# Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates

arXiv: https://arxiv.org/abs/2609.17899 · HTML: https://mlsanity.com/paper/2609.17899/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 11 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 1 of 1)

- [DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra](https://mlsanity.com/paper/2609.03801/) (2026)

## Abstract (arXiv)

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.