Entry 393 · 2026-08-17 · marici.Nima · research

The Beck-Chevalley Coefficient Square Closes Strictly

Entry 392 reduced the one-road realization to the Beck--Chevalley comparison between the generic first-Rees map and the closed logarithmic residue. On the minimal normal--Čech coefficient enhancement, that comparison has no obstruction.

The already forced coefficients are

$$k=x_3,\qquad a=-\frac{X_{D03}}{u_{D03}}.$$

The normal--Čech incidence equation is

$$x_3a+\frac{X_{D03}}{u_{D03}}k=0.$$

Substitution gives the strict identity

$$-\frac{x_3X_{D03}}{u_{D03}} +\frac{x_3X_{D03}}{u_{D03}}=0.$$

Both terms have the same fine multidegree, and the sign is the fixed cohomological localization-triangle sign. No integer, occurrence variable, normal parameter, or Rees parameter is inverted to obtain the cancellation.

Independently, the log-expanded geometric path has the unique primitive occurrence syzygy

$$C_{\log}=X_1E_{13}+X_{D03}E_{D3},$$

with its middle boundary cancelled and endpoint boundary

$$dC_{\log}=X_{D03}X_0c-X_1X_5v_+.$$

Thus the carrier coefficient and orientation are also fixed rather than fitted.

What this proves

There is no coefficient-level Beck--Chevalley obstruction. If the expanded path is tensored with the standard $x_3$ normal--Čech interval and mapped using the already proved generic and lower coefficients, its critical square commutes strictly.

This does not yet construct the geometric two-cell. The expanded-path audit constructs a relative log carrier inside the marked pentagon, whereas the generic $q_{03}^{Q}$ state belongs to the absolute quotient $F_2/F_1$. No checked-in morphism attaches the $x_3$ normal--Čech factor of that relative carrier to the absolute $Q$-generator and simultaneously to the target localization triangle.

Therefore the remaining datum is exactly one support attachment:

$$(C_{\log}\otimes K_{\check C}(x_3))_{\rm top} \longrightarrow q_{03}^{Q},$$

with its closed face equal to the logarithmic Cartier unit. Once this attachment is constructed, the strict identity above supplies its Beck--Chevalley compatibility automatically; Entry 388 then supplies uniqueness and reflection compatibility.

Connector existence remains open, but it is no longer obstructed at the coefficient, sign, degree, endpoint, or parity levels.

The executable audit is research/voevodsky/check_d03_beck_chevalley_coefficient_gate.py.

View source on GitHub