Relativistic Shedding as Threshold Emission into Hidden Sectors

TWe develop a laboratory–only test of hidden–sector emission based on a deterministic, tunable threshold into the mixed in–medium eigenmode. The key signature is an on/off contrast obtained by bracketing the condition β nA′ (ω) > 1 with two otherwise identical runs (“Off” just below; “On” just above), so that smooth detector effects cancel by design. We keep the method simulation–light: the incremental threshold yield Nth is built from a transparent scaling |εeff (ω)|2× (Frank–Tamm spectrum) × S(ω) A(ω, θ), while the selected background Nbrem is reused from a vetted missing–momentum baseline (e.g. LDMX). The on/off contrast C and expected significance Z follow in closed form in the Gaussian regime, with profile– likelihood CLs and Asimov construction used for exclusions. Dataset–native falsifiers are built in: a step only at the predicted crossing, thickness scaling (linear → plateau), material swaps that move the threshold, an angular check (cos θ ≃ 1/(β ℜ[nA′ ])), band–limited excess, null samples, and a calorimetric null. Formation–length and damping effects are handled analytically near turn–on. We position the method within the current/prospective landscape (NA64, BaBar/Belle II, LDMX) and provide a reuse–first adapter to interface directly to published projections. Complete derivations (effective mixing, dispersion, threshold, statistics) are collected in Apps. A–C. Cosmology and model–building implications are deferred to a separate companion paper. We release minimal scripts for Nth, a medium–model card for εr (ω), and a template Combine workspace to reproduce the results.