Synthesis: Irreversibility and the Entropy Chain
The Irreversibility Chain
The complete logical thread linking Perelman’s monotonicity to observable spin memory:
$$\underbrace{\frac{d\mathcal{W}}{dt} \geq 0}_{\text{Perelman}} \longrightarrow{\text{sigma model}} \underbrace{\frac{dc}{d\log\mu}\leq 0}_{c\text{-theorem}} \longrightarrow{\text{celestial CFT}} \underbrace{[L_n,L_m]=\cdots}_{\text{Virasoro}} \longrightarrow{\text{BMS}} \underbrace{\Delta\Psi \neq 0}_{\text{spin memory}}$$
Each arrow represents a precise mathematical correspondence developed in the preceding sections.
Irreversibility of Spin Memory
The Perelman $\mathcal{W}$-functional is non-decreasing under Ricci flow. Through the sigma-model correspondence, this maps to irreversibility of the RG flow. At the level of asymptotic symmetries, this means the BMS vacua before and after a gravitational wave burst are generically inequivalent: the transition between them cannot be undone.
Define the spin vacuum angle as the net spin memory:
$$\theta_{\rm spin} = \Delta\Psi$$
The vacuum transition amplitude for any operator $\mathcal{O}$ is controlled by the superrotation charge $\mathcal{Q}_Y$:
$$\langle 0^+|\mathcal{O}|0^+\rangle - \langle 0^-|\mathcal{O}|0^-\rangle = \langle 0^-|[i\mathcal{Q}_Y,\mathcal{O}]|0^-\rangle \neq 0$$
The non-vanishing of this commutator is the physical statement that gravitational radiation permanently alters the vacuum structure — a direct observable consequence of Perelman’s monotonicity theorem.
Combined Perelman–BMS Functional
The three threads — Ricci flow, Chern–Simons gauge theory, and Bondi radiation — unify into a single variational principle:
$$\mathcal{F}_{\rm combined}[g,A,f] = \underbrace{\int_M(R+|\nabla f|^2)e^{-f}d\mu}_{\text{Perelman}} + \underbrace{\frac{k}{4\pi}\int_{\partial M}\mathrm{tr}(A\wedge dA + \tfrac{2}{3}A^3)}_{\text{Chern-Simons}} + \underbrace{\int_{\mathscr{I}^+}(m_B + \tfrac{1}{8}C_{AB}N^{AB})du\,d^2\Omega}_{\text{Bondi mass}}$$
The Euler–Lagrange equations of $\mathcal{F}_{\rm combined}$ yield the Ricci flow equation, Chern–Simons flatness, and the Bondi mass-loss formula simultaneously. This is the geometric content of the infrared triangle: three seemingly disparate structures emerge as critical points of a single functional.
Open Problem
Can one define a rigorous Perelman entropy for gravitational radiation — a monotone functional on BMS vacua whose fixed points are Minkowski and Kerr, and whose gradient flow is the Bondi news equation?
Such a functional would unify the second law of black hole thermodynamics with Perelman’s $\mathcal{W}$-entropy, providing a single irreversibility principle governing both the formation and evaporation of black holes and the permanent displacement of gravitational wave detectors by spin memory.