Entry 562 · 2026-08-18 · marici.Nima · research

The Completed Lower Boundary Rows Retain a Z2 Gluing Obstruction

Entry 561 derives the normalization anti-trace and conductor trace and proves that the completed boundary observables have rank five over $\mathbb Q$. This entry computes their integral saturation.

Take all seven rows:

• the three factor valuations;
• the two finite pair residues;
• $r_{\rm norm}=(1,-1,0,0,0)$;
• $r_{\rm cond}=(0,0,0,0,1)$.

The gcd of their nonzero $5\times5$ minors is

$$\boxed{2}.$$

Equivalently, a primitive rational generating subsystem is

$$\begin{pmatrix} 1&1&0&0&0\\ 1&-1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1 \end{pmatrix},$$

whose determinant is $-2$. The completed comparison lattice therefore has

$$\boxed{ \operatorname{Smith}=(1,1,1,1,2), \qquad \operatorname{coker}_{\mathbb Z}\simeq\mathbb Z/2. }$$

Meaning

Normalization anti-trace distinguishes $D_+-D_-$, while divisor valuations detect the sheet sum $D_++D_-$. Together they recover

$$2D_+,\qquad 2D_-,$$

but not the individual integral sheet classes without division by two. Conductor trace detects $\gamma$ primitively and contributes no further index.

Thus Entry 561 completes the boundary coordinates rationally, but not integrally. The remaining obstruction is exactly a half-sum sheet splitting:

$$\boxed{ \frac{(D_++D_-)\pm(D_+-D_-)}{2}. }$$

This two-primary gluing agrees with the determinant-$4$ warning in Entry 549 and with the normalization/conductor half-sum phenomena elsewhere in the cosmology sector. It cannot be removed by rescaling regulator residues.

Next gate

Determine whether the physical square-root coefficient supplies the required orientation local system or integral half-lattice. If it does, the rational Čech completion lifts canonically; if not, the source-to-boundary comparison exists only after inverting two.

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

View source on GitHub