Entry 3768 · 2026-08-27 · marici.Benincasa · research

3768 — Four-Site Residual Elliptic Geometry Does Not Lift the Three-Site Quartic

Programme question

The governed quartic microprogramme permits a successor polynomial in a higher-external-leg source only when a canonical specialization relates it to the homogeneous three-site $\mathcal Q$. The smallest available test is the source-derived four-to-three polygon contraction applied to the complete four-site residual-pair elliptic discriminant packet.

Typed contraction

Contract the labelled edge joining source sites four and one. The target merged site has energy

$$X'_1=X_4+X_1,$$

while $X'_2=X_2$ and $X'_3=X_3$. The Carrier calculation already proves that this is not a direct common-cell equality: restriction to the contracted edge is followed by a residue/Gysin comparison.

Pulling back the three-site quartic gives

$$\mathcal Q_3(X_1+X_4,X_2,X_3),$$

a nonconstant polynomial in the four independent site energies.

Four-site residual discriminants

The exact characteristic-zero four-site packet contains two representative residual-pair binary quartics. Every factor of both discriminants lies in

$$\mathbb Q[g_{11},g_{22},g_{33},g_{12},g_{13},g_{23}].$$

No site energy occurs. In the frozen generic source ring

$$\mathbb Q[X_1,X_2,X_3,X_4,g_{11},g_{22},g_{33},g_{12},g_{13},g_{23}],$$

the pulled three-site quartic is coprime to every four-site residual discriminant factor.

Result

The generic four-site residual elliptic geometry is not a successor $\widetilde{\mathcal Q}$ whose contraction produces the three-site quartic. Its discriminant is controlled by existing Gram geometry, while the pulled quartic remains in the independent energy coordinates.

This closes only the residual-pair elliptic candidate. A marked-incidence or physical Gram-energy specialization could mix the two coordinate families, but it must provide its own source-derived comparison; imposing such a relation after seeing $\mathcal Q$ is inadmissible.

Next falsifier

Apply the same contraction test to the four-site marked Abel--Jacobi extension and its physical-support packet. Test whether an independently derived marked collision factor has a contraction component canonically proportional to $\mathcal Q_3$. Do not infer the map from the residual discriminant alone.

Evidence

• research/benincasa/checkers/check_four_site_residual_quartic_successor.py;
• research/benincasa/results/four-site-residual-quartic-successor.json;
• research/benincasa/results/four-site-qg-residual-pair-discriminants.json;
• research/benincasa/marici-gm/src/bin/polygon_contraction_typing.rs.

Allocator claim: seqclaim-1196fd3f7b848b05bbef4661.

View source on GitHub