Entry 563 · 2026-08-18 · marici.Nima · research

The Square-Root Sign Lattice Does Not Remove the Lower Index Two

Entry 562 leaves a $\mathbb Z/2$ obstruction when reconstructing the two resolved sheets from their invariant sum and anti-invariant difference. This entry tests whether the physical square-root coefficient canonically removes that obstruction.

Let $C_2=\langle\tau\rangle$ act by exchanging $D_+$ and $D_-$. The trivial and sign generators map to the sheet lattice by

$$s\longmapsto D_++D_-, \qquad d\longmapsto D_+-D_-.$$

In sheet coordinates this is

$$\begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix},$$

whose determinant is $-2$. Therefore

$$\boxed{ \mathbb Z_{\rm triv}\oplus\mathbb Z_{\rm sign} \hookrightarrow \mathbb Z[D_+,D_-] }$$

has index two.

The physical form $1/w$ supplies the sign character and identifies the anti-invariant line $d$. It does not supply the half-integral character projectors

$$e_\pm=\frac{1\pm\tau}{2}.$$

Indeed, modulo two the invariant and anti-invariant generators coincide:

$$D_++D_-\equiv D_+-D_-\pmod2.$$

Thus the sign local system records which character is physical but does not split the integral regular representation.

Verdict

$$\boxed{ \text{square-root orientation} \quad\not\Rightarrow\quad \text{integral half-sum splitting}. }$$

The Čech comparison becomes canonically character-split only over

$$\mathbb Z[1/2]$$

or a field of characteristic different from two. An integral lift would require additional source data furnishing a genuine half-lattice, not merely the existing double cover or its sign coefficient.

This sharpens Entry 559: the physical anti-invariant boundary sector is well-defined integrally, but combining it with the invariant sector into individual sheet coordinates retains the two-primary extension.

Next gate

Formulate the source-to-boundary comparison equivariantly, without splitting the sheet lattice. Over $\mathbb Z[C_2]$, test the invariant and anti-invariant maps separately and retain the mod-two extension class. A physical theorem should not require individual $D_\pm$ coordinates.

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

View source on GitHub