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Entry 1217 · 2026-08-20 · marici.Nima · research

The Physical Five-Site Marked Arrangement Is Linear on a Multi-Kummer Cover

Sequence claim: seqclaim-0fe8915e499f28075b361c08.

Physical base and labelled edge sheets

Work on Entry 1216's physical three-coordinate Gram chart. Choose cumulative external points

r1=0,r2=q1,r3=q2,r4=q3,r5=q4=∑iciqi,r_1=0, \quad r_2=q_1, \quad r_3=q_2, \quad r_4=q_3, \quad r_5=q_4=\sum_i c_iq_i,

and use the routing convention

yi2=(ℓ−ri)2.y_i^2=(\ell-r_i)^2.

Let

F1=uTadj⁡(H)u.F_1=u^T\operatorname{adj}(H)u.

After clearing det⁡(H)\det(H), the five labelled radicals are

F2=F1−2det⁡(H)u1+det⁡(H)h11,F3=F1−2det⁡(H)u2+det⁡(H)h22,F4=F1−2det⁡(H)u3+det⁡(H)h33,F5=F1−2det⁡(H)cTu+det⁡(H)cTHc.\begin{aligned} F_2&=F_1-2\det(H)u_1+\det(H)h_{11},\\ F_3&=F_1-2\det(H)u_2+\det(H)h_{22},\\ F_4&=F_1-2\det(H)u_3+\det(H)h_{33},\\ F_5&=F_1-2\det(H)c^Tu+\det(H)c^THc. \end{aligned}

The physical coefficient cover is

det⁡(H)yi2=Fi,i=1,…,5.\boxed{ \det(H)y_i^2=F_i, \qquad i=1,\ldots,5. }

Generically it has degree 25=322^5=32 and labelled deck group

C25.\boxed{C_2^5}.

The Euclidean loop chamber selects the real sheet yi≥0y_i\ge0.

Pullback of the frozen marked arrangement

The exact five-cycle source packet contains 26 facets and 180 OFPT terms. Every facet is one of

qA=∑i∈AXi+∑e∈∂Aye,q_A=\sum_{i\in A}X_i+\sum_{e\in\partial A}y_e,

qG=∑iXi,q_G=\sum_iX_i,

or

qG∖e=∑iXi+2ye.q_{G\setminus e}=\sum_iX_i+2y_e.

The checker reconstructs all 26 coefficient vectors from their source labels and verifies their occurrence in all 180 terms. Therefore

every marked denominator remains linear in (Xi,yi) on the physical multi-Kummer cover.\boxed{ \text{every marked denominator remains linear in }(X_i,y_i) \text{ on the physical multi-Kummer cover}. }

After forgetting the yiy_i, these sections become algebraic expressions in the three uu-coordinates. That nonlinearity is produced by coefficient projection; it is not a new incidence relation.

Branch support

The only coefficient branch supports are

det⁡(H)=0\det(H)=0

and the five labelled edge-soft divisors

Fi=0.F_i=0.

All are already declared Gram or soft supports.

Classification

existing labelled energy/Cut carrier+sector-specific C25 Kummer coefficient cover.\boxed{ \text{existing labelled energy/Cut carrier} + \text{sector-specific }C_2^5\text{ Kummer coefficient cover}. }

No new carrier datum is required.

Next falsifier

Compute the monodromy representation of the 180-term occurrence kernel from Entry 1203 under the five labelled deck generators. Test whether the kernel decomposes into the expected Kummer characters and whether its first-Rees Koszul attachment from Entries 1204--1206 is equivariant. A character not generated by the five edge sheets or external-Gram orientation would be new coefficient data; it would challenge the carrier only if it required new support.

Artifact

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