Entry 920 · 2026-08-19 · marici.Nima · research

920 — Reflection Strictly Exchanges the Two Off-Diagonal Rees Lines

Frozen comparison

Entry 919 globalizes the (y=s_{35}) off-diagonal line on

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

as

$$\mathcal L_y = \mathbf Q\langle r_y\rangle \otimes \mathcal O_{E_y}(D_a-D_y).$$

The source reflection

$$\tau_{\rm off}=(23)$$

fixes (a=s_{14}) and (q=s_{235}), and exchanges

$$y=s_{35} \longleftrightarrow z=s_{25}.$$

The comparison is made by transporting this frozen source object. No independent normalization is chosen for the (z)-line.

Exceptional divisor transport

On homogeneous exceptional coordinates, reflection acts by

$$[n_a:n_y:n_z:n_q] \longmapsto [n_a:n_z:n_y:n_q].$$

Therefore

$$\tau^*\!\left(\frac{n_a}{n_y}\right) = \frac{n_a}{n_z},$$

and

$$\tau(D_a-D_y)=D_a-D_z.$$

Both divisors have degree zero. The (q)-component remains zero.

Orientation audit

In the source-normalized transition, $\tau_{\rm off}$ reverses both serialized variance bases:

$$\chi_{\rm sparse}(\tau_{\rm off})=-1, \qquad \chi_{\rm dense}(\tau_{\rm off})=-1.$$

It preserves the ordered residue orientation and the normal-line orientation. Hence

$$\chi_{\rm total}(\tau_{\rm off}) =(-1)(-1)(+1)(+1) =+1.$$

The exact checker consequently finds no residual sign or rational unit:

$$\boxed{ \tau^*\mathcal L_y=\mathcal L_z, \qquad \mathcal L_z = \mathbf Q\langle r_z\rangle \otimes \mathcal O_{E_z}(D_a-D_z). }$$

Narrow conclusion

The two order-dependent off-diagonal flags are not separate fitted coefficient objects. They are a single source-labelled occurrence orbit. Reflection transports both the constant projective line and its Hecke divisor strictly:

$$\boxed{ D_a-D_y \xleftrightarrow{\ \tau\ } D_a-D_z, \qquad \text{transition unit}=1. }$$

Thus the entire (y/z) asymmetry is occurrence labelling on the existing Rees carrier. It supplies neither a new carrier stratum nor an additional coefficient character.

Scope and next falsifier

This proves pairwise reflection covariance. It does not yet assemble the diagonal (x)-flag and off-diagonal (y/z)-flags into one normal-crossing coefficient complex: the (x)-flag lives in first conormal grade, whereas the (y/z)-flags live on Rees exceptional lines.

The next test is to construct the smallest typed comparison between these two filtration types at their common deeper corner. A map may be proposed only if it is induced by the frozen marked-incidence differential or a Rees specialization map; rank matching alone is inadmissible.

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