IndisputableMonolith.MaxwellDEC
The MaxwellDEC module sets up the discrete exterior calculus needed to express Maxwell's equations on simplicial complexes. It introduces oriented k-simplices as abstract identifiers together with differential forms, coboundary operators, and Hodge stars. Computational physicists and discrete geometers would cite the module when discretizing electromagnetic fields. The entire module consists of definitions and class declarations with no proofs.
claimThe module defines the oriented $k$-simplex as an abstract identifier, the type $DForm$ of discrete differential forms, the class $HasCoboundary$ supplying the coboundary operator, and the class $HasHodge$ supplying the discrete Hodge star, together with supporting structures for media, sources, and the discrete Maxwell equations.
background
The module works inside discrete differential geometry applied to electromagnetism. Its central object is the oriented $k$-simplex, which serves as the basic cell for building simplicial complexes. It then equips these complexes with $DForm$ for discrete forms, $HasCoboundary$ for the exterior derivative, and $HasHodge$ for the Hodge star. Additional definitions introduce material media, charge and current sources, and the resulting discrete Maxwell system. The setting is purely algebraic and does not yet invoke the Recognition Science forcing chain or the $J$-cost function.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
MaxwellDEC supplies the geometric primitives required for any discrete treatment of Maxwell's equations inside the Recognition Science monolith. It directly supports the energy functionals, admissibility conditions, and perfect-electric-conductor declarations that appear among its siblings. By furnishing the DEC infrastructure the module prepares the ground for later steps that connect the eight-tick octave and spatial dimension $D=3$ to concrete electromagnetic observables.
scope and limits
- Does not prove any convergence result to the continuum Maxwell equations.
- Does not introduce numerical discretization parameters or mesh refinement.
- Does not reference the $J$-cost function, the golden-ratio ladder, or Recognition Science constants.
- Does not specify boundary conditions beyond the abstract coboundary operator.