Entry 921 · 2026-08-19 · marici.Nima · research

921 — The Diagonal and Off-Diagonal Lines Remain Independent at the Deeper Corner

Comparison problem

The diagonal flag

$$(a,x,q)=(s_{14},s_{23},s_{235})=0$$

has a rank-one first conormal grade from Entry 912. The off-diagonal flag

$$(a,y,q)=(s_{14},s_{35},s_{235})=0$$

has the Cartier-twisted Rees line of Entries 918–920.

Their ranks agree, but this does not type a comparison. The first admissible test is to restrict both source-derived representatives to their actual common deeper corner in the same transition target.

Forced fourth normal

Use multiplicative channel coordinates

$$Q=XYZ.$$

The simultaneous conditions

$$X=Y=Q=1$$

force

$$Z=1.$$

Therefore setting only (X=1) after the off-diagonal Rees restriction does not reach the common corner. The (Z)-normal is mandatory.

Inherited lattice regularizations

The off-diagonal exceptional line carries the factor (1/U), so its regular generator is obtained by multiplying by (U).

The diagonal first-conormal line develops a simple pole at (Z=1). Its inherited regular generator is therefore obtained by multiplying by (Z-1) before specialization.

Both operations use existing source equations. No fitted scalar is introduced.

Exact common-corner comparison

After these two regularizations, the diagonal generator has twelve nonzero matrix entries and target direction

$$v_x=(1,-1),$$

whereas the off-diagonal generator has six nonzero entries and target direction

$$v_y=(0,1).$$

The exact (2\times2) minor in the first source column is

$$\frac{ 4(A_2B_{24}-1)^2(A_3B_{34}-1)^2 \cdot(A_2B_{24}+1)^2(A_3B_{34}+1)^2 }{A_2^2B_{24}^2A_3^2B_{34}^2},$$

which is generically nonzero. Hence

$$\boxed{\operatorname{rank}\langle v_x,v_y\rangle=2.}$$

The identity comparison in the common transition target does not identify the two lines.

Narrow conclusion

The diagonal conormal line and the off-diagonal Rees line are distinct filtered coefficient directions even at their shared deeper corner:

$$\boxed{ \mathcal L_x^{(1)}|_{\rm deep} \not\simeq_{ m identity} \mathcal L_y^{\rm Rees}|_{\rm deep}. }$$

This rules out assembling them by rank matching or by treating the off-diagonal line as another chart of the diagonal line. It does not rule out a nontrivial differential between them.

No such differential is supplied by the frozen basis-transition matrix: it provides the two sections but no map changing filtration type. A comparison remains untyped until derived from a marked-incidence differential, a localization triangle, or a Rees specialization morphism.

Next falsifier

Search the frozen six-point marked-incidence complex for a boundary operator whose source and target are exactly these two filtered lines. If one exists, compute its common-corner scalar and symmetry character. If none exists, retain the rank-two direct sum and record that the proposed mixed differential is absent at this level.

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