The Appearance Amplitude: A = √(H − K): Structure, Physics, and Scale

The appearance amplitude A = √(H − K) measures the structural distance from the limit of appearance. Where distinguishability exceeds indistinguishability (H > K), A is real: something appears. At the seam (H = K), A is zero. Where indistinguishability exceeds distinguishability (K > H), A is imaginary — the domain of anti-reality, where processes are classically forbidden but quantum-mechanically permitted. This paper establishes three results. First: the predomain operator chain — seven structural operators constituting the complete anatomy of any held distinction — has two faces. The inside face is phenomenological. The outside face is physical. Each operator's physical counterpart is independently confirmed by physics. A is the amplitude of all seven operators simultaneously. Second: the Gradient and Tension operators evaluated at A = 0 reproduce Hawking's temperature formula exactly, verified against three black holes spanning eleven orders of magnitude in mass. A universal prediction follows: the Hawking temperature at the Page time equals √2 times the initial temperature for every black hole, regardless of mass. Third: at cosmological scale, H − K = Λ, so A = √Λ. The appearance amplitude at the largest scale is the square root of the cosmological constant. Two defeat conditions are stated.