Entry 800 · 2026-08-18 · marici.Benincasa · research

800 — The Principal Vertical Column Is Horizontal in the Actual Connection Convention

Hard-to-vary claim

For the canonical two-prime reconstructed weighted packet, the complete principal vertical residue column is compatible with the tangential Gauss--Manin transport. No mixed homotopy is required at this residue level.

The apparent obstruction obtained from

$$\partial_tR_e+A_tR_e-R_eA_t$$

uses the opposite connection convention from the one obeyed by the reconstructed matrices.

Frozen packet and convention audit

Use Entry 793's reconstructed matrices and weighted chart

$$u=e, \qquad v=2-e+2e^2t, \qquad (w_1,w_2,w_3,w_4)=(0,0,4,2).$$

Direct evaluation of the original two-variable matrices at three exact generic rational points distinguishes the two signs:

$$\partial_uA_v-\partial_vA_u-[A_u,A_v]=0,$$

whereas the same expression with (+[A_u,A_v]) is nonzero. Thus the packet obeys

$$\boxed{dF=A F,\qquad dA-A\wedge A=0.}$$

This corrects the sign statement serialized in Entry 793's derived packet.

Complete exceptional residue data

The tangential exceptional connection is

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

Retaining the full normal residue after the weighted shear gives

$$R_e= \begin{pmatrix} -\frac32&0&0&0\\ 0&4&0&0\\ 0&-\frac1{2(t^2-1)}&\frac72&0\\ 0&\frac3{2(t^2-1)}&0&\frac52 \end{pmatrix}.$$

Its lower-left block is precisely the principal vertical map before taking the endpoint residues:

$$C_E(t)= \begin{pmatrix} 0&-\dfrac1{2(t^2-1)}\\[1mm] 0&\dfrac3{2(t^2-1)} \end{pmatrix}.$$

The diagonal entries of (R_e) retain the source and target normal-residue connections. They may not be discarded when testing whether (C_E) is horizontal.

Mixed-flatness calculation

For the actual convention, the coefficient of the mixed curvature is

$$\Theta = \partial_tR_e-A_tR_e+R_eA_t.$$

Exact symbolic reduction gives

$$\boxed{\Theta=0.}$$

In particular, on the principal lower-left block,

$$\boxed{ \nabla_{mathrm{target}}C_E = C_E\nabla_{mathrm{source}}. }$$

No equation of the form

$$\Theta=d_{\rm vert}H+Hd_{\rm vert}$$

is needed at this level: one may take (H=0).

Using the opposite sign instead produces the spurious defect

$$\begin{pmatrix} 0&\dfrac{2t}{(t^2-1)^2}\\[1mm] 0&-\dfrac{6t}{(t^2-1)^2} \end{pmatrix}.$$

This is exactly the earlier apparent obstruction.

Interpretation

There is no need to manufacture a rank-five connection. The smallest typed object is the two-term relative de Rham complex

$$\left( P^\bullet,\nabla_P \right) \xrightarrow{\ C_E\ } \left( E^\bullet,\nabla_E \right),$$

or its cone with the existing Čech orientation. The vertical map is already a morphism of the reconstructed connections.

This does not make the constant vector

$$(0,1,0,-3)$$

horizontal. That vector is only a coordinate presentation obtained after forgetting the source and target normal-residue actions. The invariant object is the map (C_E(t)), not its constant projective numerator.

Classification

• carrier: unchanged resolved infinity/soft closure;
• coefficient object: a typed relative de Rham--Čech cone;
• mixed obstruction: zero in the two-prime reconstructed packet;
• new carrier datum: none;
• remaining qualification: exact identification of the two-prime lift with the source characteristic-zero connection, as already stated in Entry 793.

Verification

• exact Symbolica checker: research/benincasa/marici-gm/src/bin/weighted_mixed_flatness_convention.rs;
• machine-readable packet: research/benincasa/weighted-mixed-flatness-convention.json;
• allocator claim seqclaim-d55e388485c91adb7b8d48c8.

Next falsifier

Insert this horizontal principal map into the oriented three-divisor supported Čech complex. Verify all signed vertex-to-edge squares as morphisms of connections, then compute the horizontal cohomology of the supported cone. Only if that cohomology has a canonical rank-one factor may it replace the nonhorizontal constant projective line in the physical pairing.

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