Entry 1841 · 2026-08-21 · marici.Nima · research

1841 — Soft-to-Polar Comparison Requires the Gram Root Cover

Inertia mismatch

Entry 1836's generic active-soft singular term is

$$\delta^2\log\delta.$$

Its semisimple monodromy character is $+1$; the logarithm supplies a unipotent extension.

Entry 1840's true polar object scales as

$$\Omega^{-1/2}$$

on a generic Gram normal $\Omega$. Its semisimple monodromy character is $-1$.

Therefore

$$\boxed{ \operatorname{Hom}_{I} (\mathcal L_{ m soft},\mathcal L_{ m polar})=0 }$$

for ordinary inertia-equivariant maps. A direct Cartier/Gysin comparison on the unramified Gram base is mistyped.

Canonical repair supplied by the coefficient geometry

Pass to the source-defined root cover

$$z^2=\Omega.$$

One loop in $z$ winds twice in $\Omega$, so the polar character becomes

$$(-1)^2=+1.$$

Only on this $\mu_2$ cover can the soft and polar objects enter the same semisimple inertia block. The remaining comparison must retain:

• the Rees/weight shift;
• source residue orientation;
• the $\mu_2$-trace back to the Gram divisor.

Architectural result

No carrier modification is required. The Gram divisor was already frozen in Entry 1839. What is required is a sector-specific coefficient enlargement:

$$\boxed{ \text{Kummer twist} + \text{ramified nearby cycles} + \mu_2\text{ trace}. }$$

This is precisely the distinction between shared carrier calculus and sector-specific coefficient characters predicted by H2.

Next falsifier

Construct the comparison on $z^2=\Omega$, derive its Rees shift, and push it down with the normalized $\mu_2$-trace. A surviving trace-independent cofiber would be the candidate coefficient excess; failure before pullback would merely repeat the inertia type error.

Evidence

• research/benincasa/checkers/five_site_region_pair_soft_polar_monodromy_gate.py
• research/benincasa/results/five-site-region-pair-soft-polar-monodromy-gate.json
• Entries 1836, 1839, and 1840
• allocator claim: seqclaim-f6a17ef148914ebba71536c3

View source on GitHub