The Electroweak Hierarchy from First Principles: Complete Derivation of v = 245 GeV in the Canvas Model

The electroweak scale v = 246 GeV is sixteen orders of magnitude below the Planck scale M_P = 1.22 \times 10^{19} GeV. In the Standard Model, this hierarchy is unexplained—the Higgs mass parameter is a free input. This paper derives the electroweak scale from first principles in the canvas model. The hierarchy emerges from the balance of two competing pressures in the Higgs effective potential, selected by the Pillar IV Feed dynamics. The threshold pressure arises from the vacuum energy of the ten structural fields and favors large Higgs amplitude. The information bound pressure arises from the finite information capacity of the observable universe and favors small Higgs amplitude. The threshold pressure potential V_{\text{threshold}}(\phi) = -10\pi^2 M_P^4 \exp(-1/(2\phi^2)) is derived from the zero-point energy of the ten structural fields on the discrete voxel lattice with a physical Planck-scale cutoff. The information bound pressure potential V_{\text{info}}(\phi) = B_0 I_{\text{max}} \phi^6/M_P^6 has coefficient B_0 = 5\pi/6 determined by self-consistency of the dimensional reduction. The balance at the attractor minimum gives \phi^8 \propto I_{\text{max}}^{-1}, yielding the characteristic (H_0 t_P)^{1/4} scaling of the hierarchy. The complete formula is: \boxed{v = \frac{M_P}{\sqrt{n}} \cdot (H_0 t_P)^{1/4} \cdot \sqrt{\frac{c_{\text{eff}}}{T_{ST}}} \cdot \sqrt{\frac{10}{\pi}} = 245 \text{ GeV}} where n=3 is the number of spatial dimensions, c_{\text{eff}} = 0.00446 is the effective Acceleration weight, T_{ST}=4 is the spacetime threshold, and \kappa = \sqrt{10/\pi} \approx 1.784 is derived from the self-consistency of the dimensional reduction for a scalar field—in direct analogy to the gauge coupling result g_3 = 5\pi/32. Why this matters: Every factor in the hierarchy formula traces to the eight primitives and four pillars. The derivation uses no experimental inputs beyond the cosmological boundary condition H_0. The electroweak scale is a cosmological parameter—it depends on the age of the universe through the fourth root of the dimensionless cosmic time (H_0 t_P)^{1/4}. The Standard Model hierarchy problem is resolved: the smallness of v/M_P is not a fine-tuning. It is a consequence of the large age of the universe, encoded in the fourth root of the dimensionless cosmic time. No supersymmetry, no warped dimensions, no anthropic selection are required. The hierarchy coefficient \kappa = \sqrt{10/\pi} is the scalar analog of the gauge coupling result g_3 = 5\pi/32. Both emerge from the same self-consistency principle applied to different sectors: the gauge sector yields a dimensionless coupling, the Higgs sector yields a mass scale. Keywords: electroweak hierarchy, Higgs mechanism, canvas model, threshold pressure, information bound, cosmological boundary condition, H_0, hierarchy problem, vacuum energy, structural fields