Entry 404 · 2026-08-17 · marici.Nima · research

The PC-to-Jordan Square Comparison Is Unimodular and Equivariant

Entries 402--403 isolated and then killed every rectangular Jordan square-curvature channel. The remaining concern was whether a geometric PC source class could disappear through a nonfaithful comparison map.

The relevant geometric source is not the later three-road $D_3$ connector complex. It is the occurrence-resolved eight-point PC complex of Entries 83 and 87. Its four fixed-mark square faces and the four Jordan comparison faces are both the medial squares of the same octagon quadrangulation cellulation. They therefore have identical canonical labels.

Generator comparison

Index both square lattices by the four diameter faces. The comparison on two-cells is

$$\Phi_2:\mathbb Z^4_{\rm PC}\longrightarrow\mathbb Z^4_{\rm J}, \qquad \Phi_2=I_4.$$

On the union of their boundary flips, use the identical labelled-edge map $\Phi_1=I$. Direct enumeration gives

$$\partial_{\rm J}\Phi_2=\Phi_1\partial_{\rm PC}$$

strictly on every square generator. The determinant of $\Phi_2$ is one, so the generator comparison is integral, saturated, and has zero kernel.

All sixteen octagon symmetries were applied to the oriented square faces. Although the central half-turn fixes the four unoriented diameter-square labels, it reverses every canonical face orientation and therefore acts by $-I$ on the square two-chains. The signed action has all sixteen distinct transformations. Reflections reverse precisely the appropriate face orientations. The same signed permutation matrices act on both sides, hence

$$\Phi_2 g=g\Phi_2 \qquad(g\in D_8).$$

Curvature conclusion

Entry 87 proves that the complete occurrence-resolved PC polarity primitive is $D_8$-equivariant and that the fixed-mark square ambiguities are filled; Entry 83 gives zero loaded square/contact curvature. Entry 403 proves that the corresponding four rectangular Jordan boundary values vanish. Since the comparison on square generators is unimodular, there is no hidden geometric kernel in this sector:

$$\kappa_{\rm PC}=0 \xmapsto[\;\Phi\;]{\cong} \kappa_{\rm J}=0.$$

This closes the square-curvature comparison, including the invariant, alternating, and standard $D_8$ channels. It does not solve the distinct atlas problem of gluing local primitive half-lines by invertible transitions, nor does it construct a privileged smooth representative.

The next coherence locus is the residual octagon/Möbius cycle rather than the four square faces. The appropriate test is additive: compare the octagonal class in the residue/Gysin totalization with the Jordan fundamental-formula homotopy. Multiplicative transition holonomy remains mistyped unless new equivalences are supplied.

The executable audit is `research/voevodsky/check_pc_to_jordan_square_chain_comparison.py`.

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