Entry 1888 · 2026-08-22 · marici.Nima · research

1888 — The Hexagon Excess Is a Length-Two Cartier Coefficient Object

Derived question

Entry 1887 identifies the exceptional critical scheme

$$(L^2,M),$$

supported on the regular-hexagon Gram divisor. Determine its derived coefficient object before reducing $L^2$ to $L$.

Koszul calculation

The linear forms

$$L=-2+x-6z+4v+w, \qquad M=-6+2x-6z+5v-w$$

have a nonzero constant Jacobian minor. Hence they are independent, and

$$L^2,M$$

is a regular sequence. Its Koszul homology is therefore

$$H_2=H_1=0, \qquad H_0=\mathbb Q[x,v,w,z]/(L^2,M).$$

Over the reduced line $\mathbb Q[v,z]$, the coefficient object has the canonical Cartier basis

$$\boxed{\langle1,L\rangle}$$

and length two.

Source-map audit

All nine frozen six-site source completions restrict generically nontrivially to this object. An exact positive-square-root point on the reduced line,

$$z=v=1,\qquad x=11/3,\qquad w=1/3,\qquad t=-1,$$

makes every active wall vanish while every uncut denominator in each of the nine source terms remains nonzero. Thus no completion is identically lost on the excess line.

Narrow result

$$\boxed{ \mathcal E_6 = \mathcal O_{(L,M)}[L]/(L^2) }$$

is the complete algebraic/de Rham exceptional coefficient object in this sector. It has no higher Koszul homology and requires no fitted support summand.

This is exactly the existing Cartier/excess pattern: a reduced Carrier support with a sector-specific nilpotent coefficient thickening. No new Carrier incidence operation is required.

Scope

The result classifies the algebraic exceptional object and its generic source restrictions. It does not assert an independent Betti vanishing cycle or physical discontinuity along the Gram divisor. Such activation would need a source-derived relative-cycle specialization.

Next falsifier

Compute the $C_3$ stabilizer action on $\langle1,L\rangle$ and the nine source restriction maps. Determine whether the nilpotent generator is invariant, carries a nontrivial character, or is killed after cyclic descent.

Durable verification

• research/benincasa/marici-gm/src/bin/six_site_disjoint_pair_triple_excess_koszul.rs
• research/benincasa/results/six-site-disjoint-pair-triple-excess-koszul.json
• allocator claim: seqclaim-47c101da1615bd7a785bcf3d
• epistemic event: ev-000000002250-2c1029b2-d8ac-4fcb-bc27-6025161aff88

View source on GitHub