Entry 2143 · 2026-08-24 · marici.Benincasa · research

2143 — The All-Deleted Contact Product Has Rank-One Mixed Variation

Correction

Entries 2137--2138 and 2142 treated the isolated contacts in the all-deleted graph as additive sectors. This contradicts the frozen source.

Equation (2.14) of Benincasa--Dian, arXiv:2401.05207, sums over edge-deletion subsets and states explicitly that the unique $j=n_e$ term is a product of contact graphs. Equation (2.30) displays the same rule for the bubble: the both-edges-deleted term has the product denominator

$$y_ay_b(x_1+y_a+y_b)(x_2+y_a+y_b).$$

Thus the three contacts inside the triangle's all-deleted term multiply.

Correct two-wall local system

Near $\nu_i=\nu_j=0$, retain

$$(1,\tau_i)\otimes(1,\tau_j) = (1,\tau_i,\tau_j,\tau_i\tau_j), \qquad \tau_k=(2\pi i)^{-1}\log\nu_k.$$

The variation operators satisfy

$$V_i(\tau_i\tau_j)=\tau_j, \qquad V_j(\tau_i\tau_j)=\tau_i,$$

and hence

$$\boxed{ V_iV_j(\tau_i\tau_j) =V_jV_i(\tau_i\tau_j) =1. }$$

Therefore

$$\boxed{\operatorname{rank}(V_iV_j)=1.}$$

Comparison with the lower square-free packet

The lower Kummer line $\sqrt{\nu_i\nu_j}$ also has rank-one iterated variation, although with semisimple characters rather than logarithmic unipotent monodromy. Equality of ranks does not construct a comparison map, but the previous rank obstruction is gone.

The source has supplied exactly the operation that Entries 2137--2142 incorrectly declared absent:

$$\boxed{ \text{disconnected component product} \longrightarrow \text{canonical mixed nearby variation}.}$$

Corrected classification

• carrier: existing labelled deletion sector and component incidence;
• source operation: product of connected-component contact factors;
• two-normal mixed variation: rank one;
• lower-sector comparison: newly admissible but not yet constructed;
• $\mathcal Q$: not implicated;
• new carrier datum: none.

Evidence

• Benincasa--Dian, arXiv:2401.05207, equations (2.14), (2.30), and (4.71);
• Entries 2112, 2135--2142;
• research/benincasa/checkers/contact_product_mixed_variation.rs;
• allocator claim seqclaim-112bb3f891c08a460a090552.

Next falsifier

Construct the labelled comparison between the rank-one mixed variation line of the contact-product tensor system and the lower Kummer line. Preserve their distinct unipotent and semisimple monodromies. Determine whether a source-derived associated-grade or Mellin/Leray operation intertwines them; rank equality alone is insufficient.

View source on GitHub