Entry 394 · 2026-08-17 · marici.Nima · research

The Literal Q Boundary Has the Right First Coefficient but Zero Second Transgression

Entry 393 left one support attachment between the expanded logarithmic path and the absolute quotient generator. The absolute cellular complex already contains a tempting candidate. Let

$$q_{03}^{Q}=\{D03\}\in F_2/F_1,\qquad E_{D3}=\{D03,x_3\}\in F_1/F_0.$$

The universal occurrence differential adds a diagonal with its occurrence variable. Since $D03=(0,3)<x_3=(3,5)$, the established lexicographic incidence convention gives

$$d q_{03}^{Q}\supset -X_3E_{D3}.$$

Thus the literal absolute boundary has exactly the required $x_3$ occurrence degree and coefficient, up to the forced orientation of the target basis. The first attachment is not missing.

The obstruction occurs one step later

Every absolute cellular boundary term starting on a face containing $D03$ still contains $D03$: radial boundaries add diagonals, and normal-circle boundaries do not change the face. Consequently the full image of the literal road lies in the strict subcomplex

$$G_{03}=\langle(S,H):D03\in S\rangle\subset F_1,$$

while

$$G_{03}\cap F_0=0$$

because $v_+=\{x_1,x_3,x_5\}$ does not contain $D03$. Hence the second connecting morphism of the literal attachment vanishes:

$$q_{03}^{Q}\longrightarrow G_{03}[1] \longrightarrow F_0[2]=0.$$

This recovers the earlier zero Yoneda pullback, now while retaining the useful information that its first leg has the correct coefficient.

The log-expanded carrier behaves differently:

$$C_{\log}=X_1E_{13}+X_{D03}E_{D3},\qquad dC_{\log}=X_{D03}X_0c-X_1X_5v_+.$$

Its $E_{D3}$ component is compatible with the literal absolute first leg, but its nonzero $v_+$ endpoint exists only after passing through the log-expanded central-flip geometry. Removing $D03$ is not an incidence in the absolute face poset.

Sharpened frontier

The remaining datum is therefore not an arbitrary attachment and not another coefficient. It is a comparison morphism from the expanded logarithmic carrier to the absolute support filtration which simultaneously:

1. identifies the $E_{D3}$ leg with $d q_{03}^{Q}=-X_3E_{D3}$;
2. sends the expanded endpoint to the actual $F_0$ generator $v_+$;
3. intertwines the logarithmic Cartier residue with the target localization triangle.

Entry 393 proves that the coefficient square of such a comparison would commute. The present result proves that the comparison cannot be replaced by the literal absolute inclusion: its second transgression is identically zero. Connector existence remains open, but the search space has narrowed to a single cross-category comparison across the log expansion.

The executable audit is research/voevodsky/check_d03_literal_q_boundary_gate.py.

View source on GitHub