KXcoeff_orient_odd

This module stages honest intermediate targets for the Steve baryogenesis loop. The first invariant is sphaleron zero-protection: electroweak sphalerons conserve $B-L$, so a vanishing sourced $B-L$ with equilibrated sphalerons forces surviving baryon number to zero.

The CKN source coefficient $K_X$ packages the only allowed $\chi$ coupling to $B-L$. Because $B-L$ is gauge-anomaly-free, the interaction is the derivative coupling $(\partial_\mu\chi/f_\chi)\cdot J^\mu_{B-L}$, which yields the chemical potential $\mu_{B-L}=\varepsilon,\dot\chi/f_\chi$. Thus $K_X=\varepsilon/f_\chi$: the sign $\varepsilon$ is the eight-tick orientation, and the magnitude is set by the decay constant $f_\chi$. No $\eta_B$ is inserted by hand.

The present lemma records the elementary oddness of that prefactor under orientation reversal.

Unfold the definition $K_X(\varepsilon,f_\chi)=\varepsilon/f_\chi$. The goal becomes $(-\varepsilon)/f_\chi=-(\varepsilon/f_\chi)$, which ring discharges over $\mathbb{R}$. No external lemmas are required.

The sign of $\eta_B$ in the Recognition baryogenesis lane is not an external input; it is inherited from the eight-tick orientation $\varepsilon$ through $K_X$ into the $B-L$ source and then into the yield. The doc-comment flags this lemma as the structural origin of that sign, later carried by the oddness statements for the $a_3$ source and for $\eta_B$ from yield.

In the broader framework this sits under the T7 eight-tick octave: $\varepsilon=\pm 1$ is the discrete orientation of the period-$2^3$ ledger cycle. Keeping $K_X$ odd under $\varepsilon\mapsto -\varepsilon$ prevents the staging file from smuggling a preferred baryon sign. Downstream used-by edges are not yet wired in this snapshot, but the intended consumers are the oddness lemmas named in the doc-comment.

The module charter forbids fake physics conditions; this pure algebraic identity is exactly the kind of small proved target that charter demands.