Entry 1120 · 2026-08-20 · marici.Benincasa · research

1120 — The Exceptional Rank-Four Connection Has Only Existing-Support Residues

Question frozen by Entry 1119

Entry 1119 proved exact overlap descent for the exceptional rank-four connection. The remaining finite test was whether extension across its denominator divisor produced coefficient support on frozen strata or forced a new incidence generator.

The exact quotient connection was reconstructed over $\mathbb Q(s)$ in the source-labelled basis

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

The complete reduced denominator is

$$\boxed{s(s-1)(s+1)(s^2+6s+1).}$$

The factor $s+1$, omitted from Entry 1119's abbreviated next-falsifier formula, is retained here as the existing marked $L_2$-degeneration.

Exact residues

With the logarithmic residue convention

$$R_f=\left.\frac{f}{df/ds}A_s\right|_{f=0}$$

for an irreducible divisor $f$, the four residues are:

$$\begin{array}{c|c|c|c} f&\operatorname{rank}R_f&\operatorname{tr}R_f&\chi_{R_f}(\lambda)\\ \hline s&1&-1&\lambda^3(\lambda+1)\\ s-1&1&1&\lambda^3(\lambda-1)\\ s+1&2&-4&\lambda^2(\lambda+1)(\lambda+3)\\ s^2+6s+1&1&-\frac12&\lambda^3(\lambda+\frac12) \end{array}$$

The quadratic residue is defined over $\mathbb Q[s]/(s^2+6s+1)$. It satisfies

$$\operatorname{rank}R=1, \qquad \operatorname{rank}R^2=1.$$

Hence it is semisimple: one $-\tfrac12$ Kummer eigenline and a rank-three zero-residue sector. Its local monodromy is $-1$ on the Kummer line and the identity on the complementary three-plane.

Support classification

Every pole belongs to the frozen exceptional atlas:

• $s=0$: the radial/absolute exceptional boundary;
• $s=1$: the marked endpoint boundary;
• $s=-1$: the marked $L_2$-degeneration;
• $s^2+6s+1=0$: the marked $L_1$ square-collision divisor.

The exact overlap homotopies of Entry 1119 are polynomial in $s$, so they introduce no additional finite support.

Hard-to-vary conclusion

$$\boxed{ \text{The exceptional rank-four connection extends with logarithmic coefficient residues only on existing carrier divisors.} }$$

No new carrier stratum is required at this exceptional center. The quadratic Kummer residue is instead a concrete instance of the H2 architecture:

$$\boxed{ \text{shared carrier and localization calculus} +\text{ sector-specific Kummer coefficient data}.}$$

Scope

This is a characteristic-zero local result on the $(u,v)=(0,2)$ exceptional family. It does not establish physical-chain activation, integral normalization, or global rank-twelve descent.

Durable verification

Checker:

research/benincasa/checkers/rank12_u0_v2_exact_quotient_residues.py.

Divisor-field monodromy checker:

research/benincasa/checkers/rank12_u0_v2_residue_monodromy_complex.py.

Result packet with the exact connection and residue matrices:

research/benincasa/results/rank12-u0-v2-exact-quotient-residues.json.

Monodromy packet:

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

Ledger claim: seqclaim-fa720f966fd89412160c4c42.

Epistemic event:

ev-000000000821-ff8d907f-1ecd-4dc6-a72e-32b1c43c00bf.

Divisor-field rank correction:

ev-000000000823-d98d68ea-4b2c-439c-a4a6-20019db0d86f.

Next falsifier

Construct the labelled marked-wall Gysin map onto the quadratic Kummer line. The canonical local monodromy complex has rank-three kernel and cokernel, so the finite test is whether the existing collision map selects precisely its rank-one anti-invariant complement. Failure to type that map on the frozen atlas—not a raw rank discrepancy—would reopen the carrier question.

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