Beyond the Fontaine-Wintenberger theorem

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript p Baseline left-parenthesis t right-parenthesis Superscript h"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>h</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {F}}_p(t)^h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an elementary substructure of the perfect hull of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript p Baseline left-parenthesis left-parenthesis t right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {F}}_p(\!(t)\!)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .