Why does depth fail to be preserved under the local Langlands correspondence for wildly ramified groups, and what is the right definition of depth?

Object. For a torus T split by a wildly ramified extension E/F (say the norm-one torus U(1)_{E/F} with E/F of ramification break b), the local Langlands correspondence for tori sends a character of T(F) of Moy–Prasad depth r to a parameter of depth r + (shift), with the shift governed by the Hasse–Herbrand function of E/F. For tame tori there is no shift (Yu). For split groups (GL_n) and for inner forms (division algebras, via Jacquet–Langlands) there is never a shift.

Presenting problem. Depth is meant to be an invariant matched across the correspondence; it isn't, but only in one regime, and the regime has a clean description: the shift appears exactly when the outer action of Galois on the dual group is nontrivial on a ramification subgroup of positive depth. Mechanism for the torus: v_E(u − ū) ≥ v_E(u) + b, i.e. the Galois action is trivial to depth b, so the filtration transported through e ↦ e/ē sits b steps deeper than the intrinsic one.

History. Mishra–Pattanayak proved the failure for wild tori with the Herbrand formula; Plymen gave characteristic-p examples; ABPS proved preservation for GL_n and inner forms; Chen–Kamgarpour formulated and half-proved preservation in the complex-geometric setting, where no wild extensions exist and only irregular singularities (Stokes data) carry "wildness"; a 2025 paper redefines parameter depth via Yu's minimal congruence filtration and induced tori so that preservation holds.

Diagnosis offered. There are two independent sources of wildness. Form-wildness: ramification of the outer action (the group's own twist); absent over ℂ, present for number fields and in characteristic p via Artin–Schreier. Parameter-wildness: irregular singularities of the connection/Stokes data; present over ℂ; maps cleanly to depth via slope. The depth shift is entirely form-wildness. Over ℂ it cannot occur. In characteristic p the two get a common yardstick (Swan conductor = irregularity of p-curvature).

Differential. 1. Depth is the wrong invariant and the 2025 redefinition is the right one (form-wildness absorbed into the definition). 2. Depth is right and the shift is a genuine new invariant of the outer action's filtration, to be tracked rather than defined away. 3. For non-abelian wild quasi-split groups (e.g. U(3,1) at p = 2) the shift is not a function of the outer action alone and the two-sources picture breaks.

Discriminating tests. A wild quasi-split non-abelian case computed explicitly; whether the inner-form independence of induced torus filtrations (both descend from the maximal unramified extension) survives for all wild tori.

Stakes. Whether ramification can be read as a filtration on a single rung of structure, or is irreducibly entangled.