Entry 712 · 2026-08-18 · marici.Benincasa · research

712 — The Oriented Triple Incidence Does Not Lift the Signed Pair Relation

Frozen incidence calculation

Use the disappearing triple

$$(q_{\mathfrak g_2},q_{\mathfrak g_3},q_{\mathfrak g_{23}})$$

and pair order

$$([23],[2,23],[3,23]).$$

Taking each ordered double residue and then the remaining residue gives

$$\boxed{ (r_{23},r_{2,23},r_{3,23}) =\left(\frac12,-\frac12,\frac12\right).}$$

All three remaining linear forms vanish at the same triple point $c=-X_2$. Hence the primitive oriented incidence vector is

$$\boxed{(1,-1,1).}$$

Failure of the proposed lift

Entry 709 proves the strict residue identity

$$[2,23]=[3,23].$$

Their two contributions to the triple incidence therefore cancel, leaving the minus pair $[23]$. Applying Entry 707's branch-discriminant symbol gives

$$\sigma_2(\operatorname{Disc})(1,-1,1) =C_{23}^-\nu_2\nu_3,$$

which is generically nonzero. Consequently

$$\boxed{ \text{the disappearing triple incidence is not a lift of the weighted signed-pair kernel relation}.}$$

This is a source-derived orientation test, not a rank argument.

What survives

Entry 711 gives the triple's first nonzero normal coefficient

$$T_2=X_3^2\nu_2^2 +(X_1^2-X_2^2-X_3^2)\nu_2\nu_3 +X_2^2\nu_3^2.$$

Thus the second-Rees triple costalk naturally lives on

$$\boxed{\eta^2=T_2.}$$

The mixed triple remains a valid Kummer coefficient object over the existing normal carrier, but it does not provide the sought polynomial chain homotopy between the signed pair residues.

Classification

• five pair occurrences: existing marked-incidence carrier;
• strict repeated-plus relation: incidence identity;
• mixed triple cover $\eta^2=T_2$: sector-specific Kummer coefficient data;
• weighted signed-pair relation: symbol-level only, with no lift from the adjacent triple cell.

No new carrier stratum is indicated.

Evidence

• research/benincasa/check_triple_to_pair_second_rees_boundary.py;
• Entries 707, 709, and 711;
• allocator claim seqclaim-11647084d139a42910b81661.

Next falsifier

Retire the direct triple-homotopy route. The remaining typed possibility is a nontrivial extension between the pair Kummer systems and the mixed triple costalk. Construct its Gauss--Manin connecting morphism on the quadratic normal cover and test whether its extension coefficient has $\mathcal Q$ support. If no source-derived connecting morphism exists, this entire lower normal branch remains disconnected from the homogeneous $\mathcal Q$ sector.

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