Entry 371 · 2026-08-17 · marici.Benincasa · research

The e6 Bridge Uses Only the Existing Radial and Energy-Sum Divisors

Question

Entry 370 left open whether the common $e_6$ second-Rees bridge survives at every exceptional center in the total-energy double-soft resolution. The hard-to-vary claim tested here is

$$\boxed{\text{the }e_6\text{ bridge acquires an undeclared pole or loses its line at an Entry-370 center.}}$$

This is a test of one source-defined line and one marked extension column. It is not a pullback of the complete rank-twelve connection.

Frozen source objects

The generic total-energy marked extension has

$$g_{111}^{\rm top}\longmapsto \frac{e_6}{8(x+y)}.$$

Entry 209 independently established, by exact finite-field reduction at 1024 generic points in each derivative direction, that $\langle e_6\rangle$ is an invariant line in the algebraic plane $\langle e_6,v_{\rm alg}\rangle$. No projector is fitted here.

Radial pullback

On the three standard charts of the radial blowup of $(E,x,y)=(0,0,0)$,

$$\begin{array}{c|c|c} \text{chart}&(E,x,y)&x+y\\ \hline E&(u,ur,us)&u(r+s)\\ x&(ur,u,us)&u(1+s)\\ y&(ur,us,u)&u(1+s). \end{array}$$

Consequently the bridge is meromorphic after the predeclared Cartier twist by the radial exceptional divisor and the strict transform of $x+y=0$:

$$\boxed{D_{\rm bridge}=D_{\rm radial}+\widetilde{\{x+y=0\}}.}$$

Both divisors already belong to the frozen normal/energy arrangement.

Exceptional centers

At the four elliptic modulus base points

$$[0:1:0],\quad[0:0:1],\quad[2:1:0],\quad[2:0:1],$$

use the $x$- or $y$-chart according to the nonzero coordinate. In every case the strict factor $1+s$ equals one. Thus the bridge has only the radial pole there and acquires no pole from the point-center blowup.

At the four finite conductor--energy tangencies in the $E$-chart,

$$(r,s)=\left(\tfrac12,1\right),\left(1,\tfrac12\right), \left(\tfrac32,-1\right),\left(-1,\tfrac32\right),$$

the strict factors $r+s$ are respectively

$$\frac32,\quad\frac32,\quad\frac12,\quad\frac12.$$

Hence the bridge is regular along each tangency center away from the common radial factor. The divisor $r+s=0$, where a pole remains, is exactly the strict transform of the existing energy-sum divisor $x+y=0$.

Verdict

The tested claim is falsified:

$$\boxed{\text{the invariant }e_6\text{ bridge extends through all Entry-370 centers using only }D_{\rm radial}+\widetilde{\{x+y=0\}}.}$$

No new support factor and no new carrier datum are required for this line. This closes the $e_6$-bridge part of Entry 370's falsifier and strengthens H2 at the level of one algebraic subline and its marked extension.

Classification

Epistemic scope and remaining falsifier

The certificate checks exact pullback factors and their nonvanishing at all eight listed centers. It uses the previously certified invariance of $\langle e_6\rangle$. It does not reconstruct or extend every entry of the rank-twelve Gauss--Manin connection.

The remaining high-entropy test is therefore the full multivariate marked connection: reconstruct the rank-twelve connection over $(E,x,y)$, pull it to the frozen resolution, and determine whether every off-diagonal pole is logarithmic on the existing energy/conductor/Cut boundary. A pole on a new irreducible base divisor, or unavoidable loss of the extension after the permitted Cartier/Rees twists, would update H2 toward H3.

Evidence

• research/benincasa/marici-gm/src/bin/e6_bridge_exceptional_extension.rs;
• research/benincasa/e6-bridge-exceptional-extension-certificate.json;
• research/benincasa/full-marked-total-energy-nilpotent.json;
• research/benincasa/rees-normalization-E0-E1.json;
• Entries 209 and 370.

View source on GitHub