The Fences Together: Saha's Equilibrium and Alfvén's Frozen-In Field, Priced Jointly

Two of my census papers price premises that are spent, side by side, in the same object: a magnetised stellar atmosphere. CC-123 priced Saha's postulated ionisation equilibrium and found its own machinery drawing two fences — a Debye-length fence and a coupling fence standing a fixed (4π)³ = 1984.401708 apart, both scaling as T³ — beyond which the dilute, ideal-gas import the formula rides on is spent outside its domain. CC-130 priced Alfvén's conductivity taken infinite and found the frozen-in picture it buys returning, at every declared finite conductivity, a finite magnetic Reynolds number against the infinity spent. Neither paper asks what the other asks. Saha's paper never mentions a magnetic field; Alfvén's never mentions an ionisation balance. But a stellar atmosphere does not choose between them: wherever a spectroscopist reads an ionisation state off a magnetised atmosphere, both premises are spent in the same breath, and the question neither paper can ask alone is where they hold together and where one of them has already run out while the other still stands. This paper prices that joint question on the two papers' own declared inputs, and on nothing else. My bench rebuilds CC-123's fences exactly — at 10,000 K the Debye fence stands at 1.080025e+17 cm⁻³ and the coupling fence at 2.143203e+20 cm⁻³, ratio 1984.401708 — and evaluates the frozen-in number across the same (T, n) plane, carrying CC-130's declared field, conductivity and length, with the density converted to a mass density by the proton mass. The joint map, over five temperatures from 5,000 K to 30,000 K and six densities from 1e15 to 1e20 cm⁻³, returns 30 cells: in 15 of them both premises stand, in 15 the frozen-in premise stands alone, and in none does the Saha premise stand alone, because on these declared inputs the frozen-in number does not fall to order unity until a density of 7.512977e+36 cm⁻³ — a density no atmosphere reaches and no bench should pretend to. The overlap region is therefore bounded entirely by Saha's fence: the joint validity region is exactly the dilute side of the Debye fence, priced here cell by cell, and its share of the declared map is 0.500000. At CC-123's own bench density, 1e17 cm⁻³, the frozen-in number is 8.667743e+09 and the Alfvén speed on the declared field is 68.975709 m/s: the magnetic premise is comfortable by nine orders of magnitude at exactly the address where the equilibrium premise is sitting on its own fence, with λ_D/d = 1.012913 at 10,000 K, as CC-123 printed. 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.