Black Holes as Circulation Interfaces: Extending the First Law with Φ∞

This paper extends the First Law of Thermodynamics to curved and expanding spacetimes by introducing an additional flux term, Φ∞, representing energy circulation through the compactified boundary at infinity. While the local law ΔE = Q - W remains intact, global conservation requires the extension ΔE=Q−W+Φ∞. Applied to black holes, this framework shows that event horizons are not endpoints but circulation interfaces: mass-energy falling inward contributes to $\Phi_{\infty}$, balanced as flux across conformal infinity. This reframes black holes not as voids but as conduits in an infinite energy exchange. The theory is grounded in established GR formalisms (Bondi mass loss, ADM energy, Penrose compactification), consistent with Hawking radiation and Bekenstein entropy bounds, and resonates with Susskind’s holographic principle. Concrete tests are outlined: gravitational-wave energy accounting (LIGO/Virgo), photon redshift and CMB spectral distortions, multi-messenger (GW+EM+$\nu$) event closures, and laboratory analogues in Bose–Einstein condensates. The Modified First Law unifies these disparate phenomena under a single principle: energy is not absolutely conserved, but circulates through infinity.