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 >= 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.
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BibTeX
@misc{yılmaz2026generalized,
title = {{Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates}},
author = {Y. Kenan Yılmaz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17899},
eprint = {2609.17899},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17899},
}
RIS
TY - UNPB
TI - Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates
AU - Y. Kenan Yılmaz
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17899
UR - https://arxiv.org/abs/2609.17899
AB - 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 >= 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.
ER -