Entry 1818 · 2026-08-21 · marici.Nima · research

1818 — The Active Transverse Pairs Assemble into a Rank-110 Physical Coefficient Object

Construction

Combine:

• Entry 1809's exact nonzero source coefficients;
• Entry 1810's 110 labelled cyclic occurrences;
• Entry 1811's rank-one rational quadratic quotients $\mathcal L_{A,B}$;
• Entry 1813's normal orientation lines;
• Entry 1817's physical intersection pairings;
• Entry 1216's external-Gram Kummer density.

On the nonsingular external-Gram chart, the supported physical object is

$$\boxed{ \mathcal P_{\rm tr} = \mathcal K_{\det(H)^{-1/2}} \otimes \bigoplus_{(A,B)\in\mathcal O_{110}} i_{A,B*} \left( \mathcal L_{A,B}\otimes\det N^*_{A,B} \right). }$$

Here $\mathcal O_{110}$ is the complete labelled active occurrence set.

Normalization

For each ordered occurrence, the physical real-chain intersection index cancels the sign of the determinant-line frame. The remaining scalar is exactly the source double-residue coefficient certified in Entry 1809. Every such scalar is nonzero.

Therefore

$$\operatorname{rank}\mathcal P_{\rm tr}=110, \qquad \chi_{C_5}=(110,0,0,0,0).$$

Monodromy and support

The two coefficient factors remain distinct:

$$T_{\rm quadratic}=+1, \qquad T_{\det H}=-1.$$

On their generic normal-crossing locus these monodromies commute. At the central threshold node, Entry 1812 shows that the quadratic factor splits over the constant field $\mathbb Q(i)$; it introduces no new kinematic character.

Result

The five-site transverse-pair lane yields a source-normalized, physically activated, rank-110 supported coefficient object. Its support and twists are compiled entirely from:

• existing threshold and marked-wall incidence;
• the existing external-Gram Kummer density;
• normal-orientation/Gysin data;
• source-specific rational quadratic quotients.

No new carrier generator is required.

Scope

This is a local supported physical object on the nonsingular external-Gram chart. Global continuation through $\det(H)=0$ and simultaneous deeper soft corners is not asserted.

Next falsifier

Push $\mathcal P_{\rm tr}$ to the external-Gram divisor. Compute whether the Kummer twist and oriented quadratic quotient extend by the existing Gram nearby-cycle/Gysin calculus or leave a supported excess.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_supported_physical_object.py
• research/benincasa/results/five-site-g5-transverse-supported-physical-object.json
• Entries 1216 and 1809--1817
• allocator claim: seqclaim-61b6cd114a1f001e721a634f

View source on GitHub