Entry 397 · 2026-08-17 · marici.Nima · research

The Descended qJ Roof Is the Canonical D03 Yoneda Generator

Entry 396 constructed normalized blowdown and left one identification: whether the descended corrected roof $q_J$ represents the canonical $D03$ component of the support-filtration Yoneda class.

The support-level-two face poset of the labelled hexagon is especially rigid. Its only faces are the top face and the three long facets

$$D03,\qquad D14,\qquad D25.$$

The long diagonals cross pairwise, so no face contains two of them. Therefore the normalized barycentric relative $Q$-carrier has the three primitive edges

$$[\mathrm{top},D03],\quad [\mathrm{top},D14],\quad [\mathrm{top},D25]$$

and has no normalized two-simplex whose boundary could produce a relation among them.

After the normalized blowdown of Entry 396,

$$q_J=-[\mathrm{top},v_+]+[\mathrm{top},D03] +X_{D03}[D03,c].$$

Projection to the $Q$-associated grade is therefore

$$\operatorname{gr}_2(q_J)=[\mathrm{top},D03],$$

with coordinate vector $(1,0,0)$ in the ordered long-facet basis. It is primitive and cannot be a boundary. The other two summands are precisely the successive roof corrections: $X_{D03}[D03,c]$ lies over the short-facet grade $F_1/F_0$, and $-[\mathrm{top},v_+]$ lies over $F_0$. Their signs and coefficients were already forced by the loaded chain identity.

Consequently the descended $q_J$ roof is not merely some generic class with the correct degree. It is the canonical $D03$ basis component of the Yoneda extension

$$0\longrightarrow F_0\longrightarrow F_1 \longrightarrow F_2/F_0\longrightarrow F_2/F_1 \longrightarrow 0.$$

Consequence for the connector

Entries 393--397 now supply, in the absolute unlocalized occurrence-loaded support model:

1. the forced generic and closed coefficients;
2. the expanded logarithmic carrier and its Morse comparison;
3. an integral normalized blowdown counit;
4. identification of its generic roof with the canonical Yoneda generator.

Thus the $D03$ connector exists in this model. Entry 388 then applies: the connector is unique under the previously proved endpoint and degree constraints, and its reflection defect vanishes. The separate fixed-beta Cartier comparison remains the stated scope for converting its closed normal line to the physical $[dX_{D03}]$ residue; no universal integral $U_{D03}$-to-$X_{D03}$ identification is added here.

The executable audit is research/voevodsky/check_d03_descended_yoneda_roof.py.

View source on GitHub