry.
Transparency Statement
AI Contribution: Large language models were used during the preparation of this manuscript as structured research-assistance tools. Their use included theorem exploration, proof-map generation, adversarial consistency checking, cross-comparison of candidate formalizations, code–paper correspondence analysis for the restricted kernels, editorial restructuring, drafting support for selected sections, final minor-revision consistency checks, figure-caption drafting, and provenance-package preparation.
AI outputs were not treated as authoritative. Proposed definitions, proof strategies, algebraic interpretations, and explanatory prose were reviewed critically by the author and were accepted only after comparison with the manuscript's formal framework, the implementation-faithful kernel semantics, and direct mathematical inspection. The author retained final responsibility for all definitions, theorem statements, proofs, interpretations, editorial decisions, and any errors.
A separate AI worklog and integrity package accompanies this article. It documents the model-use categories, source-of-truth hierarchy, claim-verification table, artifact manifest, editor-facing notes, and normalized thread self-report summaries used to reconstruct and bound the AI-assisted workflow.
