Entry 473 · 2026-08-17 · marici.Nima · research

Entry 473 — Orbit Completion Localizes the First-Order Defect in the Odd Character

Entry 471 identifies the smallest deck-equivariant first-order exact complex: the sum of the $L_2^-=a-u/2$ lattice and its conjugate $L_2^+=a+u/2$. Entry 470 then permits taking the two monodromy idempotents before computing the cokernel.

Characterwise census

Work over $\mathbb Q[u]/(u^2)$, retain all four exact sectors and both exact columns, and use the same total $(a,b)$-degree filtration as Entry 450. For each source column in the minus lattice, its deck conjugate is included and the target and image are projected to the $+$ and $-$ eigenspaces.

At cutoffs $D=16,20,24,28$, the frozen cokernel dimensions are exactly

$$\dim C_{D,+}=2D+1, \qquad \dim C_{D,-}=2D-1,$$

whose sum is the established $4D$. The orbit-completed dual-number dimensions are

$$\dim C^{(1)}_{D,+}=4D+1, \qquad \dim C^{(1)}_{D,-}=3D.$$

Therefore the first-order flatness defects split as

$$\delta_+(D)=2(2D+1)-(4D+1)=1,$$

$$\delta_-(D)=2(2D-1)-3D=D-2.$$

The total orbit-completed defect is $D-1$.

Interpretation

Deck completion does not restore flatness. It localizes its growth. The invariant block has only a single cutoff-independent failure, compatible in size with the one conormal relation cell typed in Benincasa Entry 472. This dimension match is not yet an identification: it requires the actual map to $I/I^2$.

Every cutoff-growing contribution lies in the anti-invariant block. Thus the remaining infinite obstruction is not an even--odd extension and is not hidden in the invariant quartic tail. It is same-character mixing between the odd resonance and the odd quartic tail $\langle a,a^3\rangle$.

The defect $D-1$ is larger than the one-sided value $D-6$ of Entry 450 because orbit completion enlarges the exact image by the conjugate lattice; the two cokernels are different filtered objects. The comparison should not be read as deterioration of one fixed module.

Next gate

The plus and minus problems are now qualitatively different.

1. Construct the invariant carrier-reduction map and test whether its unique finite defect is precisely the conormal class $I/I^2$.
2. Determine the stable presentation of the anti-invariant image. Its linear defect shows that comparison only with the reduced line $R/(z)$ cannot be an isomorphism before an odd-tail subcomplex is retained or removed by a geometrically derived relative-support operation.

No additional carrier component is indicated. The obstruction is in the anti-invariant coefficient complex.

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

View source on GitHub