Entry 796 · 2026-08-18 · marici.Benincasa · research

796 — Generic Infinity Marks Meet Elliptic Branches Only on Soft Support

Frozen marked sector

In the $q_{\mathcal G_{12}}$-residue sector, put

$$q_{\mathcal G_{12}}=E+c=0, \qquad c=-E.$$

The five remaining source denominators are

$$\begin{aligned} q_{\mathfrak g_1}&=b+X_1-E,\\ q_{\mathfrak g_2}&=a+X_2-E,\\ q_{\mathfrak g_3}&=a+b+X_3,\\ q_{\mathcal G_{23}}&=a+E,\\ q_{\mathcal G_{31}}&=b+E. \end{aligned}$$

No marked section is added after seeing the elliptic boundary.

Restriction to infinity

On the chart

$$b=\frac1s, \qquad a=\frac ts,$$

the primitive strict transforms restrict at $s=0$ as

$$\begin{array}{c|ccccc} &q_{\mathfrak g_1}&q_{\mathfrak g_2}&q_{\mathfrak g_3} &q_{\mathcal G_{23}}&q_{\mathcal G_{31}}\\ \hline s=0&1&t&t+1&t&1. \end{array}$$

The two unit entries are visible in the complementary chart

$$a=\frac1r, \qquad b=\frac\tau r,$$

where both restrict to $\tau=0$. Thus the complete labelled marking is

$$\boxed{ \begin{aligned} t=0 &: q_{\mathfrak g_2},q_{\mathcal G_{23}},\\ t=-1 &: q_{\mathfrak g_3},\\ t=\infty &:q_{\mathfrak g_1},q_{\mathcal G_{31}}. \end{aligned} }$$

The doubled marks at $0$ and $\infty$ are source-derived occurrence multiplicities, not fitted cells.

Collision with the elliptic branch divisor

Entry 794 gives

$$F_P(t) =P_1^2t^4-(P_1^2+P_2^2-P_3^2)t^2+P_2^2.$$

The three marked-point resultants are simply

$$\boxed{ F_P(0)=P_2^2, \qquad F_P(-1)=P_3^2, \qquad F_P(\infty)=P_1^2, }$$

where the last expression is the leading coefficient in the $\tau=1/t$ chart.

Therefore every marked-section/branch collision at infinity lies on soft momentum support:

$$\boxed{ P_1P_2P_3=0. }$$

No residual irreducible divisor appears.

Classification

• elliptic coefficient geometry: generic genus-one infinity boundary;
• marked divisor: source incidence at $t=0,-1,\infty$;
• repeated marks: occurrence multiplicity already present in the source;
• collision support: soft momentum only;
• new carrier datum: none.

Together with Entry 794, this shows that the generic infinity-relative elliptic object is supported on the frozen Cayley--Menger and marked-energy geometry. It strengthens H2 beyond the homogeneous locus.

Verification

• exact checker: research/benincasa/marici-gm/src/bin/generic_marked_infinity_collisions.rs;
• machine-readable packet: research/benincasa/generic-marked-infinity-collisions.json;
• allocator claim seqclaim-72002d1bae386b402f537512.

Next falsifier

Compute the finite-chart intersections among the five marked sections and the branch divisor before taking $s=0$. Eliminate $(a,b)$ from each source-labelled collision. Test whether their complete resultant support is generated by the existing Cayley--Menger, energy-incidence, and soft divisors. Only a residual irreducible factor outside that union can indicate new generic carrier structure.

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