muBL_from_KXcoeff

This module stages honest, small targets for the Steve baryogenesis loop. The governing invariant is sphaleron zero-protection: electroweak sphalerons conserve $B-L$, so a vanishing sourced $B-L$ charge forces vanishing relic baryon number once sphalerons equilibrate.

The CKN source coefficient is defined by $K_X=\varepsilon/f_\chi$. Because $B-L$ is gauge-anomaly-free, the only $\chi$ source is the derivative coupling $(\partial_\mu\chi/f_\chi)\cdot J^\mu_{B-L}$. The sign $\varepsilon$ comes from eight-tick orientation; the magnitude is set by the decay constant $f_\chi$. No baryon asymmetry $\eta_B$ is fed in by hand.

The chemical potential is the linear map $\mu_{B-L}(K_X,\dot\chi):=K_X\cdot\dot\chi$. Composing the two definitions yields the textbook source law used downstream for freeze-out windows and washout exponents.

Term-mode proof: unfold the definitions of $\mu_{B-L}$ and $K_X$, then close the resulting rational identity by ring. No external lemmas are required; the equality is definitional once both abbreviations are expanded.

Locks the coefficient-level source law $\mu_{B-L}=\varepsilon,\dot\chi/f_\chi$ that the baryogenesis staging lane needs before any freeze-out or sphaleron reprocessing argument. Its sole downstream consumer is the source-off limit muBL_from_KXcoeff_zero, which specializes to $\dot\chi=0$ and obtains $\mu_{B-L}=0$. That zero feeds the sphaleron zero-protection obstruction: if sourced $B-L$ vanishes and sphalerons equilibrate, the surviving baryon number is zero. The sign $\varepsilon$ is the eight-tick orientation (T7 landmark), so the identity also ties the chemical-potential source to the discrete ledger clock rather than to an ad-hoc continuous parameter.