Entry 731 · 2026-08-18 · marici.Benincasa · research

731 — The Rational Weighted Crossing Has No Exceptional Extension Resonance

Frozen crossing

At the rational intersection $D_2\cap D_3$, put

$$y=(u+v)/2-1, \qquad D_2=(y-u^2), \qquad D_3=(y+u^2).$$

Entry 726 found unequal raw pole orders $(-3,-2)$. The Newton equality

$$\operatorname{wt}(y)=2\operatorname{wt}(u)$$

therefore forces the weighted blowup with weights $(1,2)$.

Charts

The two charts are

$$U_u:\quad u=e,\quad y=e^2t,$$

and the stack chart

$$U_y:\quad u=es,\quad y=e^2,$$

with its intrinsic $\mu_2$-action. On their overlap $t=s^{-2}$.

Forced exceptional transform

Pullback of the complete adapted rank-four connection, including its algebraic--elliptic extension block, gives the exceptional valuation matrix

$$\begin{pmatrix} -1&\infty&\infty&\infty\\ 3&-1&\infty&\infty\\ -1&-5&-1&-2\\ 1&-3&2&-1 \end{pmatrix}.$$

The unique minimum-sum normalized nonnegative integral shear making every entry logarithmic is

$$\boxed{(0,0,4,2).}$$

The frame transition on the overlap is consequently

$$\boxed{\operatorname{diag}(1,1,s^{-4},s^{-2}).}$$

Its even exponents prove $\mu_2$-descent; no square-root trivialization was chosen.

Exceptional residue

Both charts give

$$\operatorname{rank}R_{23}=4, \qquad \ker R_{23}=\operatorname{coker}R_{23}=0.$$

For the algebraic--elliptic extension indicial operators,

$$\boxed{\ker L_m(R_{23})=0\quad(1\le m\le10).}$$

Meanwhile the strict transforms (t=1) and (t=-1) remain logarithmic and retain two-dimensional first indicial kernels. Thus the loss of resonance is specific to the new rational exceptional component.

Relation to Entry 728

The constant resolved dual graph has the invariant cycle

$$\gamma_0=(e_{12}^++e_{12}^-)-(e_{13}^++e_{13}^-)+2e_{23}.$$

The rational edge $e_{23}$ does not acquire an exceptional algebraic--elliptic extension resonance from the weighted crossing. This is a local coefficient statement, not yet a proof that the global invariant cycle is exact: that requires the actual restriction maps from vertex and edge coefficient objects.

Narrow conclusion

$$\boxed{ D_2\cap D_3\text{ contributes no local exceptional extension-resonance class after the Newton-forced transform.} }$$

No new carrier component and no pairwise physical obstruction are inferred.

Evidence

• research/benincasa/gysin_weighted_crossing_blowup.py;
• research/benincasa/marici-gm/gysin-weighted-crossing-conventions.md;
• research/benincasa/marici-gm/src/bin/gysin_weighted_crossing_certificate.rs;
• allocator claim seqclaim-7d16f08f97e1c708a87cfa61.
• epistemic event ev-000000000344-f9226ca3-5c10-4dee-ac04-142d898cfa51.

Next falsifier

Construct the coefficient Čech differential on the resolved five-edge graph using the chart transition matrices from Entries 729 and 731. Project it to

$$\mathbf1\oplus\chi_{-3}\oplus\chi_5.$$

Only a nonzero cofiber in the invariant block can support a rational pairwise obstruction.

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