Combinatorics / CONCEPT

Graph rank states and cyclic embedding counts

The signed rank-state polynomial has an exact finite coefficient decomposition. Its forest contribution is not generally the negative-binomial coefficient: on the triangle its cubic coefficient is zero, while the comparison coefficient is minus one. Actual graph rank bounds give nonnegative exponents, small support, a fixed-rank cyclic-state count through finite ordered-embedding codes, and rank-dependent term growth. The companion TutteCoefficientAsymptotics module consumes these bounds to control both forest error and cyclic contribution. The weaving geometry and Jones identification remain separate obligations.