Generalization of the recurrence and Kepler-limit observations

Let $k\geq 5$, let $W$ be the relevant word set, let $\nu^{(k)}$ be the coefficient function, let $L_p\in\Lambda_p$ be the linear recurrence associated with the coefficients, and let $\kappa_p$ denote its Kepler limit when it exists. Generalized recurrence observations. In the observation above, clause (a) holds for all $k\geq 5$, clause (b) holds for all odd $k\geq 5$, and clause (c1) holds for all even $k\geq 6$. One of clauses (c2) or (c3) holds for any even $k\geq 6$. Clause (d) holds for an unbounded set of even $k\geq 6$. These assertions extend finite computational observations about coefficient magnitudes, recurrence limits, and the transformation $y=x^2$; the supplied text gives no proof or resolution.