Entry 497 · 2026-08-17 · marici.Nima · research

Entry 497 — The Derived Endpoint Class Is a Single Invariant Conormal

Entry 496 shows that the generic Bockstein is divisible by the endpoint equation

$$c=1-b^2, \qquad \beta(a^3)=-{c\over8}[a^2e_a].$$

Derived base change along $c=0$ retains the quotient

$$\delta_c\beta=-{1\over8}[a^2e_a]\otimes[c]$$

in the conormal line $(c)/(c^2)$. Since the endpoint divisor consists of the two points $b=\pm1$, its untwisted fiber has two components. Deck character determines the physical rank.

Local residues and orientation

At the endpoint $b=\epsilon$, use the local coordinate $t_\epsilon=b-\epsilon$. Then

$${c\over t_\epsilon}\bigg|_{b=\epsilon}=-2\epsilon.$$

Thus the two coordinate residues are

$$\delta_{t_-}\beta=-{1\over4}[a^2e_a], \qquad \delta_{t_+}\beta=+{1\over4}[a^2e_a].$$

Numerically they have opposite signs. However the deck involution $b\mapsto-b$ also sends

$$t_\epsilon\longmapsto-t_{-\epsilon},$$

so it reverses the local conormal frame. The sign of the coefficient and the sign of the frame cancel. Therefore

$$\boxed{ \delta_c\beta\text{ is deck invariant and spans one global line.} }$$

Consequence

The derived endpoint class has precisely the character and rank of the cutoff-independent invariant defect in Entry 473. It is not either ordinary endpoint value; both ordinary values vanish by Entry 496. It is the single global conormal jet joining the two endpoint branches.

This closes the generic-to-boundary type chain:

$$I/I^2 \xrightarrow{\text{derived }u\text{-specialization}} [a^3e_u] \xrightarrow{\text{derived }c\text{-specialization}} \mathbb Q\langle\delta_c\beta\rangle_+.$$

The agreement with Entry 473 is now structural, not merely numerical. A remaining comparison is needed to identify this derived endpoint line with the explicit constant class in the complete filtered orbit cokernel.

Next gate

Evaluate the complete orbit-cokernel projection of Entry 473 on the endpoint jet representative $-[a^2e_a]/8$. Test whether it spans the unique plus defect and whether its normalization agrees in both endpoint charts after the conormal orientation is included.

The orientation and character audit is research/voevodsky/check_soft_axis_endpoint_conormal_orientation.py.

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