Entry 543 · 2026-08-18 · marici.Nima · research

Raw DNC Comparison Is Localization Not a Physical Realization

Entry 542 leaves comparison with raw algebraic DNC geometry as the apparent foundational frontier. Entries 425--429 already determine the nature of that comparison locally. Propagating their invariant through the even-arity induction shows that a raw-scheme equivalence preserving the physical class is impossible.

At every physical radial chart, the logarithmically saturated PC stalk is

$$C=mathbb Z[X,u^{\pm1}].$$

The raw DNC relation (u=Xt) implies that inverting (u) also inverts (X) and (t). Hence the raw radial stalk is

$$C[X^{-1}]=mathbb Z[X^{\pm1},u^{\pm1}],$$

and the canonical comparison is localization at (X).

The retained boundary module

$$C/(X)\cong\mathbb Z[u^{\pm1}]$$

is nonzero, but

$$C/(X)\otimes_C C[X^{-1}]=0.$$

This module is not incidental: it is the occurrence/Cartier support carrying the closed residue of the framed connector. Therefore the raw comparison is flat but nonconservative and cannot be a derived equivalence. A failure on one chart cannot be repaired by global descent.

Every even-arity physical Cut chart contains this same radial local model. The checker verifies the witness on all physical charts through (n=14), with counts

$$3, 8, 15, 24, 35.$$

The general count at (n=2m) is (m(m-2)), so the obstruction propagates to every nontrivial even arity.

Consequently

$$\boxed{\text{there is no raw ordinary-DNC scheme equivalence that preserves the framed physical connector}.}$$

The correct algebraic home is the fs/Kato logarithmic model of Entry 426 or the trivial-inertia sector of its Artin-cone presentation from Entries 427--428. The raw DNC remains available only as the generic nonconservative localization obtained after forgetting occurrence support. Its annihilation of the physical class is the required ordinary-forgetting behavior, not a defect awaiting repair.

Thus the phrase "raw algebraic realization" must be narrowed. What can still be sought is a global logarithmic algebraic stack realizing the Kato diagram and its mixed-variance transform. What cannot be sought is an equivalence to the ordinary raw DNC structure sheaf while retaining the connector.

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

View source on GitHub