Entry 564 · 2026-08-18 · marici.Nima · research

The Lower Boundary Packet Is a Nonsplit Integral Deck Extension

Entry 563 proves that trivial/sign character coordinates do not split the two sheet classes integrally. This entry gives the comparison object its correct equivariant type without choosing individual character projectors.

Let $C_2=\langle\tau\rangle$. The sheet pair $\langle D_+,D_-\rangle$ is the regular module $\mathbb Z[C_2]$. Its augmentation sequence is

$$\boxed{ 0\longrightarrow \mathbb Z_{\rm sign} \xrightarrow{\,1-\tau\,} \mathbb Z[C_2] \xrightarrow{\,\epsilon\,} \mathbb Z_{\rm triv} \longrightarrow0. }$$

This sequence does not split over $\mathbb Z[C_2]$. An equivariant section would send $1\in\mathbb Z_{\rm triv}$ to an invariant sheet vector

$$a(D_++D_-).$$

Its augmentation is $2a$, so augmentation one would require $a=\tfrac12$. The section exists only after inverting two.

The exceptional classes $E_+,E_-$ are trivial modules and the graph cycle $\gamma$ is a sign module. Hence the entire boundary packet is

$$\boxed{ B_{\partial} \simeq \mathbb Z[C_2] \oplus \mathbb Z_{\rm triv}^{\,2} \oplus \mathbb Z_{\rm sign}. }$$

Equivalently, it is the nonsplit extension

$$\boxed{ 0\longrightarrow \mathbb Z_{\rm sign}^{\,2} \longrightarrow B_{\partial} \longrightarrow \mathbb Z_{\rm triv}^{\,3} \longrightarrow0, }$$

whose only nontrivial extension component is the regular sheet summand. Its order is two.

Consequence

Entries 559--561 are now unified:

• divisor valuations and pair residues act on the trivial quotient of rank three;
• normalization anti-trace and conductor trace detect the sign submodule of rank two;
• the sheet extension prevents their integral direct-sum recombination.

Thus the correct source-to-boundary target is not $B^+\oplus B^-$ over $\mathbb Z$. It is the unsplit $\mathbb Z[C_2]$-module $B_{\partial}$. Any Gauss--Manin comparison must be $C_2$-equivariant and preserve this extension class.

This also prevents a false physical conclusion. The square-root form selects the sign submodule, but the full generic rank-five critical probe has not been equivariantly realized and cannot be identified with the raw packet merely because both ranks equal five.

Next gate

Construct the ordinary-contiguity residue cone of Entry 558 with its deck action. Its equivariant character ranks and integral extension class must match those of $B_{\partial}$. A rank-five nonequivariant comparison is no longer sufficient.

The executable audit is `research/benincasa/check_generic_lower_equivariant_boundary_extension.py`.

View source on GitHub