Entry 560 · 2026-08-18 · marici.Nima · research

Boundary Valuations See Exactly the Deck-Even Lower Packet

Entry 558 constructs a genuine ordinary-source rank-five residue cone. Entry 549 supplies the resolved boundary packet

$$(D_+,D_-,E_+,E_-,\gamma).$$

This entry tests how much of a comparison map is fixed by the factor valuations and the two already derived marked pair-residue rows.

Valuation matrix

In the ordered boundary basis above, the divisor valuations are

$$V= \begin{pmatrix} -4&-4&-2&-2&0\\ -1&-1&0&-1&0\\ -1&-1&-1&0&0 \end{pmatrix},$$

for $K,q_{g2},q_{g3}$, respectively. Pairing with weights $(5,19,23)$ reproduces Entry 550:

$$(5,19,23)V=(-62,-62,-33,-29,0).$$

The finite pair residues from Entry 549 add

$$R_{\rm pair}= \begin{pmatrix} 0&0&1&0&0\\ 0&0&0&1&0 \end{pmatrix}.$$

The valuation rows themselves have rank two because

$$\nu(K)=2\bigl(\nu(q_{g2})+\nu(q_{g3})\bigr).$$

After adjoining the pair rows,

$$\operatorname{rank} \begin{pmatrix} V\\R_{\rm pair} \end{pmatrix} =3.$$

The two pair rows already lie in the row span generated by the valuation data except for one new direction, raising the total rank from two to three. The common kernel is the canonical two-plane

$$\boxed{ \left\langle D_+-D_-, \gamma \right\rangle. }$$

Entry 559 independently computes the deck characters and identifies exactly this plane as the anti-invariant sector

$$B^-=\langle D_+-D_-,\gamma\rangle,$$

while the detected quotient is the rank-three invariant sector

$$B^+=\langle D_++D_-,E_+,E_-\rangle.$$

Consequence

Current regulator and incidence data determine exactly the deck-even rank-three part of the boundary packet. They cannot distinguish:

• the anti-invariant difference of the two normalization sheets;
• the cycle in the $K_{2,2}$ conductor graph.

Therefore the rank-five comparison map is still underdetermined even though both its source and target are now legitimately typed. No choice of regulator weights or recombination of the existing pair rows can repair the missing rank, because both directions vanish identically under all those rows.

The missing operations are geometrically specific:

$$\boxed{ \text{deck/normalization transition} \quad+\quad \text{conductor trace}. }$$

These are precisely the two operations not encoded by divisor valuations. The next calculation must derive their rows from the resolved double-cover Čech complex. If the two rows are independent modulo the present rank-three span, the comparison data become full rank five; otherwise the boundary packet cannot realize the contiguity residue cone.

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

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