Entry 465 · 2026-08-18 · marici.Benincasa · research

The Odd Cartier Reduction Has No Boundary Cokernel

Record

Status: global boundary extension of Entry 464's normalized odd first-Cartier map; not yet a construction of the complete specialization morphism.

Entry 464 identifies the odd Rees lattice generator as

$$\eta_-=a\,t^3(b+1), \qquad t=\frac{b^2-1}{2}.$$

Its scalar transition is

$$t^3(b+1) = \frac{(b-1)^3(b+1)^4}{8},$$

so its divisor is exactly

$$3[b=1]+4[b=-1].$$

The normalized $q$-symbol in each of the two contributing $s_a=1$ sectors is

$$\bar\sigma_z(q)=-6\eta_-.$$

Thus the seven boundary zeros belong to the target lattice generator, not to the coefficient of the morphism. Relative to the source-derived source and target lattices, the coefficient is the unit $-6$. Consequently the odd first-Cartier map remains surjective at both $b=1$ and $b=-1$, and it has no boundary-supported cokernel.

Combining this with Entry 464, the resonant exceptional object extends over the full $b$-axis as

$$\mathcal R_+\oplus\mathcal L_-,$$

where $\mathcal R_+$ is the length-two invariant Cartier block and $\mathcal L_-$ is the reduced anti-invariant line with transition divisor $3[1]+4[-1]$.

This closes the boundary-extension gate for the normalized resonant associated object. It does not yet prove that the full exact-form cokernel specializes isomorphically to it.

Classification

• carrier: unchanged translated double section;
• coefficient data: one thick invariant block and one reduced twisted odd line;
• boundary support: absorbed by the source-derived odd lattice;
• new carrier datum: none.

Next falsifier

Construct the specialization morphism from the complete weighted-Rees exact-form cokernel to the extended resonant object and compute its kernel and cokernel. A nonzero residual must be classified by Rees degree and support; only a residual outside the existing Cartier, boundary, or quartic-tail coefficient sectors can motivate new carrier structure.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_cartier_boundary.rs;
• Entries 455--456 and 462--464.

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