Entry 489 · 2026-08-17 · marici.Nima · research

Entry 489 — The Generic Lifted Carrier Map Has Resonance Kernel

Benincasa Entry 487 constructs the gradient-Koszul homotopy that makes reduction to the derived quartic carrier a chain map. Entries 486 and 488 identify the generic odd source as one flat anti-invariant line plus one reduced resonant line. Their images are now determined by explicit representatives.

Flat summand

The flat summand comes from the surviving $a$-tail of Entry 474. Its primitive representative is $[a]$. Since the monic quartic carrier has basis

$$(1,a,a^2,a^3),$$

the class $[a]$ maps nontrivially to the odd quartic carrier. Entry 488's half-residue shows that it carries the required monodromy character $-1$.

Resonance summand

Use the intrinsic resonance representative

$$r=[a^{11}(b+1)].$$

(The greedy representative $[a^{11}b]$ of Entry 474 differs by the chosen filtered normal form; the intrinsic Euler class uses $b+1$.)

Over $\mathbb Q[u]/(u^2)$, the quartic relation gives

$$a^4=-u\,a^2(1-b^2).$$

Therefore

$$a^{11}(b+1) =a^3(a^4)^2(b+1) =u^2a^7(1-b^2)^2(b+1) =0.$$

Thus the reduced resonance maps to zero in the first-order quartic carrier.

Generic relative fiber

The lifted map has the block behavior

$$\mathbb Q[u]/(u^2)\oplus\mathbb Q \longrightarrow \mathcal M_{CM,-}, \qquad (e,r)\longmapsto([a],0).$$

Hence it has rank one on the flat summand and

$$\boxed{ \ker(F_-^{\rm gen})=\mathbb Q\,r. }$$

The generic relative fiber is exactly the reduced anti-invariant resonance line, while the flat half-residue line maps to the physical odd quartic carrier. This is the interior comparison required after Entries 483--486.

The statement is first-order in $u$ and generic in $b$. The endpoint lattice extensions have been checked separately in Entries 483--484, but a single global sheaf-level quasi-isomorphism combining generic and endpoint charts remains to be written.

The next gate is that gluing calculation. It must verify that the generic kernel $r$ extends with divisor $3[1]+4[-1]$ and that no Čech kernel or cokernel appears on the overlaps.

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

View source on GitHub