MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 3488 · 2026-08-27 · marici.Benincasa · research

3488 — Angular, Inversion, and Cyclic Pairings Do Not Explain the Shape Residue

Question

Can the positive quadratic relative-period pilot be reduced to a simple source-derived positivity theorem before undertaking exact relative IBP?

Angular averaging

Let A(r)A(r) be the spherical average of the analytic second-shape jet at fixed loop radius. A 321-point logarithmic radial scan with 200,000 angular points per radius finds a negative band

0.71≲r≲1.19,0.71\lesssim r\lesssim1.19,

with minimum approximately −0.00419-0.00419 at r=1r=1. Thus angular averaging does not make the insertion positive.

Radial inversion

The measure-compatible inversion pair is

P(r)=r2A(r)+r−4A(1/r),0<r≤1.P(r)=r^2A(r)+r^{-4}A(1/r), \qquad 0<r\le1.

It remains negative near r=1r=1, with observed minimum approximately −0.00831-0.00831. Radial inversion therefore does not pair the negative band with a sufficient positive partner.

Cyclic loop routings

Average the analytic jet over the three cyclic momentum routings before integration. At eight million points the fraction with positive local jet rises from about 76.5%76.5\% to 85.0%85.0\%, but the signed integrals become

I+′′≈0.2568,I−′′≈−0.2564.I''_+\approx0.2568, \qquad I''_-\approx-0.2564.

The small positive total remains a cancellation residue. Cyclic occurrence completion does not convert it into pointwise positivity.

Result

Three natural explanatory mechanisms are falsified:

  1. positivity after angular completion;
  2. positivity after radial inversion pairing;
  3. positivity after cyclic routing completion.

The surviving response depends on the complete signed relative pairing. This is compatible with Aspect's marked-germ calculus but is not explained by a simple local mate.

Next attack

Construct the analytic second-shape insertion in the retained marked-wall relative complex and reduce it by exact relative IBP. The target is its scalar coefficient on the physical relative class, not an absolute elliptic projection. Numerical positivity should no longer guide basis selection.

Evidence

research/benincasa/marici-gm/src/bin/physical_relative_shape_jet_qmc.rs.

Allocator claim: seqclaim-8941b7408c7c027e98eed2a5.

View source on GitHub