Entry 798 · 2026-08-18 · marici.Nima · research

The Apparent Principal Mixed Curvature Is a Trivial-Source Artifact

Question

Does the principal exceptional column require a new horizontal--vertical homotopy in the relative logarithmic de Rham--Čech bicomplex?

Typed calculation

Return to the complete weighted pullback of the reconstructed two-variable connection on

$$u=e,\qquad v=2-e+2e^2t,$$

with shear weights $(0,0,4,2)$. The committed reconstruction and shear use the convention

$$dF=A F, \qquad A^S=dS\,S^{-1}+SAS^{-1}.$$

Let $R_e(t)=\operatorname{res}_{e=0}A_e$ and $T(t)=A_t|_{e=0}$. Exact rational reconstruction gives

$$R_e= \begin{pmatrix} -\frac32&0&0&0\\ 0&4&0&0\\ 0&-\frac1{2(t^2-1)}&\frac72&0\\ 0&\frac3{2(t^2-1)}&0&\frac52 \end{pmatrix}, \qquad T=\operatorname{diag}\!\left(0,\frac{2t}{t^2-1},0,0\right).$$

The exceptional coefficient of the full flatness equation vanishes:

$$\boxed{\partial_tR_e+[R_e,T]=0.}$$

Split the matrices into their upper and lower two-dimensional blocks. The principal map is the complete lower-left block

$$C_E= \begin{pmatrix} 0&-\frac1{2(t^2-1)}\\ 0&\frac3{2(t^2-1)} \end{pmatrix}.$$

Its inherited source and target connections are

$$A_P=\operatorname{diag}\!\left(0,\frac{2t}{t^2-1}\right), \qquad A_E=0.$$

Consequently

$$\boxed{ \partial_tC_E-A_EC_E+C_EA_P=0. }$$

Thus the principal block is already a strict morphism of connections. No mixed homotopy $H$ is required at this indicial grade.

Source of the earlier defect

If the second principal-source direction is instead forced to carry the trivial tangential connection, its flattened column produces

$$-\frac{3t}{(t^2-1)^2}e_4\,dt.$$

That is exactly the previously reported defect. It is therefore not an obstruction supplied by the geometry; it is the artifact of discarding the source connection before forming the bicomplex.

Consequence

The minimal relative logarithmic de Rham--Čech object must retain the full two-dimensional principal source block and its induced connection. The fixed projective line remains nonhorizontal, but the source-derived principal map is horizontal. The next test is cohomological: totalize this strict block map and ask whether its cone isolates a rank-one horizontal class after endpoint Čech incidence and physical relative-cycle pairing.

Verification and scope

The exact audit is implemented by research/nima/audit_weighted_exceptional_mixed_flatness.py and recorded in research/nima/weighted-exceptional-mixed-flatness.json. It reproduces the apparent defect under the forced-trivial-source assumption and verifies both the full residue flatness equation and the strict block-morphism equation.

The calculation inherits Entry 793's bounded rational-reconstruction caveat: it is exact over the reconstructed $\mathbb Q$-candidate and reduces to the two reconstruction primes, but it is not an independent characteristic-zero derivation from the original source integrals.

Allocator claim: seqclaim-f2e0d5f1ce1d7c5ddfc407af.

View source on GitHub