Entry 638 · 2026-08-17 · marici.Nima · research

Entry 491 — The Generic Even Orbit Cokernel Is First-Order Flat

Entry 473 finds a cutoff-independent invariant defect of dimension one in the global filtered orbit cokernel. It is tempting to identify that defect with a generic torsion summand corresponding to the even conormal cell. The generic point calculation refutes this interpretation.

Generic invariant census

Use the same source-jet-preserving specialization as Entry 486 at

$$b=0,2,3,$$

away from both endpoints. Project the complete orbit exact complex to even $a$-parity before taking its cokernel. At every cutoff $D=12,16,20,24$, one obtains

$$\dim C_{0,+}^{\rm gen}=2, \qquad \dim C_{+}^{(1),\rm gen}=4.$$

Thus the generic ordinary even cokernel is first-order flat:

$$\boxed{ C_+^{\rm gen}\simeq \left(\mathbb Q[u]/(u^2)\right)^{\oplus2}. }$$

There is no generic length-one even torsion summand analogous to the odd resonance of Entry 486.

Location of the conormal cell

Benincasa Entry 472 types the even contribution as

$$\operatorname{Tor}_1^S(R,R)\cong I/I^2.$$

The present flatness result shows why this cell must not be sought as an ordinary nonflat summand of the source cokernel. It arises from the derived self-intersection in the gradient-Koszul carrier target of Benincasa Entry 487. Forgetting that derived degree leaves only the flat even quartic module.

Consequently Entry 473's constant global defect is not a generic interior torsion line. It must come from the interaction of the filtered global presentation with the derived relation or its boundary extension.

Next gate

Evaluate the lifted even carrier map into

$$[\mathcal O^{\oplus2}\xrightarrow{(K_a,K_b)}\mathcal O/(K)]$$

and compute its homotopy fiber. The degree-zero map should account for the two flat quartic directions, while the degree-minus-one fiber should recover $I/I^2$. This must be verified from the Koszul differential, not inferred from the dimension-one filtered defect.

The executable audit is the extended parity census in research/voevodsky/check_soft_axis_generic_interior_odd_cokernel.py.

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