Meridians on the Markov Surface: Geometry, Residues, and Collision Windows
Year: 2026
Type: preprint
Abstract: Meridians organize positive integer Markov triples into families with compact addresses and explicit evaluation rules. This release presents simple indexing for fast, low-storage verification and retains the catenary-based introduction, geometric descent, residue patterns, collision windows, corridor recurrences, and Fibonacci-matrix and snake-graph arguments. Completed verification: 550,693,634,649 configurations were compared, certifying uniqueness of the first 500,000,000,000 Markov values in numerical order. The larger count is the comparison workload, including possible competitors and conservative cutoff coverage; it is not a claim about the first 550 billion numerical values. All four repeated 64-bit residue groups, comprising eight exact candidates, were resolved with no equal maxima. The exact 500-billionth value was not identified. Production began on 8 November 2026 at 14:55:01.580398 UTC. The last retained band completed on 9 November 2026 at 15:50:31.046707 UTC (09:50:31.046707 CST). On eight workers, retained active elapsed band processing took 54,470.884742 seconds: 15 h 07 min 50.885 s. Summed native-child CPU time was 264,075.377305 seconds (73.354 hours); parent CPU was not separately totalled. The calendar span was 24.925 hours. Active band time excludes pauses, restoration, lost attempts, and subsequent audits; the manuscript reports these scopes separately. Maximum recorded temporary key-file storage was 754,846,136 bytes (719.877 MiB; 0.703 GiB), including the shard duplicated during partitioning. This excludes receipts, source files, backups, filesystem overhead, and unrelated work. Generator-only resident memory peaked at 28 MiB. A continuous whole-workspace disk or aggregate-memory peak was not recorded. The frozen recovery archive is 27,168,031 bytes. The complete 13,638-band receipt chain and exact candidate evidence were audited. A separate checker recomputed the numerical envelope, independently reconstructed all candidate maxima, and regenerated every repeated-key band. The final independent replay completed at 15:51:24.981257 UTC; a fresh portable extraction verified every manifested archive member and completed its replay at 15:57:38.529312 UTC on 9 November 2026. This is a receipt-and-candidate replay, not a second full enumeration of discarded singleton keys. The separate degree-bounded certificate covers all 5,067,077 Meridians of degree at most 10,000 at every integer phase. The equal-degree separation result, the author's Fibonacci-matrix normalization and fixed-numerator arguments, and his six-piece fixed-sum and fixed-denominator arguments are retained. Earlier ordering results and the 2002–2003 origins of the matching interpretation are credited in the paper. The unrestricted Markov unicity problem remains open. Version 11 removes the obsolete submission-status note and associated wording from the manuscript. Profiles and related derivations continue to use explicit powers, square roots, and logarithms. The original catenary figure and all formal mathematical statements are preserved. Files include the revised manuscript, editable source with all figure assets and the frozen verifier archive, release notes, separately downloadable recovery archive, final verifier report, machine-readable final evidence, progress chart, historical six-page fifty-billion proof-and-timing supplement, and standalone catenary figure. The historical fifty-billion run completed on 28 October 2026; its separate original archive remains available at https://doi.org/10.5281/zenodo.23874075. Historical filenames retain the original 500-billion certification target. Licensing: Original software is licensed under MIT; original documentation and data under CC BY 4.0. Third-party materials and dependencies retain their own terms. Copyright (c) 2026 Elleard Felix Webster Heffern, to the extent copyright applies. Funding: No funding. Substantial generative-AI assistance is disclosed in the manuscript; the author directed the research and is responsible for the work.
Author: Heffern, Elleard Felix Webster
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225458248