Entry 1121 · 2026-08-20 · marici.Benincasa · research

1121 — The Quadratic Kummer Line Is a Coherent Existing-Wall Residue

Corrected local coefficient object

Entry 1120's divisor-field correction gives, on

$$f=s^2+6s+1=0,$$

a semisimple residue with one $-\tfrac12$ eigenline and a rank-three zero-residue complement. Thus local monodromy is

$$T=-1\quad\text{on the Kummer line}, \qquad T=1\quad\text{on its complement}.$$

Consequently

$$\operatorname{rank}(T-I)=1, \qquad \dim\ker(T-I)=\dim\operatorname{coker}(T-I)=3.$$

Source-labelled Kummer generator

In the quotient basis

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_5),$$

the exact spectral projector is

$$P_- =4R_f^2.$$

Its image is generated over $\mathbb Q[s]/(f)$ by

$$\boxed{ k= \left(-\frac{s+7}{4},1,0,0\right)^T. }$$

No basis fitting enters this line: it is the image of a polynomial in the source-derived residue matrix.

Existing Cousin realization

Use Entry 851's frozen Poincaré-residue maps, with stacked wall basis

$$(t_1,g_1,t_2,g_2).$$

Their restriction to the rank-four quotient sends

$$k\longmapsto \boxed{ \left( \frac{s+7}{4},-1,-\frac{s+7}{4},0 \right)^T. }$$

This image is nonzero and has rank one on $f=0$. The same-sheet top map

$$\operatorname{Res}_{\rm top}=(1,0,1,0)$$

annihilates it exactly:

$$\operatorname{Res}_{\rm top}\operatorname{Res}_{W}(k)=0.$$

The cancellation uses the source-fixed opposite orientations of the two iterated top residues. It is not a later choice of sign or splitting.

Hard-to-vary conclusion

$$\boxed{ \text{The unique anti-invariant quadratic Kummer line is already a coherent class in the frozen two-wall Cousin complex.} }$$

Therefore the quadratic exceptional collision produces coefficient monodromy, but no unsupported coefficient direction and no new carrier incidence. The local result realizes H2 concretely:

$$\text{existing marked carrier and residue calculus} +\text{ one sector-specific Kummer line}.$$

Scope

This establishes algebraic/de Rham support compatibility. It does not show that the physical Bunch--Davies relative chain pairs nontrivially with the Kummer line, nor does it fix an integral normalization.

Durable verification

Checker:

research/benincasa/checkers/rank12_u0_v2_residue_monodromy_complex.py.

Packet:

research/benincasa/results/rank12-u0-v2-residue-monodromy-complex.json.

Ledger claim: seqclaim-57113259c57809d08ea22b19.

Epistemic event:

ev-000000000825-e903e6bf-c6cf-49e3-a8a3-e1b716c1dd31.

Next falsifier

Pull the source-normalized physical relative chain to the quadratic marked collision and compute its pairing with $k$. The possible outcomes are:

• nonzero, lift-independent pairing: a physical Kummer coefficient class;
• zero pairing: an algebraic class invisible to the physical chamber;
• no source-defined specialization: physical activation remains undefined.

No carrier modification is admissible in any outcome.

View source on GitHub