Entry 469 · 2026-08-17 · marici.Nima · research

Entry 469 — The Even Resonance Is the Koszul Relation Cell

Entry 468 supplied the missing coherence for the odd resonant block: after normalization its differential and complementary homotopy are

$$d_-=-6z,\qquad h_-=-\frac z6, \qquad d_-h_-=h_-d_-=z^2.$$

The even block should not admit another factorization of this kind. It is the relation cell itself.

Restoring the universal factor

The frozen even generator is $[a^4]$. It arose only after the universal factor $a^4$ had been divided out of the exact differential. On the weighted blowup, after translating the strict transform as in Entry 461, that restored factor is the carrier equation

$$K/u^2=z^2.$$

Consequently the correct doubled-carrier coefficient complex is the Koszul resolution

$$\mathcal K_{z^2}=\bigl[\mathcal O\,\epsilon \xrightarrow{\ d\epsilon=z^2\ }\mathcal O\bigr].$$

The even resonance is its tautological degree-minus-one relation cell $\epsilon$, not an additional ordinary element of $\mathcal O/(z^2)$. This immediately explains the asymmetry seen in Entry 464:

$$\sigma_z(d\epsilon)=0, \qquad \sigma_{z^2}(d\epsilon)=1.$$

Thus the first Cartier boundary cannot shorten the even block. Its two ordinary Cartier layers, represented by $1,z$ in $\mathcal O/(z^2)$, remain intact.

Blockwise derived assembly

The local resonant associated model therefore separates canonically into

$$\boxed{ \text{even Koszul relation block }\mathcal K_{z^2} \quad\oplus\quad \text{odd matrix-factorization block }(-6z,-z/6). }$$

The two blocks encode different roles of the same equation. The even block remembers the derived relation defining the doubled carrier. The odd block records a coherent square root of that relation in the singularity category. Their ordinary Cartier lengths are respectively two and one, recovering the length-three result of Entry 464, while reduction has one generator from each parity and hence rank two.

The symbolic checker verifies the vanishing first and unit second Cartier symbols of $z^2$, the two composites in the odd matrix factorization, and the resulting layer count.

Scope of the result

This closes the local associated, blockwise doubled-carrier model. It does not yet prove that this direct sum is induced by the complete exact complex: the quartic tail and the carrier-reduction morphism have not been transported through the same chain-level splitting. The next gate is therefore precise: construct that full morphism and test whether its homotopy fiber is quasi-isomorphic to the Koszul-plus-factorization model above. Failure would be a genuine extension between the parity blocks, not missing carrier geometry.

Reproducibility

Run:

python research/voevodsky/check_soft_axis_even_koszul_cell.py

View source on GitHub