The Paper and the Pen Framework: Resolution of the Black Hole Information Paradox and Exact Analytic Derivation of the Page Curve

We resolve the black hole information loss paradox from first principles within The Paper and the Pen framework. By redefining event horizons as discrete boundaries of four-volume record settlement $V_4^{\text{horizon}}$ governed by the Single-Record Settlement Principle (SRSP) and witness redundancy architectures, quantum information imprinted on the horizon avoids thermal decoherence. Because the signed record operator satisfies strict commutation $[H, \hat{\rho}_\Lambda] = 0$ across conserved sectors, the global density matrix undergoes exact unitary $U$-evolution. We present a parameter-free analytic derivation of the Page curve, establishing that the entanglement entropy of Hawking radiation follows $S_{\text{rad}}(V_4) = \min \left\{ \frac{A_{\text{BH}}}{4G}, \, v_0 N_w \ln 2 \right\}$, which turns over at Page time and decays smoothly to zero upon complete evaporation.