A finite measuring apparatus cannot record indefinitely without thermodynamic cost. This paper studies a phase-changing detector medium that serves simultaneously as sensor, memory, and operating reservoir. In the frozen phase the medium can preserve a persistent record; in the melted phase it restores reusable sensing volume but erases part of that record. This coupling produces a trade-off between persistent memory and reusable reservoir, formalized here as a finite observer accounting architecture. We derive a pressure-control law for the frozen fraction, include the fixed-volume phase density correction, and identify a stability condition requiring latent cooling to dominate phase-volume feedback. We distinguish storage capacity from recoverable temporal order by introducing a freeze-front ordering field. Optical and thermal readout are treated as apparatus-corrected measurement channels, not as a closed entropy ledger. The individual formulas are standard or engineering prior art; the proposed contribution is the architecture tying phase change, pressure control, erasure, and apparatus cost into one finite-observer model. Black-hole finite-radius physics is treated only as a boundary correction through redshift, radar delay, proper acceleration, and quasi local thermodynamic fences. The finite shell keeps an apparatus-corrected record; exact asymptotic export accounting remains at null infinity.