A 2-Adic Theory of Coronal Heating

The solar coronal heating problem asks how a plasma rooted above a comparatively cool photosphere can sustain coronal-scale temperatures. We propose that the relevant magnetic transport is governed by a reciprocal binary hierarchy rather than by a continuously diluted carrier. At hierarchical depth v, the normalised Haar weight of a designated active magnetic lineage is postulated to scale as P(v)=2−v, whilst the pressure-equivalent transport capacity associated with that lineage, conditional on survival, scales as E(v)=E02v. Their product is therefore an exact invariant, Q(v)=P(v)E(v)=E0. This reciprocal cancellation is the mathematical core of the theory. The quantity Q is defined as a pressure-equivalent transport density and is coupled to the electron pressure through Q=Pe=NekBTe. The coupling gives the profile-independent thermodynamic law Te(Ne)=E0/(NekB), so temperature rises inversely with particle density whenever the constant-Q regime applies. For the exponential atmospheric embedding Ne=N0exp(−h/H), the theory gives Te=T0exp(h/H). Assuming local proton-electron thermal equality, Tp=Te, the corresponding proton root-mean-square thermal speed obeys vth∝exp(h/2H). With T0=5800 K, N0=1012 cm−3, H=250 km, and a cooling-regime threshold Ncrit=2.5×109 cm−3, the threshold occurs at hcrit=1497.9 km. The exact switching temperature is 2.32×106 K, whilst the first 1500-km sample gives 2.340×106 K and, under Tp=Te, a characteristic proton thermal speed of 240.7 km s−1. The paper also formulates the inverse problem and a separate finite-state conjecture relating a one-day binary state tick to an approximately 11.2-year cycle with b≈12.