Structural Limits and Cyclotomic Symmetries in Diophantine Approximations of ζ(2m + 1)

We study the rational kernel β_s(N) of the Ramanujan–Berndt–Straub formula for odd zeta values as a rational function of the level parameter N. We prove an inter-weight identity β_3 − 42β_5 + 315β_7 = 0 from the cyclotomic factorisation of the denominators, and determine the Q-rank of the cross-weight system for low weights (s ≤ 9), conjecturing the general pattern. A reduction operator via alternating evaluation along full-absorption chains of a Fermat-quotient map eliminates Bernoulli-constant divisors and isolates cyclotomic factors ∏(b_i² + 1). We prove that the approximation system is arithmetically incompatible with Q-linear isolation of ζ(s): each equation involves a transcendental Lambert tail whose multiplicative rigidity (rooted in the Fundamental Theorem of Arithmetic via σ₋ₛ) and transcendental arguments (Lindemann–Weierstrass) jointly prevent rational elimination. This yields a demarcation between reductive Diophantine operators (fixed transcendental base; e.g. Ball–Rivoal–Zudilin) and additive operators (growing transcendental rank; e.g. Ramanujan–Lambert). Three negative results close the spectral and interpolation directions.