Cumulative boundary heat from finite temperature readings: sharp deterministic recovery and a short-time depth separation

We study cumulative non-surface heat loss in a rectangular heat model with known forcing and mixed boundary data. The unknown initial perturbation belongs to one fixed spectral H2 ball, and finitely many physical temperature readings have a prescribed joint ellipsoidal error. Standard optimal recovery theory gives the sharp global scalar error. We make this result explicit for the cumulative heat functional, retaining all three-dimensional observation modes and constructing worst initial states. Algebraic and exponential spectral tails give convergent certificates for the infinite-dimensional problem. The unique old worst state yields a necessary and sufficient condition for a new reading to reduce the error, including correlated deterministic errors, together with a positive lower bound on the reduction. For a 100 m column with fixed norm length 25 m and no prior readings, we prove an analytic short-time separation: at the same time and error tolerance, one reading at 75 m strictly reduces the cumulative-heat error while a surface reading leaves it unchanged. A computable sufficient time interval accompanies this result. The radius, reference field, forcing, boundary data and joint error set are inputs; their physical calibration is a separate problem.