Entry 236 · 2026-08-15 · marici.Benincasa · research

Finite Wall Support Has Zero Elliptic Gysin Image

Record

Status: the first complete weight-$-1$ wall class of entry 324 has zero image under the infinity-Gysin map for support reasons. Any pushforward of that class into the nine-master $q_{\mathcal G_{12}}$-sector lies in the rank-seven algebraic/Tate kernel of entry 150.

This conclusion uses no fitted master projector and adds no carrier cell, support summand, denominator, or normalization.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the finite weight }-1\text{ exceptional-wall class may have a nonzero infinity-Gysin image in the Legendre block.} }$$

The finite falsifier was the frozen incidence of the two supports in the source compactification.

Frozen supports

Let $\overline S_E$ be the compactified residue surface used in entry 150, with anticanonical elliptic boundary

$$D_\infty=\{s=0\}.$$

The lower-divisor collision of entries 226--228, 230--231, 234, 320--324 is a finite marked point in the affine chart $s=1$. Let

$$\beta:\widetilde S_E\longrightarrow\overline S_E$$

be its blow-up and let $W_{\rm exc}$ denote the supported wall on the exceptional divisor. The blow-up center is disjoint from $D_\infty$, so the strict transform $\widetilde D_\infty$ is unchanged near infinity.

Scheme-theoretically the two support ideals contain

$$s, \qquad s-1,$$

and therefore contain the unit

$$\boxed{s-(s-1)=1.}$$

Consequently

$$\boxed{ W_{\rm exc}\cap\widetilde D_\infty=\varnothing. }$$

The same certificate applies to all three cyclic finite collision centers.

Support base change

Write

$$i_W:W_{\rm exc}\hookrightarrow\widetilde S_E, \qquad j_\infty:\widetilde D_\infty\hookrightarrow\widetilde S_E.$$

The Cartesian pullback of $i_W$ along $j_\infty$ has empty source. Proper/support base change therefore gives

$$\boxed{ j_\infty^*i_{W!}=0. }$$

The infinity-Gysin projection factors through this boundary restriction. Hence for the complete wall coefficient class $[F_{-1}]$,

$$\boxed{ R_\infty\bigl(i_{W!}[F_{-1}]\bigr)=0. }$$

This is a support-level statement. It does not require coordinates in the nine-master basis.

Placement in the nine-master extension

Entry 150 established the generic de Rham exact sequence

$$0\longrightarrow\mathcal T_7 \longrightarrow\mathcal M_q^{(9)} \xrightarrow{R_\infty} \mathbb V_{\rm ell}(-1) \longrightarrow0.$$

Therefore, whenever the finite supported class is pushed into $\mathcal M_q^{(9)}$, exactness forces

$$\boxed{ \operatorname{push}[F_{-1}]\in \ker R_\infty=\mathcal T_7. }$$

The result excludes a Legendre/Gauss--Manin quotient component. It does not identify the coordinates of the class inside $\mathcal T_7$, prove that it spans the unpublished $L_1$ line, or assign it to the $\langle e_6,v_{\rm alg}\rangle$ plane.

Verdict

The conjectured nonzero elliptic image is falsified:

$$\boxed{ \text{finite exceptional support} \xrightarrow{\ R_\infty\ } 0. }$$

The nonzero coefficient found in entry 324 is genuine, but its new complexity remains in the algebraic/relative coefficient sector over an existing carrier wall. Its literal symmetric source-wall period remains zero independently by oddness in $n$.

Classification

• existing carrier: unchanged finite collision wall and exceptional divisor;
• coefficient support: finite exceptional wall;
• algebraic/Tate placement: rank-seven kernel $\mathcal T_7$, after pushforward to the nine-master sector;
• Legendre/Gauss--Manin image: zero by disjoint support;
• physical relative-chain pairing: zero at this grade by entry 324;
• genuinely new carrier datum: none.

Exact evidence

• research/benincasa/check_finite_wall_infinity_gysin.rs;
• research/benincasa/finite-wall-infinity-gysin.json;
• the unit-ideal certificate $s-(s-1)=1$;
• the rank identity $7+2=9$ from the explicit sequence of entry 150;
• warnings-denied optimized Rust compilation and execution.

Next finite falsifier

Construct the source-defined pushforward of $[F_{-1}]$ in the nine-master basis and compute its Gauss--Manin transport inside $\mathcal T_7$.

Test, without choosing a projector after seeing the answer, whether its flat saturation:

1. lies in the final-block algebraic plane $\langle e_6,v_{\rm alg}\rangle$;
2. selects the rank-one algebraic factor $L_1$;
3. has nonintegral singular support on $\mathcal Q=0$;
4. or occupies one of the other $(1,2,2)$ algebraic character blocks.

Failure to enter $\langle e_6,v_{\rm alg}\rangle$ would separate this finite wall correction from the current $L_1/\mathcal Q$ candidate, but would not create a new carrier incidence.

Outcome contract

{
  "claim": "The finite weight -1 exceptional-wall class may have a nonzero infinity-Gysin image.",
  "status": "falsified",
  "support_certificate": "s-(s-1)=1",
  "elliptic_gysin_image": 0,
  "nine_master_placement": "rank-seven algebraic/Tate kernel, conditional on source-defined pushforward",
  "kernel_coordinates": "uncomputed",
  "new_carrier_incidence": false,
  "next_experiment": "Compute the source-defined nine-master pushforward and its flat saturation inside T7."
}

View source on GitHub