Primitive Mellin Obstruction Packets VII: Finite Schur–Mellin Models for Boundary Resonance Splitting and Mode Conversion

We develop a finite algebraic Schur–Mellin packet calculus for selected local obstruction
modules over one-dimensional analytic local rings. The central invariant is the reduced primitive
Smith packet of a selected finite-length quotient. This invariant refines determinant order and
maximal Fitting length by recording how the finite obstruction is assembled into Smith blocks.
In particular, split and nonsplit length-two packets have the same total length but different
primitive packet types, namely (1, 1) and (2).
The paper formulates Schur-compatible selected obstruction data, proves packet consistency
for transported Schur data under local Schur cokernel equivalence, defines selected Mellin spectra
as supports of selected finite obstruction modules, and separates no-leakage of support from
finite-flat base-change conservation of fiber length. A supplied formal Mellin realization dictionary
is used to interpret primitive packets as formal logarithmic boundary towers. All boundary-mode
conversion statements are therefore dictionary-level formal displays of the finite packet calculus.
They are not analytic assertions about closed operator domains, actual kernel spaces, or physical
mode conversion without additional realization hypotheses. No global analytic PDE realization
theorem is asserted; the construction is a finite algebraic model motivated by singular Mellin
boundary problems