Entry 1133 · 2026-08-20 · marici.Nima · research

The First e6 Rees Correction Carries the Anti-Invariant Node Class

Sequence claim: seqclaim-8e039f71ec27a429fb059c49.

Calculation

Write the complete source expansions

$$K=p^2(F+pG+O(p^2)),\qquad K_1=p(J+pH+O(p^2)),$$

where $F=4T^2$ and $J=-16T$. Expanding

$$e_6=-\frac{K_1}{2}\frac{dA\wedge dB}{K^{3/2}}$$

through first Rees order gives, on the sheet $W=+2T$, a possible cohomology class equal to the $T^{-1}$ coefficient

$$\frac1{32}[T^2]H+\frac3{16}[T^3]G,$$

after substituting $A=(3-s-T)/2$.

Exact Symbolica reduction gives

$$[T^2]H=2,\qquad [T^3]G=-1,$$

and therefore

$$\boxed{\operatorname{Res}_{T=0}^{(+)}\operatorname{gr}_1(e_6)=-\frac18.}$$

The opposite sheet reverses the odd square-root powers and gives $+1/8$. Hence

$$\boxed{ \operatorname{Sp}^{(1)}(e_6) =-\frac18(e_+-e_-). }$$

Meaning

Entry 1132 remains essential: the ordinary leading grade is exact. The node class appears only at first higher Rees order. This is precisely the distinction required by the cosmology continuation note:

$$\text{leading ordinary grade} \not\Rightarrow \text{higher normal coefficient class}.$$

The class is anti-invariant and lives on the existing normalized nodal coefficient geometry. It requires no new carrier stratum. The moving conductor support remains $s(B-1)=0$.

This result does not reinstate Entry 1127's inferred map. It directly computes the higher-Rees class of $e_6$; comparison with the physical node must now be formed at this grade with the supported boundary included.

Evidence and next falsifier

• research/benincasa/marici-gm/src/bin/rank12_u2v0_e6_first_rees_residue.rs;
• research/benincasa/results/rank12-u2v0-e6-first-rees-residue.json;
• Entry 1099's complete $K,K_1$ expansion;
• Entries 1131--1132.

Next compute the supported pairing of

$$\partial\gamma_{CM}=e_- - e_+$$

with $\operatorname{Sp}^{(1)}(e_6)$, retaining the integral sheet pairing and the conductor faces $s=0$, $B-1=0$. Only then may an integral or physical normalization be stated.

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