Reconstructing the Past: Identifiability, Physical Witnesses, and Experimental Tests

Reconstructing a past is an inverse problem before it is a narrative problem. A physical history is identifiable only to the extent that accessible witnesses, observation channels, calibration state, side information, nuisance structure, and provenance distinguish it from alternatives.

We formalize this limit with a witness-relative historical quotient. Let H be a declared history space, W a physically accessible witness family, K the observation channel, C the calibration and side information genuinely available to an observer, and N the nuisance and provenance family. Histories are observationally equivalent when the declared evidence induces the same observable law after exactly the allowed treatment of C and N. The maximal identifiable object is therefore Q* = H / ~_(W,K,C,N).

A validated new witness can refine this quotient by splitting an existing equivalence class; downstream post-processing alone cannot. Local rank gain is useful but does not establish global uniqueness, because disconnected solutions, symmetries, hidden sectors, topology, or model-family alternatives can preserve the same observations. Statistical overlap imposes irreducible error, channel drift changes the inverse problem, and local reconstructions need not glue into a globally consistent history.

We combine exact fiber factorization, statistical experiment comparison, stabilizer and nullspace analysis, calibration accounting, and complementary-sensor design into a single reconstruction framework. The central result is constructive non-uniqueness: physical pasts can be reconstructed, but only to the resolution licensed by surviving evidence.