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

This release verifies Frobenius's Markov uniqueness conjecture for the first one trillion Markov values in numerical order, that is, for every Markov number z ≤ M(1228920/1426807), which has 1,021,609 digits. Here M(p/q) denotes the Markov value at the ordinary Farey address p/q, with M(1/2) = 5; the numerical ordering counts 1 first. The exact cutoff is provided as data/ordinal_1000000000000.txt in the selected-values supplement: download the selected-values supplement. It includes the revised manuscript, complete reproducible source and supplements, final verifier report and machine-readable evidence, portable recovery archive, and six selected full decimal expansions.

Version 22 correction — 12 November 2026. The manuscript now explicitly credits Arthur Baragar’s early computational benchmark: 18,906 Markoff numbers below 10¹⁴⁰, with no unicity counterexample among triples whose maximum is less than 10¹⁴⁰. See A. Baragar, On the unicity conjecture for Markoff numbers, Canadian Mathematical Bulletin 39(1) (1996), 3–9, Section 4, p. 9; doi:10.4153/CMB-1996-001-x. This corrects an omitted historical attribution. The revised title and the 1,021,609-digit endpoint are now in the PDF and abstract. The computational results and frozen evidence are unchanged.

Posting chronology. This citation revision is dated 12 November 2026. The preceding v21 record has a publication-date field of 9 November 2026 and a repository creation/upload date of 10 November 2026. The selected decimal expansions were first posted on 9 November 2026, as detailed below. Earlier versions remain available; the dates of the underlying work are not changed by this correction.

Meridian analysis—especially helm scales and degree-dependent logarithmic spacing—guided the organization of comparisons into magnitude bands. The production verifier implements this plan using reduced Farey addresses, certified positive-matrix logarithmic bounds on 162 cones, scalar Vieta residues modulo 2^64, band sorting and exact resolution. The algorithm can be formulated without Meridian terminology; its finite numerical-prefix certificate does not rely on Meridian collision-exclusion theorems. No controlled ablation establishes a speedup specifically from the coordinate system.

Completed computation. The combined certificate compares 1,101,388,071,945 configurations: 550,693,634,649 inherited and 550,694,437,296 newly computed. Its inner region contains 1,000,000,419,685 configurations. Original screening produced 1,107,352,458,539 emissions, including boundary overlaps. These workload counts are distinct from the one-trillion numerical-prefix claim. All 12 repeated 64-bit residue groups, comprising 24 member occurrences in 11 of 19,287 bands, were resolved unequal.

Timing and storage. On eight workers, retained active band processing totaled 23 h 57 min 23.146 s (15.131 inherited hours plus 8.826 extension hours). Summed native-child CPU was 116.062 h. The combined calendar span was 36.987 h, from 8 November 2026 at 14:55:01.580398 UTC (08:55:01.580398 CST) to 10 November at 03:54:13.271944 UTC (9 November, 21:54:13.271944 CST). Pauses, lost/regenerated work, interrupted attempts and overhead are reported separately. The maximum measured temporary key-file footprint was 1,065,664,392 bytes (1.066 GB; 0.992 GiB); peak generator RSS was 36,468 KiB. Neither is a whole-machine resource bound.

Final checks. The independent final replay passed at 03:56:16.801036 UTC on 10 November. A fresh portable extraction verified all 20,208 archive members and passed its independent replay at 03:59:41.149704 UTC (9 November, 21:59:41.149704 CST), without partial or skip flags. These checks validate receipts and regenerate repeated-key bands; they are not a complete second enumeration of discarded singleton keys. The 38,920,281-byte recovery archive is bound to final receipt head 624c31904f64f8d9434162396a311228f2ece95d495ff2c7dc9d053314ed0c72.

Six full base-10 values. The supplement gives the trillionth Markov value, its predecessor, and the strict predecessor/successor pairs around 10^1000000 and 10^999999. Both ordinal values have 1,021,609 digits. The six files contain 6,043,218 digits. Their expansions were first posted on 9 November 2026 in record 24093257, v12-supplement-only (manuscript v11); the public record exposes the date without a time of day. Their numerical ranks were confirmed on 10 November at 04:18:33.296330 UTC (9 November, 22:18:33.296330 CST) by the completed certificate and fresh rank checks with two Farey partitions. The decimal digits are unchanged. The first million-digit value has rank 958,142,673,903; exactly 1,916,520 Markov values have one million digits. The selected ordinal pair is not asserted to be the globally closest pair; that statistic was not recorded.

Figures 13 and 14 explain the Meridian motivation, Farey grouping and comparison with H. Tan and Y. Zhang, arXiv:2611.01007v1 / doi:10.5281/zenodo.23670834. Their reported result covers 239,536,376,556 maxima through 10^500000. Their 184.27-GB largest-job five-byte payload and our temporary key-file peak are explicitly scoped components. No total-runtime or CPU speedup ratio is claimed.

The manuscript preserves the author-approved Fibonacci-matrix and six-piece arguments, contribution statement and AI disclosure, and final-page exploratory Zeckendorf preview followed by both dedications. The historical 500-billion certificate remains unchanged. The general Markov unicity problem remains open. Author: Elleard Felix Webster Heffern, unaffiliated; no funding. Original software: MIT. Original documentation and data: CC BY 4.0. Third-party materials retain their own terms.