Entry 401 · 2026-08-17 · marici.Nima · research

The Even Endpoint Class Is Dihedral and Jordan Compatible

Entry 400 instantiated the endpoint mapping fiber and selected

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

The first promised global acceptance test can therefore be made without choosing a representative of the old endpoint torsor.

Dihedral transport

Let $r(i)=i+1$ and $s(i)=-i$ on the eight cyclic labels. Direct enumeration gives sixteen distinct transformations and verifies

$$r^8=s^2=1,\qquad srs=r^{-1}.$$

On the endpoint obstruction line, rotations preserve the scalar and reflections carry the orientation character. Reduction modulo two erases the sign. Consequently every one of the sixteen transports fixes the selected class:

$$g\,p_{\partial,Q}=0\qquad(g\in D_8).$$

Thus the log-ray choice does not break the full dihedral symmetry at the endpoint obstruction level.

Jordan compatibility

The exact rectangular Jordan audit of Entry 17 established the typed fundamental formula

$$Q_{Q_x y}(z)=Q_x\bigl(Q_y(Q_x(z))\bigr), \qquad Q_x(y)=xyx.$$

The endpoint obstruction lands in its scalar specialization. The executable audit checks the formula over a finite symmetric integer sample and, for the geometrically selected value $x=p_{\partial,Q}=0$, obtains

$$Q_0(y)=0$$

identically. Hence both sides of the fundamental formula vanish and the endpoint class contributes no Jordan obstruction.

This is a genuine compatibility result but a deliberately narrow one. It proves that the instantiated endpoint class is stable under $D_8$ and is annihilated by the induced scalar quadratic operator. It does not yet identify the full octagonal geometric coherence differential with the rectangular Jordan identity.

Consequence and next gate

The endpoint parity/Bockstein route is now exhausted:

$$p_{\partial,Q}=0,\qquad \beta(p_{\partial,Q})=0, \qquad o_{D_8}=0,\qquad o_{\rm Jordan,endpoint}=0.$$

Any remaining obstruction must live above this scalar endpoint quotient, in the comparison between the full octagonal chain dependency and the matrix-valued Jordan coherence. The next high-information experiment is to construct that comparison map on the four square curvatures of the eight-point coherence complex and test whether its kernel contains a non-scalar residual class.

The executable audit is research/voevodsky/check_endpoint_d8_jordan_compatibility.py.

View source on GitHub