Three Dimensions from a One-Bit Budget: A Termination Theorem for Pregeometric Ladders

Abstract

Before There Was Space: The Birth of Three Dimensions from a One-Bit Budget. Unlike operational reconstructions and dynamical dimension mechanisms, this finite pregeometric record model proves an exact budget-to-terminal-rank law. Here terminal rank is the strict formation-admissibility endpoint of the candidate ladder. An imported self-dual coordinate gives the assigned load L(r)=0.5 ln r, and for B>0 the strict positive-gap rule gives r_max(B)=ceil(exp(2B))-1. The endpoint equals three throughout 0.5 ln 3 < B <= ln 2. Here B is a nondepleting cumulative-load budget (capacity), r is candidate-ladder rank and candidate carrier multiplicity, L(r) is its assigned load, and q_car is carrier cardinality. The physical specialization is the Landauer-normalized one-bit budget B=ln 2, whose terminal least-sufficient carrier has q_car=3 states. At that specialization, the stated threshold-attainment and transition premises execute the binary and ternary crossings, and carrier selection with filled-simplex attachment closes the 1D-to-2D and 2D-to-3D steps. Thus the one-bit budget completes D_max(ln 2)=3 and uniquely places carrier four exactly on the zero-gap wall. The arbitrary-budget terminal-rank law governs formation admissibility; completion is established for the one-bit chain. The preprint states every load-bearing premise, supplies the full proof and dependency map, and includes source and executable checks. Files and reproducibility: The record has exactly four direct uploads: the manuscript PDF, Supplemental Material PDF, Bulmash_Three_Dimensions_Source.zip, and the standalone RIGHTS_NOTICE.txt. The source ZIP contains the complete LaTeX and bibliography sources, deterministic figure and build scripts, dependency map, and the version 1.0.0 verifier under software/. That bundled six-file verifier distribution contains its program, frozen expected output, README, citation metadata, MIT licence, and dependency declaration. It checks finite consequences of the printed definitions over the documented numerical ranges; it does not test the model premises or supply empirical evidence for them. Rights: The manuscript, Supplemental Material, figures, bibliography, dependency map, public source README, and rights notice are All Rights Reserved. BUILD_SOURCE.ps1, build_figures.py, requirements-figures.txt, and every file under software/ inside the source ZIP are MIT-licensed. The separately uploaded RIGHTS_NOTICE.txt states this mixed file-level boundary; no generic MIT licence is exposed as a record-wide direct-upload licence. Related source record: Quantum Genesis v6 is the public framework and derivation lineage for this result (DOI 10.5281/zenodo.21744228). This dedicated record is self-contained: every premise and proof used here is reproduced in the present record. This termination theorem does not use the separate retention and readable-order constructions developed in v6. Publication status: author-formatted, non-peer-reviewed preprint.