Entry 400 · 2026-08-17 · marici.Nima · research

The Log Rays Select the Normalization Branches and the Even Endpoint Class

Entry 399 produced two labelled endpoint rays with boundary

$$r_1-r_{D03}.$$

The global normalization source has two sheet costalks $e_+,e_-$, reflection exchanges both ray labels and both sheet labels, and the conductor quotient is the oriented difference

$$e_+-e_-.$$

Any integral reflection-equivariant map from the ray permutation lattice to the sheet permutation lattice has matrix

$$M(a,b)= \begin{pmatrix}a&b\\ b&a\end{pmatrix}.$$

Compatibility with the exceptional boundary requires

$$M(a,b)(-1,1)^T=(1,-1)^T, \qquad b-a=1.$$

Compatibility with the common normalized endpoint counit requires each column to augment to one:

$$a+b=1.$$

These equations have the unique integral solution

$$a=0,\qquad b=1,\qquad M= \begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Its determinant is $-1$, so the comparison is unimodular. In the fixed orientation convention,

$$r_{D03}\longmapsto e_-, \qquad r_1\longmapsto e_+.$$

No half-difference, sheet section, or inversion of two is used.

Endpoint mapping fiber

This matrix supplies the geometric connector cell absent from the earlier hemisphere-only calculation. The previously conditional endpoint equation

$$2a+b=1$$

now has its geometrically selected values $(a,b)=(0,1)$. Therefore the endpoint-fixed mapping fiber is instantiated and

$$\boxed{p_{\partial,Q}=0\in\mathbb Z/2.}$$

The polarity connecting homomorphism

$$H^1(D_3;\mathbb Z_{\rm or})\longrightarrow H^2(D_3;\mathbb Z)$$

sends this class to zero, so the corresponding conductor Bockstein also vanishes.

The earlier $\mathbb Z/2$ torsor was real before a geometric connector was supplied: the primitive hemisphere $Q$-row alone could not choose a component. The log-ray boundary plus the normalized counit provides exactly the missing second equation and selects the even component.

Scope and next test

The conclusion uses the labelled positive exceptional interval, the fixed conductor-difference orientation, and the normalized odd endpoint counit already established in the local model. Within that scope the complete three-road endpoint/$Q$ mapping fiber now exists and its parity is fixed.

The next falsification tests are the promised global ones: extend the selected component through the full dihedral $D_8$ action and evaluate the Jordan compatibility condition. Those tests are now well typed.

The executable audit is research/voevodsky/check_log_ray_normalization_branch_identification.py.

View source on GitHub