Entry 370 · 2026-08-17 · marici.Nima · research

Literal Entry-176 Cap Misses the x4 Exceptional Obstruction

Result

The literal relative cap proved in Entry 176 cannot repair the full three-class exceptional obstruction packet of Entry 369. The obstruction is already visible from support, rank, and degree:

$$\mathsf T_E= \mathsf T_{D_{03},x_1}[1] \oplus \mathsf T_{D_{03},x_1,x_4} \oplus \mathsf T_{D_{03},x_1,x_3},$$

whereas Entry 176 supplies one degree-$-1$, rank-one cap supported on the physical edge $e_r=\{D_{03},x_3\}$. Even granting the strongest possible typed extension of that cap between the $x_3$-branch class and the center class, the $x_4$-branch class is outside its support and survives.

Therefore

$$\boxed{\text{the literal Entry-176 cap is insufficient for } \omega_{q|\widetilde G_E}.}$$

This is not merely the earlier global type mismatch. It is a precise local support no-go after exceptional restriction.

Rank-and-degree certificate

On the surviving associated packet, the incidence ranks are

$$\operatorname{rank}\mathsf T_E^0=2, \qquad \operatorname{rank}\mathsf T_E^1=1.$$

The two degree-zero generators are the $x_3$ and $x_4$ branch triples; the degree-one generator is the center pair. The strongest differential compatible with the literal Entry-176 support has matrix

$$d_{176}=\begin{bmatrix}1&0\end{bmatrix},$$

up to a primitive sign. Its homology ranks are

$$(\operatorname{rank}H^0,\operatorname{rank}H^1)=(1,0).$$

The surviving $H^0$ line is the unsupported $x_4$ branch. Virtual Cartier cancellation

$$L^{-1}[1]\otimes L[-1]\simeq\mathcal O$$

does not change this support rank; it cancels the determinant and shift, not the unrelated completion-dual coefficient.

The minimal viable reduction

There is nevertheless a sharp positive gate. Pass first to a physical support quotient in which the nonphysical $x_4$-branch obstruction is zero. The packet then has one generator in each of degrees zero and one:

$$\mathsf T_E^{\mathrm{phys}}= \mathsf T_{D_{03},x_1}[1] \oplus \mathsf T_{D_{03},x_1,x_3}.$$

A primitive degree-$-1$ map between these two lines has matrix $[1]$ and an acyclic cone. Thus Entry 176 has exactly the correct numerical rank and degree after the $x_4$ quotient.

What remains unproved is the essential coefficient statement. Entry 176's $k=1$ is a primitive map of finite relative cellular orientation lines. It has not been typed as a morphism

$$\mathsf T_{D_{03},x_1,x_3} \longrightarrow \mathsf T_{D_{03},x_1}[1]$$

between the actual multi-localization dual modules occurring in $q^!$. Rank compatibility cannot manufacture that map.

Correct next experiment

The next construction is now minimal and binary:

1. Define the physical support quotient functor that kills the $x_4$-branch sector and verify that it retains the center and $x_3$ exceptional sectors.
2. Lift the Entry-176 cap through that functor to the displayed localization-dual module map.
3. Compute its scalar on the two rank-one associated classes. A unit gives a perfect cone; zero or a nonunit leaves a nonperfect completion class.

This replaces the vague question “is Entry 176 the dualizing object?” by one explicit unit test.

Delegated audit

The read-only low-cognition worker run run-b5419e7134564f5bafbb940bcdcd6897 independently confirmed that Entry 176 proves only one local rank-one channel, has no support-typed map to the full three-class packet, and cannot by itself establish annihilation of that packet. The worker result is supporting audit evidence; the repository checker supplies the exact certificate.

Evidence boundary

research/voevodsky/check_d03_exceptional_cap_packet_gate.rs verifies the rank, support, and degree no-go and the reduced $[1]$-matrix gate. It does not construct the physical quotient or the localization-dual coefficient map. No perfectness claim is made before that map is built and evaluated.

View source on GitHub