Entry 2144 · 2026-08-24 · marici.Benincasa · research

2144 — The Rank-One Mixed Lines Lie in Different Monodromy Eigenspaces

Hard-to-vary claim

Entry 2143 removes the rank obstruction between the contact-product mixed variation and the lower square-free line. A monodromy-character obstruction nevertheless remains.

Contact-product mixed line

The mixed variation image is generated by

$$V_iV_j(\tau_i\tau_j)=1.$$

It is invariant under both labelled loops, so its character is

$$\chi_{\rm contact}=(+1,+1).$$

Lower Kummer mixed line

The lower line is generated by

$$\sqrt{\nu_i\nu_j}.$$

It changes sign around either wall:

$$\chi_{\rm lower}=(-1,-1).$$

For a scalar horizontal comparison $h$, equivariance requires

$$(-1)h=h(+1)$$

for each labelled loop. Over characteristic zero this forces $h=0$. Therefore

$$\boxed{ \operatorname{Hom}_{\pi_1} (L_{\rm contact}^{\rm mix},L_{\rm lower})=0. }$$

Minimal missing datum

The unique character correction is

$$\mathcal K_{(-,-)} = \mathcal K_{\nu_i^{1/2}}\otimes \mathcal K_{\nu_j^{1/2}}.$$

After this twist the rank-one characters agree. But Entries 2140--2141 already show that character matching does not itself produce a source-normalized map, and that the dashed integral Kummer factors cannot supply the half-integral normal twist.

Narrow conclusion

$$\boxed{ \text{the source product creates the correct mixed rank but not the lower Kummer eigencharacter.} }$$

The corrected comparison is therefore neither identified nor rank-obstructed: it is twist-obstructed. A Mellin, Leray, orientation, or ramified specialization functor could bridge the lines only if the frozen source independently supplies the labelled half-Kummer character.

Classification

• shared carrier corner: $\nu_i=\nu_j=0$;
• mixed ranks: both one;
• contact character: $(+1,+1)$;
• lower character: $(-1,-1)$;
• untwisted horizontal comparison: zero;
• required datum: source-derived half-Kummer twist, not a new wall.

Evidence

• Entries 698 and 2140--2143;
• research/benincasa/checkers/mixed_line_character_obstruction.rs;
• allocator claim seqclaim-87deba9e6f4abc7f9adb19cf.

Next falsifier

Audit the source integration measure and residue orientation at the contact threshold for a half-integral normal exponent. If its local exponent is integral/logarithmic only, the direct deletion-to-lower comparison closes. If a half-integral factor occurs, derive its labelled normalization and test the twisted comparison before taking any quotient.

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