{"ID":22918815,"CreatedAt":"2026-09-17T01:02:08.507062015Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.17899","arxiv_id":"2609.17899","title":"Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates","abstract":"We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions. For m+1 categories, m is the number of independent probability degrees of freedom. Integer count vectors modulo common scaling determine rational points of the m-dimensional probability simplex. Building on standard simplex and log-ratio coordinate geometry, for m \u003e= 2 we define the full multinomial DCCQ coordinate map and show that it is a real-analytic diffeomorphism The previously established binary baseline m=1 gives a critical-line coordinate, while the ternary case m=2 gives the full open critical strip; higher multinomial models retain m-2 additional real contrasts. We also give a one-versus-rest specialization, an exact integer-lattice realization of the ternary coordinate, and a hyperbolic representation of its log-ratio. No zero-location theorem or proof of the Riemann Hypothesis is claimed.","short_abstract":"We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions. For m+1 categories, m is the number of independent probability degrees of freedom. Integer count vectors modulo common scaling determine rational points of the m-dimensional probability si...","url_abs":"https://arxiv.org/abs/2609.17899","url_pdf":"https://arxiv.org/pdf/2609.17899v1","authors":"[\"Y. Kenan Yılmaz\"]","published":"2026-09-15T22:41:28Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"math.NT\"]","methods":"[]","has_code":false}