Existential closedness and the exact parameter frontier in a perfectoid Witt–Kummer field

Fix a prime p. Let F = F_p((t))^perf, and let K be the completion of the union over s ≥ 0 of Frac W(F)(p^(1/p^s)), a mixed-characteristic perfectoid field whose rank-one p-adic valuation v has residue field F. Composing v with the t-adic valuation on F gives a henselian rank-two refinement u with residue field F_p. We study existential transfer between (K,u) and (K,v) and its dependence on parameters. We prove that (K_0,v) is existentially closed in (K,u), where K_0 is the completion of the union over s ≥ 0 of Q_p(p^(1/p^s)) inside K. Hence existential sentences with parameters from K_0 transfer from (K,u) to (K,v), whereas the reverse transfer already fails parameter-free. As a consequence, the valuation ring O_v is not existentially definable in the pure ring language with parameters from K_0. We give two proofs. One constructs an explicit embedding of tilt-side quotients and applies the relative Ax–Kochen/Ershov theorem of Jahnke–Kartas. The other embeds K into a nonstandard core of an ultrapower of K_0 so that a rank-two value (α,β) becomes α + εβ with ε infinitesimal. For every subfield B ⊆ K, existential transfer from (K,u) to (K,v) over B holds if and only if B ⊆ K_0, equivalently if and only if Bv = F_p, equivalently if and only if B ∩ O_u = B ∩ O_v. The key structural step proves that K_0 is the largest subfield of K with residue field F_p, using finite digit expansions in the roots of p.