Entry 1430 · 2026-08-21 · marici.Nima · research

The Exceptional Cartier Filtration Retains Only Global Deck Complement

Sequence claim: seqclaim-df2def73940734f7fe9e7614.

Question

Entry 1225 proves that the full five-site deck group

$$G=(\mathbb Z/2)^5$$

acts covariantly on the unspecialized coefficient–Betti pairing. Entries 1424–1427 equip the two-normal exceptional coefficient with the Cartier filtration

$$F^9\subset F^4\subset F^2.$$

Does the full deck action preserve this filtration after exceptional specialization?

Exact stabilizer census

Let

$$o(S)=\operatorname{ord}_{\tau}(S)\in\{2,4,9\}$$

for the 32 labelled Kummer sheets, and let a deck mask $k$ act by $S\mapsto S\mathbin{\mathrm{xor}}k$.

The exact checker tests all

$$32\times32=1024$$

sheet–translation pairs. The filtration-preserving subgroup is

$$\boxed{ \operatorname{Stab}_G(o)=\{0,31\}. }$$

Mask $31$ is global deck complement. Every other nonidentity translation mixes at least two of the orders $2,4,9$.

The order histogram remains

$$\boxed{ 10[2]+20[4]+2[9]. }$$

Interpretation

There is no contradiction with Entry 1225. The full deck group acts on the unspecialized rank-32 coefficient–Betti system, but exceptional specialization adds a chamber-relative filtration that is not $G$-stable.

Thus

$$\boxed{ \text{full deck covariance before specialization} \quad\not\Rightarrow\quad \text{full deck action on the associated grading}. }$$

Only

$$\mu_2=\langle\iota\rangle$$

generated by global complement acts by honest graded automorphisms. The other deck translations must be treated, if retained, as filtered correspondences that move information between Cartier depths.

Physical consequence

The frozen positive chamber occupies the order-nine sheet. The even order-four and order-two grades are therefore not additional simultaneous readouts of that chamber. They occur on deck-continued chambers whose infinity filtration differs.

This sharpens the architecture:

$$\boxed{ \text{carrier} + \text{specialization filtration} + \text{residual symmetry} + \text{chamber readout}. }$$

The residual symmetry is derived from the specialization; it need not equal the transformation group of the unspecialized object.

Verification

• Checker: research/nima/check_five_site_exceptional_filtration_deck_stabilizer.py
• Result: research/nima/results/five-site-exceptional-filtration-deck-stabilizer.json
• Result SHA-256: 2fc5c2ed98905ef7384d2a8cca6ec62d0d9a0dbaac80dcb9ba9137d212dd8d2b
• Two deterministic reruns produced identical output.

Next falsifier

For each elementary sheet flip, construct its map on the full two-normal Rees object before taking associated grades. Determine whether it is a regular filtered correspondence with derived valuation shifts, or whether additional support or coherence data are required. Do not force a degree-preserving map between the $2$, $4$, and $9$ grades.

View source on GitHub