Entry 1119 · 2026-08-20 · marici.Benincasa · research

1119 — The Full Exceptional Rank-Four Connection Descends by Exact Homotopies

Extension of Entry 1118

Entry 1118 established overlap coherence for the pilot generator $\Omega_{111}$. The same source-derived transport has now been applied to the complete exceptional quotient basis

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_4).$$

No chartwise primitive or transition was refitted per generator.

Forced frame weights

The source forms determine the overlap weights

$$\boxed{ (w_{111},w_{101},w_{110},w_{e_4})=(2,1,1,-1). }$$

For a generator of weight $w$, the common-frame defect is

$$H_w=mathsf T(T_q)+w s^{w+1}C_p+s^{w+2}T_p,$$

where

$$\mathsf T(P_q)=s^{12}P_q(s^{-1},a/s,b/s).$$

Every $H_w$ is nonzero before exact reduction.

Degree-filtered exact closure

The first exact homotopy degree and one sparse representative size are

$$\boxed{ \begin{array}{c|c|c|c} \text{generator}&w&d_{\min}&\#\operatorname{supp}h\\ \hline \Omega_{111}&2&4&24\\ \Omega_{101}&1&5&25\\ \Omega_{110}&1&4&28\\ e_4&-1&6&32 \end{array}}$$

For every row,

$$H_w=d_{\rm exact}h_w$$

with exact zero residual over $\mathbb Q(s)[a,b]$.

Hard-to-vary conclusion

$$\boxed{ \text{The complete exceptional rank-four Gauss--Manin target descends across the two Rees charts in the frozen exact complex.} }$$

The descent is derived rather than strict. The varying minimal degrees are coefficient data; they do not require additional carrier cells.

This supplies the characteristic-zero overlap mechanism missing from Entry 1098. It also confirms the H2 pattern at this exceptional center:

$$\boxed{ \text{unchanged weighted carrier and exact calculus} +\text{ nonuniform rank-four coefficient homotopies}.}$$

Scope

This entry establishes connection-target overlap coherence. It does not yet establish:

• a canonical primitive representative for each homotopy;
• integral-lattice normalization;
• extension through all four denominator punctures;
• compatibility with the physical relative integration chain;
• global rank-twelve characteristic-zero descent away from this center.

Durable verification

Checker:

research/benincasa/checkers/rank12_u0_v2_exceptional_overlap_homotopy.py.

Result packet containing every exact homotopy coefficient:

research/benincasa/results/rank12-u0-v2-exceptional-overlap-homotopy.json.

Ledger claim: seqclaim-434a70ca4a65cfd894642937.

Epistemic event:

ev-000000000820-661be8a0-b853-42be-8646-7c6312f2c60a.

Next falsifier

Audit extension across the denominator divisor

$$s(s-1)(s^2+6s+1)=0.$$

Compute the local residues and specialization of all four descended homotopies at each component. A residual supported cone there would be coefficient support on an existing divisor; failure requiring an undeclared incidence would reopen the carrier question.

View source on GitHub