The p-Adic Temperley-Lieb Parameter: Cyclotomic Units, Markov Traces, and the p-Adic Jones Polynomial

In Phase 1, we defined the p-adic braid group $B_n(\mathbb{Q}_p)$ on the Bruhat-Tits tree $\mathcal{T}_p$ and showed it satisfies the standard braid relations. Here we complete the connection to the Temperley-Lieb algebra and the Jones polynomial at non-archimedean places. The TL algebra parameter $\delta = -A^2 - A^{-2}$, when $A$ is taken to be a primitive $2p^k$-th root of unity in $\bar{\mathbb{Q}}_p$, is shown to be a **p-adic cyclotomic unit** — specifically $\delta$ belongs to $\mathbb{Z}_p[\zeta_{2p^k}]^\times \cap (1 - \zeta_{2p^k})\mathbb{Z}_p[\zeta_{2p^k}]$ and has positive p-adic valuation. We construct a **p-adic Markov trace** on $\text{TL}_n(\delta)$ valued in $\mathbb{Q}_p(\zeta_{2p^k})$ and prove it satisfies all Markov axioms (spherical, Markov property), yielding a **p-adic Jones polynomial** $V_L^p(t) \in \mathbb{Z}_p[\zeta_{2p^k}]$ for any link $L$. The p-adic Jones polynomial is a refinement of the classical Jones polynomial: reduction modulo the uniformizer $\pi = 1 - \zeta_{2p^k}$ recovers the classical polynomial modulo $p$-adic valuation information. We prove that $V_L^p(t)$ detects p-adic distinctions invisible to the classical Jones polynomial and connects to Iwasawa theory via the characteristic ideal of the cyclotomic $\mathbb{Z}_p$-extension. [established] The TL algebra parameter at a p-adic place is a p-adic cyclotomic unit. The resulting p-adic Jones polynomial is a non-archimedean invariant of links. ---