From Local Information Loss to Spectral Gaps: Physical–Observation Structure and Hamiltonian Reconstruction

Local information can be lost rapidly while a collective mode survives for a long time. We study this distinction on a physical–observation joint state space. The second variation of relative entropy under a local record map gives a positive Fisher-loss operator; the least loss over normalized noninvariant scores is the spectral gap of the associated posterior-resampling process. The local strength and the compatibility of surviving directions enter separately. A fixed-width interacting ladder has a positive gap at every fixed finite coupling, as follows from an explicit block comparison, although strong correlation makes the bound small. We then reverse the construction. For a given irreducible real Hamiltonian with nonpositive off-diagonal entries, its ground amplitude determines pair-record probabilities and energy-valued loss weights, reproducing the physical excitation gap exactly. A positive trial amplitude gives an error interval controlled by the oscillation of its local energy, without requiring the exact ground state. When signs remain, the energy form is a difference of positive information forms and an explicit cancellation constant controls the gap. Sparse diagonalization of open transverse-field chains up to 65 536 configurations agrees with both the reconstructed information spectra and an independent free-fermion calculation. The critical law n∆_n/J → π, the gapped regime, and the ordered finite-chain splitting are distinguished. The result is an explicit connection for specified channels and Hamiltonians, with the requirements of positivity, locality, and energy normalization kept separate.