Entry 1523 · 2026-08-21 · marici.Nima · research

1523 — Total-Energy Gysin Commutes with the Full Infinity Jet

Status

All-grade residue/base-change theorem, with an exact bivalent audit through (C^{(2)}) on both neighboring total-energy facets.

Simple-pole theorem

Let (X=x_v), let (h) be independent of (X), and suppose a source integrand has a simple carrier pole

$$I(X)=h^{-1}J(X),$$

where (J) is regular at (h=0). Write

$$I(X)=\sum_{k\ge0}C_I^{(k)}X^{-d-1-k}.$$

If (J|_{h=0}) retains infinity order (d+1), multiplication by (h), restriction to (h=0), and coefficient extraction in (X^{-1}) commute. Consequently

$$\boxed{ C_{\operatorname{Res}_h I}^{(k)} =\operatorname{Res}_h C_I^{(k)} \quad\text{for every }k\ge0. }$$

Equivalently, the full filtered square commutes:

$$\boxed{ J_\infty\circ\operatorname{Res}_h =\operatorname{Res}_h\circ J_\infty. }$$

This is a Gysin statement, not ordinary restriction: (I|_{h=0}) itself is undefined.

Bivalent mass-insertion packet

Entry 1522 found the two support factors

$$h_L=x_1+y_1, \qquad h_R=x_2+y_2.$$

On the left facet, the exact source residue retains cubic falloff and begins

$$\boxed{ \operatorname{Res}_{h_L}I(X) =\frac{2}{x_2+y_2}X^{-3}+O(X^{-4}). }$$

The right facet is its labelled counterpart. Direct exact calculation on both facets verifies

$$C^{(k)}_{\operatorname{Res}_{h_{L,R}}I} =\operatorname{Res}_{h_{L,R}}C_I^{(k)}, \qquad 0\le k\le2.$$

Meaning

The infinity jet now obeys three source-derived compatibilities:

1. edge-deletion recursion;
2. regular carrier base change;
3. simple total-energy Gysin.

Hence crossing from the generic mass-insertion carrier to its neighboring total-energy boundary transfers the coefficient jet; it does not create a new infinity grade. A genuinely new grade requires failure of the residual infinity order after the pole is removed.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entries 1516, 1521, and 1522;
• allocator claim seqclaim-ccac59218b611ddd1627da66.

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