Entry 647 · 2026-08-18 · marici.Nima · research

599 — The Physical Source Has Zero Mixed Occurrence Double Residue

Hard-to-vary claim

The rank-one mixed occurrence class of Entry 598 exists in the ambient five-mark coefficient system, but the source-prescribed physical $q_{G_{12}}$-residue has zero component on its codimension-two residue stratum.

Source-local calculation

Use local coordinates at the occurrence intersection

$$u=q_{g_{31}}=a-y, \qquad v=q_{g_{23}}=b-x.$$

The two source summands have the same common three-wall and Cayley--Menger factor. Their occurrence-dependent factor is exactly

$$\frac1v+\frac1u = \frac{u+v}{uv}.$$

The prospective iterated residue on $u=v=0$ is therefore proportional to

$$\left.(u+v)\right|_{u=v=0}=0.$$

Hence

$$\boxed{ \operatorname{Res}_{u=0}\operatorname{Res}_{v=0} \Omega_{\mathrm{phys}}=0 }$$

at generic kinematics, where the common coefficient factor is regular at the intersection.

This is not cancellation after choosing a master basis. It follows directly from the frozen source sum and the two source denominator equations.

Compatibility with the scattering degeneration

Entry 598 showed

$$K_E(y,x)=E_T^3\cdot\text{unit}$$

on the generic scattering boundary. Thus the common coefficient ceases to be regular at $E_T=0$, and generic double-residue vanishing alone would not exclude a filtered exceptional class.

Entry 591 independently computed the source-derived ramified chart

$$E_T=\tau^2$$

and found that the leading unsplit exceptional form is exact, with zero simple occurrence-exceptional residue. Combining the two calculations gives:

• zero generic mixed double residue;
• zero first source-derived weighted exceptional residue;
• no established vanishing theorem for every higher filtered grade.

Consequence

The rank-twenty-six ambient localization complex remains the correct deletion-closed coefficient object, but the physical source vector lies in the kernel of its mixed codimension-two residue map.

Equivalently, the extra rank-one class in Entry 598 is an available occurrence-incidence coefficient direction, not a physically occupied source component at the tested grades.

No carrier modification follows.

Next falsifier

Construct the source-unsplit localization cocycle on the remaining three shared walls, with its mixed occurrence component fixed to zero. Then compute the supported pushforward of its conductor boundary into the rank-seven algebraic/Tate kernel. This avoids introducing a spurious mixed generator into the physical comparison map.

Evidence

• research/benincasa/check_unsplit_occurrence_pair.rs;
• research/benincasa/unsplit-occurrence-pair.json;
• Entries 590, 591, and 598.

View source on GitHub