Theorem of the O₂–UV Projection in Satellite Climate Metrology: Structural Non-Identifiability of Thermal Attribution to Trace Gases

Background. The global satellite temperature series (AMSU/MSU) is retrieved from microwave radiance emitted by O₂ at 50–60 GHz; at the surface, kinetic sensors integrate the translational energy of 99.9% of the mass (N₂/O₂). The concentration series (XCO₂, OCO-2/OCO-3/GOSAT/AIRS) is computed as the ratio between the optical depth of the gas and the O₂ column of the A-band at 0.76 μm. Both chains are born from the same metrological foundation: the spectroscopic physics of oxygen. Objective. To identify the real measurand of the climate satellite observable and to determine whether the observed covariance Cov(C_obs, T_obs) can function as empirical proof of the thermal attribution of any trace gas. Method. The retrieval channels are defined with the formal equations operated by the agencies (T_obs = T_real + β·δ_O₂; C_obs = C_real + α·δ_O₂); the covariance is decomposed by error propagation; the identifiability of the parameter is analyzed under the null hypothesis Cov(C_real, T_real) = 0; the first-principles causal physics established by laboratory spectroscopy is substituted (T_real ∝ k·(I_UV·[O₂]²)). Results. (1) The observed covariance contains the non-null term αβ·Var(δ_O₂). (2) There exists at least one configuration with null physical covariance that exactly reproduces the recorded correlation: the attribution parameter is structurally non-identifiable with this network. (3) Affirmative theorem: the observed covariance adopts the closed form Cov(C_obs, T_obs) ≡ G(Var(δ_O₂), Cov(δ_O₂, I_UV)): the series interpreted as "gas effect on temperature" is the projection of oxygen and ultraviolet spectral physics onto the network. (4) The verdict is neutral with respect to the gas: substituting CO₂ for CH₄, H₂O, O₃, or N₂O in the attribution, Results 1–3 hold identically. (5) The effective sample size of the CMIP ensemble is N = 1 and the Central Limit Theorem does not apply to it. (6) There exists a unique spectroscopic failure point with retroactive recalibration of the series. (7) The detection and attribution literature that assumes independence of errors is nullified as a matter of right. Conclusion. The real measurand of the observable is not the effect of any trace gas: it is the spectro-photonic variability of O₂ and UV. Strictly speaking of the instrument, within the satellite thermal series there is no CO₂ effect, nor CH₄, nor H₂O, nor O₃ effect to attribute. The network does not fail to measure the effect of trace gases: it successfully measures something else and mislabels it. The delivered magnitude is real; the label is false.