The Non-Identifiability Limit in Structural Monitoring

Structural monitoring rests on a silent premise: that the telemetry a structure produces determines, well enough, the structural state it is in. This paper makes the premise explicit, proves its failure for a declared observation class, and states the contract an honest monitor owes in consequence. For finitely many sensors reading a damage field through bounded response kernels under a declared noise floor, we prove two constructive theorems. First, any un-sensed region of positive measure admits arbitrarily severe damage with exactly unchanged telemetry: non-injectivity with unbounded safety divergence. Second — and less comfortably — even full sensor coverage does not close the gap: bounded kernels with a positive noise floor cannot see unbounded concentration, so a flaw of arbitrary local severity, concentrated below the instrument's resolution, moves every reading by less than the noise floor. Exactly identical telemetry across a coverage gap; telemetry indistinguishable within the noise floor under full coverage — and arbitrarily different safety either way: now theorems for the declared class, not a slogan. The consequence is not that monitoring is futile; it is that monitoring's claims have a forced grammar. A telemetry verdict is a statement about the set of states consistent with the reading, never about the state itself; the instrument's identity must include its coverage and resolution declaration; where the consistent set contains both safe and unsafe states the honest verdict is unresolved, and silence may never be presented as safety. We state this contract in five clauses, with its authority class declared, as the foundation slot of the Structural Control series: the limit that the series' instrument and architecture papers inherit and must operate inside.