Entry 919 · 2026-08-19 · marici.Nima · research

919 — The Off-Diagonal Exceptional Line Glues with a Degree-Zero Hecke Modification

Globalization problem

Entry 918 constructs the off-diagonal exceptional image in the (a)-chart of

$$E=\mathbf P(n_a,n_y,n_q),$$

where

$$n_a=A_4-1, \qquad n_y=Y-1, \qquad n_q=Q-1.$$

Its scalar factor is (1/U), with

$$U=\frac{n_y}{n_a}.$$

The projective image is constant locally. The remaining question is whether this line glues over all three standard Rees charts.

Chart factors

Use the following coordinates.

On the (a)-chart:

$$U=\frac{n_y}{n_a}, \qquad V=\frac{n_q}{n_a}, \qquad f_a=\frac1U.$$

On the (y)-chart:

$$A=\frac{n_a}{n_y}, \qquad W=\frac{n_q}{n_y}, \qquad f_y=A.$$

On the (q)-chart:

$$C=\frac{n_a}{n_q}, \qquad D=\frac{n_y}{n_q}, \qquad f_q=\frac CD.$$

Exact overlap audit

The Symbolica checker verifies

$$f_y\big|_{a\cap y}=A=\frac1U=f_a,$$

$$f_q\big|_{a\cap q}=\frac{1/V}{U/V}=\frac1U=f_a,$$

and

$$f_q\big|_{y\cap q}=\frac{A/W}{1/W}=A=f_y.$$

No additional unit or sign appears. Therefore the rank-one projective image of Entry 918 glues globally on (E).

Global divisor

The common rational factor is

$$f=\frac{n_a}{n_y}.$$

Its divisor is

$$\boxed{\operatorname{div}(f)=D_a-D_y,}$$

where

$$D_a=\{n_a=0\}, \qquad D_y=\{n_y=0\}.$$

The (q)-direction has coefficient zero. The divisor has projective degree zero.

Thus the globally typed coefficient object is a constant projective line with a degree-zero Hecke/Cartier lattice modification:

$$\boxed{ \mathcal L_{\rm off}^{\rm mer} = \mathbf Q\langle r_{\rm tan}\rangle \otimes \mathcal O_E(D_a-D_y). }$$

Correction to the local description

Entry 918's (a)-chart pole at (U=0) is the (D_y) part of this divisor. The complementary zero on (D_a) lies outside that affine chart and becomes visible only after gluing. The global datum is therefore (D_a-D_y), not merely a single local pole.

Narrow conclusion

The off-diagonal order dependence is completely absorbed by the existing Rees exceptional geometry:

$$\boxed{ \text{constant projective line} + \text{degree-zero lattice modification }D_a-D_y. }$$

No new carrier cell, divisor, or fitted normalization is introduced.

Next falsifier

Test the reflected (z)-flag and the reflection overlap. The expected divisor is

$$D_a-D_z.$$

The decisive question is whether the (y)- and (z)-lines are exchanged strictly by the source reflection or require an additional sign/unit on their common exceptional atlas.

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