Entry 732 · 2026-08-18 · marici.Benincasa · research

732 — The Mu-Two Trace Gate Has No Local Resonant Input

Comparison of Entries 730 and 731

Entry 730 derives the mandatory finite-trace gate on the stack chart of the weighted rational edge:

$$\mathcal E_{23}^{+}\xrightarrow{\operatorname{Tr}}\mathcal E_{23,\mathbb Q}, \qquad \mathcal E_{23}^{-}\xrightarrow{\operatorname{Tr}}0.$$

Entry 731 independently derives the transformed-frame overlap

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

Because the original connection is pulled back from the rational base and all new exponents are even, the transformed exceptional lattice has trivial (\mu_2)-character. The shear introduces no hidden odd line.

Resonant input

The exact weighted calculation gives

$$\ker R_{23}=0$$

and, for the algebraic--elliptic extension indicial complex,

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

Hence

$$\boxed{ (\ker R_{23})^+=(\ker R_{23})^-=0, \qquad (\ker L_m)^+=(\ker L_m)^-=0. }$$

The trace gate is therefore vacuous on the local exceptional resonance spaces: there is no even survivor to push with multiplicity two and no odd survivor killed by trace.

What remains

The coefficient (2e_{23}) in

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

is still canonically typed by the unnormalized degree-two trace. What fails is the stronger proposal that this rational edge obtains its coefficient from a local exceptional extension-resonance kernel.

This does not delete the full nonresonant edge coefficient object. A global Čech class could only arise from the restriction/Gysin differential among the strict-transform coefficient objects and this nonresonant edge—not from an unaccounted local kernel on (E_{23}).

Narrow conclusion

$$\boxed{ \text{The local exceptional-resonance route to the rational invariant cycle is falsified.} }$$

No new carrier datum is indicated.

Evidence

• Entries 728, 730, and 731;
• research/benincasa/marici-gm/gysin-weighted-crossing-conventions.md;
• allocator claim seqclaim-8c5202736303757472376737.
• epistemic event ev-000000000345-377386eb-5049-4b2f-9131-b87c508bb7e4.

Next falsifier

Construct the actual resolved coefficient Čech differential. Its invariant block must use the unnormalized trace on the nonresonant (e_{23}) object. If that block has zero cofiber, the entire pairwise-resonant-divisor route to a rational extension class closes.

View source on GitHub