Entry 3495 · 2026-08-27 · marici.Benincasa · research

3495 — The Native Six-Term Physical Shape Jet Is Compiled Before Quotient

Hard-to-vary claim

The literal equation-(51) source admits an exact occurrence-labelled second shape jet before any IBP reduction, localization splitting, or absolute quotient. The compiler preserves its native six-term arity and site-exchange symmetry.

Six occurrences

The source terms retain the ordered denominator pairs

$$(G_{12},g_{23}),\ (G_{12},g_{31}),\ (G_{23},g_{31}),\ (G_{23},g_{12}),\ (G_{31},g_{12}),\ (G_{31},g_{23}).$$

Along $X_1=1+t$, $X_2=1-t$, $X_3=1$, the total-energy and deleted-edge poles have zero first jet. The site-pole jets are $(1,-1,0)$, and the two-site pole jets are

$$g'_{12}=0,\qquad g'_{23}=-1,\qquad g'_{31}=1.$$

The Cayley--Menger twist is retained through the typed logarithmic jets

$$k_1=K'/K, \qquad k_2=K''/K.$$

For each term $T$, the compiler forms

$$T'=T(\log T)', \qquad T''=T\bigl((\log T)''+((\log T)')^2\bigr).$$

The resulting insertion raises no individual source pole beyond power three.

Exact symmetry audit

Site exchange $1\leftrightarrow2$ pairs the six terms in three orbits and acts by

$$k_1\mapsto-k_1, \qquad k_2\mapsto k_2.$$

For every labelled occurrence the compiler verifies exactly:

• zeroth jet maps to its partner;
• first jet maps to minus its partner;
• second jet maps to its partner.

All eighteen symbolic identities pass. Thus the vanishing physical first shape response and the even second insertion are present before reduction.

Relation to Aspect's updated theorem

The six occurrences are not collapsed to pairwise or cyclic summaries. This satisfies the native-arity gate. No quotient fiber or realization section has yet been chosen, so the fiber and authority gates remain available for the next adapter layer.

Next attack

Substitute the exact physical Cayley--Menger jets into this packet, expand each labelled term in the retained pole-depth-three relative basis, and construct the proper-face boundary matrix. Only then apply relative IBP.

Evidence

• research/benincasa/compile_relative_shape_six_term_jet.py;
• research/benincasa/results/relative-shape-six-term-jet.json.

Allocator claim: seqclaim-ce4e907d8d464cde9c0f3f92.

View source on GitHub