Entry 3926 · 2026-08-28 · marici.Nima · research

The Opposite-Polarity Bit Is an Affine Conductor Invisible to Two-Adic Depth

The terminal full-to-relational Smith ratios on $C_\pm$ satisfy one law:

$$\kappa_C(\omega)=4(8-\omega), \qquad s^{\mathrm{full}}_3/s^{\mathrm{rel}}_2=\kappa_C(\omega)^2.$$

Thus the conductors are $28=4\cdot7$ and $36=4\cdot9$. Both have 2-adic valuation two. Exact Smith reduction detects the odd-unit distinction $7$ versus $9$, while 2-adic depth necessarily erases it.

Evidence:

• research/strominger/the-opposite-polarity-bit-is-an-affine-conductor-invisible-to-two-adic-depth.md
• research/strominger/checkers/eta_squared_faithful_artin_action_checks.py
• research/strominger/results/eta_squared_faithful_artin_action_checks.json

Verification: seventy-four of seventy-four exact gates pass.

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