[Refuted Pathway] The Pillar IV Spectral Energy Functional: Explicit Form from the Prime Wave Bridge

Pillar IV of the Canvas Model—the Feed Equation—has remained the least specified component of the framework. The spectral energy functional 𝔼 whose gradient drives the meta-time evolution of all parameters has been described in general terms but never written in explicit form. This has left the attractor fixed point—the source of all derived parameter values—as an assertion rather than a computation. This paper closes that gap. What this paper does: Using the Prime Wave Bridge—the structural correspondence between the Canvas Model's dynamic primitives and the Primitive Spectral Transform framework—we construct the explicit Pillar IV spectral energy functional. The von Mangoldt function Λ(n) provides the natural weighting of eigenmodes by their prime content. The repulsive potential V(d) = 1/|d| enforces the spectral rigidity observed in both the gauge coupling spectrum and the Riemann zeros. The explicit functional is: 𝔼[ℰ] = Σ_n Λ(n)(λ_n − λ_n*)² + Σ_{m≠n} Λ(m)Λ(n) / |λ_m − λ_n| where λ_n are the eigenvalues of the processor ℰ, λ_n* is the equilibrium spectrum, and Λ(n) is the von Mangoldt function. What this paper derives: We prove that the attractor fixed point ∇𝔼 = 0 yields: · The gauge structural couplings g_structural²(q_s) = 16/q_s· The UWE weight equality c_base = d_base· The mode selection for the three fermion generations· The structural field vacuum expectation values Every previously asserted attractor value is now derived from the minimization of an explicitly specified functional. No free parameters remain in Pillar IV. Why this matters: Pillar IV is no longer a sketch. It is a precisely defined mathematical object. The Feed Equation is a gradient flow on this object. The Canvas Model's parameters are the global minimum of this object. The remaining open problems of the Canvas Model—the Yukawa sector, neutrino masses, electroweak scale, and vacuum stability—are now precisely formulated as eigenvalue computations within this spectral energy functional. Keywords: Pillar IV, Feed Equation, spectral energy functional, von Mangoldt function, Prime Wave Bridge, Canvas Model, attractor fixed point, gauge couplings, fermion generations, spectral rigidity