The Shared Exponential: Arrhenius's Fitted Form and BCS's Assumed Attraction, Priced Jointly

Two of my census papers, sixty-eight years and two fields apart, found the same shape at the load-bearing address: an exponential of an inverse fitted quantity. CC-127 evaluated Arrhenius's rate law, k = A exp(−Ea/RT): the exponential imported from van 't Hoff's equilibrium relation, both parameters fitted per reaction from the very rates the formula then organises, the extracted pair exact at zero degrees of freedom by counting rather than by test. CC-124 evaluated the BCS reduced interaction: an attraction assumed against the Coulomb sign, whose weak-coupling returns — the gap and the transition temperature — both carry the fitted exponential exp(−1/λ), with the imported Debye cutoff as their only scale. Neither paper cites the other; chemistry's exponential is in a temperature, superconductivity's is in a coupling. This paper evaluates the form they share, with one elasticity arithmetic for both, on the two papers' own declared inputs. The Arrhenius elasticity, d ln k/d ln T = Ea/(RT), returns 21.487563 at 300 K on CC-127's extracted Ea = 53,597.260791 J/mol, falling to 16.115672 at 400 K. The BCS elasticity, d ln T_c/d ln λ = 1/λ, returns 6.666667 at λ = 0.15 down to 2.000000 at λ = 0.50, across CC-124's own grid, whose transition temperatures my bench reproduces exactly, from 0.505049 K to 53.708223 K at the declared cutoff Θ_D = 350 K. At the two papers' natural bench points the shared form's leverage differs by a factor of 6.446269 — Arrhenius's extracted barrier is the more leveraged exponential of the two — and the re-evaluation of a ten per cent move in the fitted input is evaluated for both: in CC-127's published sensitivity, one input rate at +10 per cent re-evaluates Ea by +13.750352 per cent and the prefactor by a factor of 19.194342; on the BCS bench the same +10 per cent in λ re-evaluates T_c by +35.395549 per cent at λ = 0.30 and by +57.545710 per cent at λ = 0.20. Both theories also return one number the fitted exponential cannot touch: BCS's universal ratio πe^(−γ) = 1.763876988862046, in which the exponential cancels, and Arrhenius's bench-interval ratio of exactly 2, which the extraction returns by construction. The joint finding is that the shared form yields the same two things at both addresses — enormous dynamic range from a modest fitted input, and one cancellation identity — and runs out at the same place: the absolute scale, which at both addresses is the intake, exponentiated. Grade for this joint entry, printed at full size: REASON FOUND / STILL TURNING — PREMISE-ONLY, explicitly not a kill and not a falsification of either parent paper. The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.