An Explicit GCD Evaluation for Three-Direction Torus Arrangements
Year: 2026
Type: preprint
Abstract: This version includes the original HV27 preprint, An Explicit GCD Evaluation for Three-Direction Torus Arrangements, its supplementary code and validation reports, and a new mathematical appendix by Rushikesh Murlidhar Walke. The original paper (v08) counts the geometric regions formed by three families of closed curves on a two-dimensional torus. It derives an explicit formula using greatest common divisors, a count of triple intersections and Euler’s relation. For the common-origin grid with N = 27, R = 12, Y = 27 and K = 60, the arrangement has 6,264 regions. The author-supplied validation report records 284 completed checks of the v08 implementation, all passing. The original preprint is available at https://doi.org/10.5281/zenodo.24014501. The v09 appendix, Counting five-label records and detecting one-step errors on the Sun–Moon torus, answers a question left open in the original paper. It proves that each of the 6,264 regions corresponds to a distinct realizable record of five angular labels: nakshatra, lunar rashi, yoga, karana and tithi. The tithi label records the ordinal within a fortnight. For each of two fixed J2000 lunar-mansion partitions (H20 and H21), the count is 6,597. Shifting the equal mansion grid by +0.1° also gives 6,597 records. The appendix then counts how often a single edit can be detected. A realizable record is selected uniformly, one of its five labels is chosen uniformly, and that label is moved one step forward or backward in its cyclic order with equal probability. An edit is detected when the resulting record is impossible. The exact detection probability is 1223/1740 (70.2874%) for the common-origin equal grid and 508/733 (69.3042%) for each of the other three configurations. These probabilities describe the specified edit model. Rushikesh Murlidhar Walke formulated the research direction and initial hypotheses. The manuscripts document AI assistance with proof development, calculations, literature comparison, internal checks and writing. Independent human peer review is pending.
Author: Walke, Rushikesh Murlidhar
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225425449