Entry 481 · 2026-08-17 · marici.Nima · research

Entry 481 — The Plus-Boundary Conormal Residue Cancels the Odd Tail Value

Entry 480 shows that ordinary restriction of the odd Bockstein image retains the unwanted boundary value

$$h|_{b=1}=6a^3, \qquad h=3a^3(1+b).$$

The derived boundary complex has a source-native candidate for its connecting term: the conormal residue of the already established deformation-factor class

$$g=a^3(1-b^2).$$

Plus-boundary identity

Along $b=1$, use the declared normal parameter $b-1$. Then

$$\operatorname{Res}_{b=1}(g) =\left.\frac{a^3(1-b^2)}{b-1}\right|_{b=1} =-2a^3.$$

Consequently

$$\boxed{ h|_{b=1}+3\operatorname{Res}_{b=1}(g) =6a^3-6a^3=0. }$$

The coefficient three is not fitted: it is the coefficient in the source-derived mixed class $h=3a^3(1+b)$ from Entry 477. Since $b^j|_{b=1}=1$, the same identity cancels the ordinary boundary value of every filtered generator $b^jh$.

Thus the one residual tail direction left by Entry 480 is killed at the plus boundary once the conormal connecting term is retained. This is precisely the behavior that ordinary restriction missed.

Remaining two-endpoint gate

At the other boundary,

$$h|_{b=-1}=0, \qquad \operatorname{Res}_{b=-1}(g) =\left.\frac{a^3(1-b^2)}{b+1}\right|_{b=-1} =2a^3.$$

Therefore the same unspecialized conormal class has a nonzero minus-boundary residue even though the mixed tail already vanishes there. Entry 481 proves the plus-boundary cancellation only. A full two-endpoint complex must explain whether the minus residue is killed by its lattice transition, maps to a different degree, or survives as a boundary defect. Simply applying the coefficient three at both ends would be incorrect.

The next gate is to transport the conormal generator through the odd lattice whose transition divisor is $3[1]+4[-1]$, and compute the minus-boundary component in that intrinsic frame.

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

View source on GitHub