math.NT — Number Theory
Verification of the Markov Uniqueness Conjecture for the First One Trillion Markov Values: Meridians on the Markov Surface and Farey Addresses
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.