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Verification of the Markov Uniqueness Conjecture for the First One Trillion Markov Values: Meridians on the Markov Surface and Farey Addresses

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CO — Combinatoricscs.MS — Mathematical Software

Contributed by Elleard Felix Webster Heffern ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

The Markov unicity problem asks whether the largest entry of a positive integer solution determines the other two. We organize the triples into families called Meridians, with compact addresses and explicit evaluation rules. The construction starts with catenary profiles: a fixed host sets the curve, and two heights one unit apart solve the Markov equation with that host. Selected positions give integer triples. Reversing geometric descent gives an exact indexing of these triples. Remainder patterns, growth bounds, corridor paths, and Fibonacci-matrix and snake-graph arguments then restrict possible collisions. The Meridian description of starting scales and logarithmic spacing guided the organization of a large verification into magnitude bands. The implementation uses Farey addresses and certified matrix bounds to select each band; its correctness does not require the Meridian collision theorems. Production finished on 10 November 2026 at 03:54:13 UTC (9 November, 21:54:13 CST). The combined computation compared 1,101,388,071,945 configurations and certified uniqueness of the first one trillion Markov values in numerical order. The trillionth value has 1,021,609 decimal digits. All 12 repeated 64-bit residue groups were resolved by exact arithmetic, with no equal maxima. On eight workers, retained active band processing took 23 h 57 min 23.146 s, including the inherited 500-billion certificate; the maximum recorded temporary key-file footprint was 1,065,664,392 bytes (0.992 GiB). This elapsed processing time is distinct from summed CPU time and the calendar span. Both final replays passed. A separate computer-checked certificate covers all 5,067,077 Meridians of degree at most 10,000, at every integer phase. The general unicity problem remains open.

Author comments: Version 22 (12 November 2026), 44 pages: revised title and abstract, including the 1,021,609-digit trillionth value; added Arthur Baragar's 1996 benchmark (M < 10^140). The proofs, figures and verification evidence are unchanged. Chronology (distinct announcements and deposits): 21 October 2026, 15:18:25 UTC: X announcement of an earlier search exceeding one billion values: https://x.com/webystar/status/2112926487091282350 23 October 2026, 14:54:57.771 UTC: LinkedIn announcement of the earlier 9.3-billion search in 24 minutes: https://www.linkedin.com/posts/elleard-felix-webster-heffern-joseph-by-confirmation-2953a3104_markovnumbers-activity-7519411048115003392-4NaJ 2 November 2026: 50-billion verification and degree-10,000 computational supplements: https://zenodo.org/records/23874075 9 November 2026: selected full decimal expansions first deposited: https://zenodo.org/records/24093257 10 November 2026: v21 trillion-verification deposit (publication-date field: 9 November): https://zenodo.org/records/24096109 12 November 2026: corrected v22 manuscript; Zenodo DOI reserved (deposit pending): https://doi.org/10.5281/zenodo.24167894 12 November 2026: ResearchGate full-text mirroring began: https://www.researchgate.net/publication/417066856_Meridians_on_the_Markov_Surface_Growth_Structure_and_Verification_of_One_Trillion_Markov_Values_Using_Farey_Addresses The earlier X and LinkedIn announcements are not announcements of the trillion result. The source releases preserve the original verification timestamps and separate software licenses. The general unicity conjecture remains open.

Provenance statement

Elleard Felix Webster Heffern conceived and directed this project and closely oversaw the detail of each paragraph and proof throughout its development. Although he did not directly type the manuscript, its production involved sustained author engagement, revision, and review. The author confirms that all parts of the contribution have been checked and understood by a human. As detailed in the manuscript's Author contributions and AI disclosure, the author's contributions include the catenary construction and geometric descent, the identification of Meridians, phasors and corridors, the indexing framework, the Fibonacci-matrix and six-piece snake-graph approaches, residue and growth investigations, and the proposal to organize verification by scale. ChatGPT (OpenAI) provided substantial mathematical, computational and editorial assistance within that author-directed framework, including derivations and proof construction, examples and counterexamples, verifier implementation, checking routines, computational analysis, figures, literature assistance, and drafting and revision. Earlier geometric-descent work also used Claude (Anthropic). Established results retain their cited attributions. The author retains responsibility for the manuscript.

Tools used

OpenAI
ChatGPT
Anthropic
Claude (model unspecified)

References

  1. E. F. W. Heffern, Geometric Descent on the Markov Surface , working draft v32, archived October 2026, pp. 38--40, historical computational reports. The earlier search was assisted by Claude.
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  6. E. F. W. Heffern, Certified magnitude-band, residue-first verification of the first 500 billion Markov maxima , completed 9 November 2026. Final evidence: v1_Band_500_Billion_Final_Evidence.json ; frozen recovery archive: v20_Band_500_Billion_Recovery.zip . The comparison includes 550,693,634,649 configurations. The final evidence and reconciliation accompany this manuscript's source; the recovery archive is identified by SHA-256 in Section sec:verification-evidence .
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Version history

  1. v1 · 44 pagesSubmitted by Elleard Felix Webster HeffernVersion 22 (12 November 2026), 44 pages: revised title and abstract, including the 1,021,609-digit trillionth value; added Arthur Baragar's 1996 benchmark (M < 10^140). The proofs, figures and verification evidence are unchanged. Chronology (distinct announcements and deposits): 21 October 2026, 15:18:25 UTC: X announcement of an earlier search exceeding one billion values: https://x.com/webystar/status/2112926487091282350 23 October 2026, 14:54:57.771 UTC: LinkedIn announcement of the earlier 9.3-billion search in 24 minutes: https://www.linkedin.com/posts/elleard-felix-webster-heffern-joseph-by-confirmation-2953a3104_markovnumbers-activity-7519411048115003392-4NaJ 2 November 2026: 50-billion verification and degree-10,000 computational supplements: https://zenodo.org/records/23874075 9 November 2026: selected full decimal expansions first deposited: https://zenodo.org/records/24093257 10 November 2026: v21 trillion-verification deposit (publication-date field: 9 November): https://zenodo.org/records/24096109 12 November 2026: corrected v22 manuscript; Zenodo DOI reserved (deposit pending): https://doi.org/10.5281/zenodo.24167894 12 November 2026: ResearchGate full-text mirroring began: https://www.researchgate.net/publication/417066856_Meridians_on_the_Markov_Surface_Growth_Structure_and_Verification_of_One_Trillion_Markov_Values_Using_Farey_Addresses The earlier X and LinkedIn announcements are not announcements of the trillion result. The source releases preserve the original verification timestamps and separate software licenses. The general unicity conjecture remains open.CurrentNov 12, 2026