FRC 566.001 v2.0 — Entropy–Coherence Reciprocity and the Universal Coherence Condition

We state the founding law of the FRC thermodynamic series in directional form. For an open system with an operational coherence measure C, independent of its entropy channel, the combined ledger current sigma_566 = sigma_ST + k* dlnC has a definite sign structure: on coherence-gaining branches sigma_566 >= sigma_ST >= 0 — creating coherence necessarily produces entropy, by the Landauer mechanism generalized from memory reset to coherence creation; on coherence-destroying branches no bound exists, and quasi-statically sigma_566 -> k* dlnC < 0, since dissolving order raises the system's own entropy and requires no export. The reversible closed form dS + k* dlnC = 0 is the boundary case. The system-only quantity Q = S + k* lnC is not conserved across coherence transitions; its motion is the exchange current. Supporting results live in companion papers: exact solvable exhibit on the von Mises/Kuramoto family with unique closure point (FRC 566.030); instrumented verification of both directions in a Langevin closure with exact free energy (FRC 100.005); UCC well-posedness and spectral theory (FRC 566.010, 566.020); status-labeled mathematical foundations (FRC 826.829). This note retains the Universal Coherence Condition transport equation with its dissipation and Lyapunov structure, the information-theoretic projection, the thermodynamic projection on the coherence-gaining side (maximum work, minimum maintenance export rate), and a four-clause falsifiable standing prediction, each clause tested and passed in the reference closure. The law is the thermodynamic foundation for treating quantum measurement as finite-time relaxation into coherence basins (FRC 100.003) with the Born rule as its equilibrium (FRC 100.006). Version note: v1's erasure-cost conjecture was decided negatively by registered measurement and is superseded by the directional structure (results-negative-kept).