Entry 416 · 2026-08-17 · marici.Nima · research

Cartier Gysin Transfer Is the Unique Amplitude-Correct Regrading

Entry 415 proved that the independent Tate--Cartier tensor totalization cannot embed strictly into the facewise PC target: its amplitude is six rather than three. The purity theorem of Entry 131 indicates the correct repair. Every Cartier degree carries a codimension-one Thom orientation and a compensating Gysin shift.

Let $p$ be Tate carrier degree and $q$ Cartier exterior degree. Before transfer the total degree is $p+q$. If a uniform Gysin correction $c$ is applied per Cartier degree, with an overall shift $s$, then

$$\deg_{c,s}(p,q)=p+q-cq+s.$$

Requiring all sixteen occupied bidegrees $0\leq p,q\leq3$ to land exactly in the target degrees $0,1,2,3$ has the unique integral solution

$$\boxed{c=1,\qquad s=0.}$$

Thus

$$\boxed{\deg_G(p,q)=p.}$$

All three external Cartier degrees must be transferred; transferring only the exceptional interval or only one marked normal cannot repair the amplitude.

Collapsed profile

After this Gysin regrading, each Tate generator carries the full eight-dimensional Cartier coefficient packet without acquiring additional chain degree. The degree profile becomes

$$(1,3,3,1)\cdot(1+3+3+1) =\boxed{(8,24,24,8)}.$$

This fits degreewise inside the fixed endpoint quotient

$$\operatorname{rk}(F_K/F_V)=\boxed{(12,57,87,43)}.$$

Hence the amplitude and raw rank obstructions disappear simultaneously. This is only a capacity statement, not yet an embedding theorem.

The regrading also determines the type of the two differentials:

• the Tate/carrier differential remains a chain differential of degree $-1$;
• the Cartier differential becomes degree zero after Gysin transfer and must be represented as a filtered Bockstein/can--var operator.

This explains why the desired realization is a filtered PC map rather than an ordinary map from the seven-degree tensor complex. The vertical Cartier direction survives as coefficient and filtration structure; it is not another cellular boundary direction in Entry 143.

Next gate

The next finite construction should work degree by degree with the collapsed profile $(8,24,24,8)$. It must place the eight Cartier states over each Tate carrier generator using Entry 131's purity maps, and verify:

1. the carrier boundary gives $N,1-r,\epsilon$;
2. the transferred degree-zero operator gives the three first Cartier symbols;
3. the normalized blowdown connector commutes with that operator; and
4. the nonzero generic $Q$ roof and both endpoint residues survive.

The executable audit is `research/voevodsky/check_cartier_gysin_regrading.py`.

View source on GitHub