Entry 375 · 2026-08-17 · marici.Nima · research

The x4 Quotient Leaves an x3 Localization-Dual Cone

Result

The support reduction isolated in Entry 370 is necessary but not sufficient. After quotienting the unsupported $x_4$-branch, there is indeed a canonical map in the direction required by the Entry-176 cap. However, that map is not an equivalence over the unlocalized physical coefficient ring. Its cone is the dual of the additional $x_3$-localization quotient and is nonzero.

Consequently Entry 176's primitive cellular coefficient $k=1$ cannot by itself make the reduced exceptional packet perfect. The remaining problem is not to manufacture a scalar. It is to construct a physical coefficient functor—such as a justified residue, nearby-cycle, or relative-boundary operation—that kills or evaluates the $x_3$-localization cone.

The reduced coefficient types

Write

$$R=A[(D_{03}x_1)^{-1}],\qquad x=x_3.$$

This is the base change that makes the two common normal factors units. It is a valid obstruction test: an equivalence over the original coefficient ring would remain an equivalence after this base change.

After the $x_4$-support quotient, the center and physical branch coefficients are

$$\mathsf T_{\mathrm{ctr}}=R\operatorname{Hom}_R(R,R), \qquad \mathsf T_{x_3}=R\operatorname{Hom}_R(R[x^{-1}],R),$$

up to the already recorded common incidence and Cartier shifts. The localization map

$$\iota:R\longrightarrow R[x^{-1}]$$

therefore gives, by contravariance, a canonical typed arrow

$$\phi_{\mathrm{loc}} =R\operatorname{Hom}_R(\iota,R): \mathsf T_{x_3}\longrightarrow\mathsf T_{\mathrm{ctr}}.$$

This closes the bare existence and arrow-direction question left open in Entry 370. Entry 176's positive normal orientation can multiply $\phi_{\mathrm{loc}}$ by $+1$; it does not change its cone.

Exact cone calculation

Let

$$Q_x=R[x^{-1}]/R.$$

The localization triangle

$$R\longrightarrow R[x^{-1}]\longrightarrow Q_x\longrightarrow R[1]$$

dualizes to

$$R\operatorname{Hom}_R(Q_x,R) \longrightarrow \mathsf T_{x_3} \xrightarrow{\phi_{\mathrm{loc}}} \mathsf T_{\mathrm{ctr}} \longrightarrow R\operatorname{Hom}_R(Q_x,R)[1].$$

Hence

$$\boxed{\operatorname{Cone}(\phi_{\mathrm{loc}}) \simeq R\operatorname{Hom}_R(Q_x,R)[1].}$$

This cone is nonzero. For the normal parameter $x$, the standard telescope resolution identifies the relevant derived dual with the $x$-adic completion sector. After localization at the height-one prime $(x)$, its cohomology contains the nonzero completion contribution (equivalently the extension generated by the tower $x^{-1},x^{-2},\ldots$). In particular, the map is not a quasi-isomorphism and the cone is not removed by multiplying the incidence generator by the unit $+1$.

This is the same coefficient mechanism detected for one-normal sectors in Entry 367, now applied to the exact center-to-$x_3$ localization arrow of the reduced exceptional packet.

Meaning of the physical quotient

A quotient that merely deletes the $x_4$ incidence summand leaves $Q_x$ untouched. Therefore it produces the two-term shape

$$\mathsf T_{x_3}\xrightarrow{\phi_{\mathrm{loc}}}\mathsf T_{\mathrm{ctr}},$$

but not the scalar complex $R\xrightarrow{1}R$. Replacing the former by the latter without an additional coefficient functor would conflate a primitive cellular coefficient with an invertible map of localization-dual modules.

The minimal viable physical reduction must do two distinct things:

1. remove the nonphysical $x_4$-support summand;
2. kill, take a residue of, or otherwise canonically evaluate $R\operatorname{Hom}_R(Q_x,R)$.

Only after the second operation can Entry 176's $k=1$ become the unit matrix $[1]$ on the resulting rank-one coefficient lines.

Correct next experiment

Construct the coefficient part of the physical-support functor on the localization triangle

$$R\to R[x_3^{-1}]\to Q_{x_3}.$$

The acceptance test is binary:

• if the functor canonically sends the dual $Q_{x_3}$-term to zero and carries $\phi_{\mathrm{loc}}$ to the identity of the retained residue line, the Entry-176 cap gives an acyclic reduced cone;
• if the dual $Q_{x_3}$-term survives, the exceptional correction remains nonperfect despite the $x_4$ quotient.

No perfectness claim is made before this coefficient functor is constructed.

Evidence boundary

`research/voevodsky/check_d03_localization_dual_cap_gate.rs` verifies the support reduction, the forced direction of the localization-dual arrow, and the unbounded negative-power tower retained by its cone. The nonvanishing and completion interpretation follow from the displayed localization triangle and standard telescope resolution. The checker does not construct the required physical coefficient functor.

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