Entry 380 · 2026-08-17 · marici.Nima · research

The One-Road Target Cell Exists Coefficientwise and Leaves One Purity Comparison

Result

The three-face target demanded by Entry 379 already exists in the established finite coefficient/Cousin model. The Entry-129 $x_3$ occurrence Koszul--Cech Gysin, externally tensored with the fixed Entry-97/100 $D03$ packet, simultaneously supplies:

1. a nonzero degree-zero lower Cech component;
2. the degree-one $x_3$-edge component;
3. both endpoint faces $v_{00}$ and $v_{10}$;
4. both repeated-normal $\operatorname{Tor}_0$ and $\operatorname{Tor}_1$ grades; and
5. the positive physical line $[dX_{03}]$.

Deleting the lower component breaks its chain equation, exactly as predicted by Entry 379. Thus there is no remaining target-shape or coefficient-level endpoint obstruction for the one-road cell.

The remaining gap is one categorical comparison, not three new maps. For each endpoint $v_i$, one needs an occurrence-loaded purity/costalk map

$$\boxed{ \operatorname{pur}^{\rm PC}_{i3}: \mathcal C^{\rm fin}_{(x_i,x_3)}\widehat\otimes D_3 \longrightarrow i^!_{(v_i,x_3)} \mathcal Q^{\rm PC}_{03,\partial,\rm lf}, }$$

and the two maps must be restrictions of the same $x_3$-edge comparison. Here $\mathcal C^{\rm fin}_{(x_i,x_3)}$ is the already constructed loaded finite Koszul--Cech corner, not a newly postulated target.

Exact coefficient cell

For $i=0,1$, the two-normal occurrence corner has the full comparison

$$K(x_i,x_3)\longrightarrow C_{(x_i,x_3)}$$

in all degrees. Entry 129's extraordinary edge map is

$$g_0(r)=(r/x_i,0), \qquad g_1(t)=t/x_i.$$

Its chain equation is

$$d g_0=g_1d,$$

and the endpoint top is

$$\frac1{x_ix_3}.$$

Tensoring with the four-normal Cech residue, the repeated-normal packet, and the positive physical orientation gives

$$v_{00}: +\frac{[dX_{03}]}{x_0x_3u_0u_1u_3u_5}, \qquad v_{10}: +\frac{[dX_{03}]}{x_1x_3u_0u_1u_3u_5}.$$

Every inverse occurs only in the target Cech summand that localizes that parameter. No occurrence or support normal is inverted in the base.

Combination with the Rees bridge

Entry 378 fixes the generic coefficient as $x_3$ and its first Cartier symbol as $+1$. Entry 379 proves that the lower Cech and endpoint faces must survive in the same cell. The finite endpoint construction satisfies precisely that coefficient-level requirement at both ends.

What is not yet justified is identifying its coefficient codomain with the ringed PC extraordinary costalk. The source and target still have distinct reciprocal-regular and locally-finite/Borel--Moore variance. The required comparison must be geometric and extraordinary; neither finite duality nor ordinary coherent restriction supplies it.

Consequence

The next experiment should construct only $\operatorname{pur}^{\rm PC}_{i3}$, first at $v_{10}$, where the old typing failure was explicit, and then transport it along the same oriented $x_3$ edge to $v_{00}$. Its acceptance tests are already fixed:

• commute with the quotient $\pi$ and excess trace $\operatorname{tr}_{\rm ex}$;
• preserve the graph-Cartier Bockstein;
• preserve every lower Cech term;
• yield the two displayed positive endpoint residues; and
• agree with Entry 377 on the closed finite Cartier packet.

If such a comparison exists, the coefficientwise three-face cell becomes a typed one-road Beck--Chevalley cell. If it does not, the obstruction lies in the passage from finite Cousin coefficients to the actual extraordinary PC costalk, not in incidence, signs, endpoints, or Rees coefficients.

Evidence boundary

research/voevodsky/check_d03_x3_loaded_pc_endpoint_boundary.rs was rerun after repairing three edition-sensitive array iterations. It verifies both endpoint quotient complexes, the full lower occurrence-Cech map, all sixteen normal Cech degrees, both Tor grades, the graph-Cartier square, and the two positive residues. It continues to report the actual ringed PC extraordinary map as untyped. No such map or full primal trace is claimed here.

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