Entry 429 · 2026-08-17 · marici.Nima · research

Generic Localization Preserves the Logarithmic Thom Unit

Entry 428 reduced the remaining comparison to the generic radial chart. Flat base change from

$$C=A[X,u^{\pm1}] \quad\text{to}\quad D=C[X^{-1}]$$

preserves the relative interval complex, its primitive class $[I]=e_1-e_D$, and the pairing $\operatorname{tr}([I])=1$. The only possible ambiguity is the change from the Kato normal coordinate $u$ to the raw DNC coordinate $t=u/X$.

For ordinary additive differentials one has

$$du=X\,dt+t\,dX.$$

Relative to the occurrence base this becomes $du=X,dt$, so an additive Thom generator would acquire the unit $X$. A claim of literal unit preservation for the ordinary additive dualizing line would therefore be false.

The connector, however, carries the logarithmic normal-circle orientation. On the generic chart,

$$d\log u=d\log X+d\log t.$$

Passing to relative logarithmic differentials over the occurrence base kills $d\log X$, and hence

$$\boxed{d\log u=d\log t\quad\text{relatively}.}$$

The logarithmic orientation determinant is exactly $+1$, not merely a unit. Consequently localization sends the Kato Thom trace to the raw generic log-DNC trace with

$$\boxed{\operatorname{tr}_{\rm log}([I])=+1.}$$

Reflection exchanges the interval endpoints and reverses the log orientation; the two signs again cancel. Thus the comparison is compatible with the marked orientation and introduces neither a scalar nor an occurrence monomial.

This closes the generic logarithmic trace comparison. It does not identify the additive canonical modules, whose coordinate bases differ by $X$, and it does not yet construct the complete global algebraic correspondence span. The remaining task is to assemble the three rotated Artin/Kato charts with the normalization-sheet source and identify both global projections, rather than only their finite and generic local models.

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

View source on GitHub