The Coherence Lagrangian: Explicit Free Energy Potential V(K), Symmetries, Noether Currents, and Seven Testable Predictions

The Spektre corpus has referenced the coherence landscape V (K) in 57 papers without deriving its explicit form. This paper fills that gap. Starting from the GCT dynamical equation, the free energy potential V (K) is derived by integration: V (K) = -α (K²/2 - K³/3) + (β/γ) K - (1/γ) ln (1+γK) + DK. This is a Landau-Ginzburg potential with cubic self-interaction and logarithmic correction. Kcrit ≈ 0. 127 is the barrier maximum — the tipping point between coherent and incoherent basins, a dimensionless geometric constant analogous to critical exponents in statistical mechanics. The spatiotemporal Lagrangian density L = ½∂μK∂μK - V (K) generates a scalar field theory whose Noether current is the stress-energy tensor sourcing gravity via Einstein’s equation. The multiplicative structure K = ρ·I_Φ·F is derived from series reliability theory. Quantum mapping made explicit: ρ → normalization, I_Φ → unitarity fidelity, F → 1 - SᵥN/Sₘax. Seven testable predictions: Born rule deviation at degraded detector coherence, precursor signal before quantum emission, F-first institutional collapse ordering, DESI DR3 thawing match, no dark matter particle, Casimir deviation at large separation, BEC coherence time correlation. External anchors: Landau-Ginzburg, Eigen error threshold, Scheffer critical transitions, Ryu-Takayanagi, Higgs mechanism, Jacobson thermodynamic gravity, Coleman-De Luccia vacuum decay. Part of the Spektre research corpus.