The Restored Minus Conormal Residue Is Exact Below Resonance
Record
Status: source-typed minus-endpoint completion of Entry 483.
Entry 483 shows that the universal quartic factor must be restored before a conormal identity is compared with the odd resonance lattice. Put
[ c=b+1, \qquad g=a^3c(2-c). ]
Restoring the frozen factor (a^4) and dividing by the declared minus normal (c) gives
[ a^4\frac{g}{c}
a^7(2-c). ]
Therefore the minus conormal residue is
[ \boxed{ operatorname{Res}_{c=0}(a^4g)=2a^7. } ]
Degreewise typing
The residue lies in bidegree
[ (I,J)=(7,0), ]
whereas the odd resonance lies in
[ (I,J)=(7,1). ]
Using Entry 460's source-derived boundary assignment,
[ B(I,J)= left( leftlfloor\frac I2 ight floor, leftlfloor\frac I2 ight floor+J
ight), ]
one obtains
[ B(7,0)=(3,3), \qquad B(7,1)=(3,4). ]
Thus the restored residue is regular in its own lattice and sits exactly one minus-incidence step below the resonance. It cannot be identified with the resonance class by bare evaluation.
Exactness
For the degreewise operator
[ D_b=a(1-cpartial_c), ]
the scalar into target degree ((I,J)) is (1-J). At ((7,0)),
[ 1-J=1. ]
Hence the incoming block
[ D_b:(6,0)longrightarrow(7,0) ]
is a unit map, and
[ \boxed{ 2a^7 in operatorname{im}D_b. } ]
The restored minus conormal residue is therefore exact in the complete degreewise exceptional complex. It creates no endpoint-supported odd class.
Combined boundary result
Entry 483 makes the plus-boundary cancellation regular by restoring the universal quartic factor. The present calculation shows that the independently typed minus residue lands in an exact block. Together they remove the endpoint obstruction to separating the extensive conjugate-(L_2) tail from the reduced odd resonance line.
This is a boundary-complex result. It does not yet prove that the global specialization morphism has no interior kernel or cokernel arising from the quartic tail.
Classification
- plus endpoint: typed cancellation in the ((7,1)) lattice;
- minus endpoint: exact restored residue in ((7,0));
- odd resonance: distinct reduced class in ((7,1));
- new carrier datum: none.
Next falsifier
Construct the global specialization map from the complete orbit-completed weighted-Rees exact cokernel to the extended resonant object. Compute its kernel and cokernel away from the endpoints. Any remaining extensive residual must be classified as interior quartic-tail coefficient data rather than boundary support.
Evidence
research/benincasa/marici-gm/src/bin/soft_axis_minus_quartic_restoration.rs;- Entries 460, 469, and 481--483.