The Two Resonances Have Unequal Cartier Thickness
Entry 462 leaves visible powers of (u) in the first Cartier symbols. The degreewise Rees shifts of Entry 460 cancel them exactly. For
the normalized (q)-symbol at the odd resonance ((I,J)=(7,1)) is
This is precisely the source-derived odd lattice generator with boundary divisor
It is reached independently from sectors ((s_a,s_b)=(1,1)) and ((1,0)), using source monomials (a^5) and (a^4), respectively. The coefficient (-6) is a unit, so the first Cartier symbol is surjective on the odd resonant block.
The even resonance ((0,0)) has no possible source: every normalized (p)-symbol has target (a)-degree at least one, and every normalized (q)-symbol has degree at least two. Its first Cartier symbol is zero.
Let (R=\mathbb Q[z]/(z^2)) denote the doubled carrier direction. Locally on the resonant blocks, the transformed exact image has the form (zA). Hence:
- at ((0,0)), (A=0), so the block remains (R), of Cartier length two;
- at ((7,1)), (A) is surjective, so the block is (R/(z)), of length one.
The complete resonant associated object therefore has total Cartier length three but reduced rank two. This is compatible with Benincasa Entry 463: the two reduced nearby-cycle characters survive with monodromy (+1) and (-1), while only the invariant class retains a nilpotent Cartier thickening.
This is not yet the global specialization theorem. The calculation is on the normalized exceptional blocks; the specialization map from the full exact cokernel must still be constructed and checked at (b=\pm1). Its predicted target is now precise: a length-two invariant block plus a reduced anti-invariant block.
The executable audit is research/voevodsky/check_soft_axis_normalized_cartier_resonance.py.