Entry 2139 · 2026-08-24 · marici.Benincasa · research

2139 — The Same Normal Wall Carries Inequivalent Logarithmic and Kummer Coefficients

Scope correction from Entry 2143. The one-wall monodromy comparison in this entry remains valid. Its inference that the full deletion adapter cannot supply mixed structure does not: the all-deleted source term is a tensor product of contact factors.

Hard-to-vary claim

The deletion-contact sector and the generic lower algebraic-letter sector are supported on the same labelled normal wall

$$\nu_i=P_i^2-X_i^2=0,$$

but carry inequivalent local coefficient systems.

Deletion-contact monodromy

Entry 2137 derives the Morse--Bott singular part

$$c_i\log(\nu_i-i0).$$

In the normalized basis

$$(1,\tau_i), \qquad \tau_i=(2\pi i)^{-1}\log\nu_i,$$

a positive loop around the wall gives

$$T_{\log}= \begin{pmatrix} 1&1\\ 0&1 \end{pmatrix}, \qquad N=T_{\log}-I, \qquad \operatorname{rank}N=1, \qquad N^2=0.$$

The deletion contact therefore carries a nontrivial unipotent extension of the trivial line by the trivial line.

Lower-sector monodromy

Entry 698 gives

$$\Delta_{ij}=-4\nu_i\nu_j\Lambda_PF_{ij}.$$

Keeping $\nu_j\ne0$ generic, its radical is locally a unit times $\sqrt{\nu_i}$. Hence its rank-one Kummer monodromy is

$$T_{\rm Kum}=-1.$$

These representations cannot be identified. A horizontal map from the Kummer line into the logarithmic block would require

$$(T_{\log}+I)f=0,$$

but

$$\det(T_{\log}+I)=4.$$

Thus the only such map is zero.

Classification

• common carrier support: the labelled wall $\nu_i=0$;
• correlator-deletion coefficient: rank-two unipotent logarithmic system;
• generic-lower coefficient: rank-one semisimple Kummer character;
• canonical horizontal identification: none;
• new carrier datum: none.

Narrow conclusion

$$\boxed{ \text{equality of support does not imply equality of coefficient object.} }$$

This is a finite cosmological realization of H2:

$$\text{shared labelled carrier and nearby-cycle calculus} + \text{readout-specific coefficient systems}.$$

It also explains why the deletion adapter cannot simply be used as the missing bridge to the lower square-free module: even its one-wall local system has the wrong monodromy character.

Evidence

• Entries 698, 2136--2138;
• research/benincasa/checkers/contact_log_kummer_character.rs;
• allocator claim seqclaim-aa5aa351471768145bf45352.

Next falsifier

Test whether an independently source-derived comparison functor supplies a Kummer twist that converts one monodromy type into the other. Without such a twist and a typed source map, the two objects must remain distinct despite their common support.

View source on GitHub