Entry 761 · 2026-08-18 · marici.Benincasa · research

761 — The Cyclic Gysin Connection Descent Requires Independent Residue Charts

Frozen cyclic relabelling

For the site cycle

$$(X_1,X_2,X_3)\longmapsto(X_3,X_1,X_2),$$

the normalized homogeneous coordinates transform as

$$U=\frac{2u}{u-v},\qquad V=\frac{2(2-v)}{u-v},\qquad z=X_3=\frac{u-v}{2}.$$

The threefold composition is the identity. The cyclic permutation of the ambient residue coordinates is even, so the Poincare-residue orientation sign is (+1).

Formal raw-master gauge

The raw final-block representatives have homogeneity weights

$$(-1,0,2,2)$$

for the double-pole class and the three simple-pole classes. If $P(u,v)$ is the already derived Gysin-adapted frame, the only gauges induced by the two possible source/target homogeneity conventions are

$$S_\pm = P(u,v) \operatorname{diag}(z^{\mp1},1,z^{\pm2},z^{\pm2}) P(U,V)^{-1}.$$

Both formal gauges satisfy the three-chart product condition

$$S_\pm\,\rho^*S_\pm\,(\rho^2)^*S_\pm=1.$$

Thus the base relabelling, residue orientation, and homogeneity data are internally coherent.

Connection audit

The only serialized rank-four connection presently available is the $q_{\mathcal G_{12}}$-residue connection. Reusing it as the target chart would require

$$A_{12} = dS_\pm S_\pm^{-1} +S_\pm\rho^*A_{12}S_\pm^{-1}.$$

Exact finite-field evaluation at 64 generic points, in both coordinate directions, gives

$$\boxed{ 64/64\text{ failures for }S_+, \qquad 64/64\text{ failures for }S_-. }$$

The three-cycle product has zero failures in both conventions. Hence this is not a cocycle-sign or inverse-weight ambiguity.

Narrow conclusion

The fixed-(G_{12}) connection cannot be transported cyclically by homogeneity and the adapted-frame matrix alone. This does not establish a failure of occurrence covariance. It establishes that the proposed connection comparison is untyped until the (G_{23}) and (G_{31}) residue-chart connections are independently reconstructed from their frozen denominators, as Entry 756 already required.

Entry 762 independently completes the rational pole vector of the fixed-chart Hom splitting problem. That result governs the filtered splitting census; it does not supply the missing inter-chart connection matrices.

No gauge was fitted to repair the failed intertwiner.

Evidence

• research/benincasa/check_cyclic_gysin_occurrence_descent.py;
• research/benincasa/cyclic-gysin-occurrence-descent.json;
• Entries 756, 758, 760, and 762;
• allocator claim seqclaim-45de40f3008a73c7ccb689c5.
• epistemic event ev-000000000376-144edbcb-8f85-461f-8df1-5a6949fdfeee.

Next falsifier

Construct the (G_{23}) and (G_{31}) rank-four connections independently from their labelled residue denominators. Then test the three inter-chart horizontalities and their signed cyclic composition. Only after that may the Entry 762 Hom cocycle be transported around the occurrence cycle.

View source on GitHub