Anatomy of the Power Layer: Discrete Geometry, Parity Invariants, and the Reflection Principlel: An Introduction to the Power Layers Method

This monograph provides an accessible and self-contained introduction to the discrete geometry of power layers (C_n(k) = k^n - (k-1)^n), synthesizing the foundational results of the Power Layers Method (Proyecto Capas y Bolas). Anatomía de las Capas de Potencia: Base Teórica, Monografías de Investigación (Revisión corregida) | Zenodo Unlike monolithic algebraic formulations, this work analyzes power structures by peeling discrete cubes layer by layer, recovering the spatial memory of growth. Key aspects covered in this concise edition include: - Discrete geometry of the shell: Cardinality formulas, disjoint coordinate partitions, and non-flattening dimensional constraints (degree vs. dimension). - The two fundamental invariants: The pronic coordinate u = k(k - 1) and the linear baseline L = 2k - 1, establishing that odd-dimension layers depend strictly on u. - The reflection principle (the mirror): Structural symmetry under k -> 1 - k, the mirror ground state at k = 1/2, and dual divisor pairs (rooms) modulo odd primes. - The Lucas recurrence and the gold floor: Layer sequences as generalized Lucas sequences and their algebraic invariants. - Universal telescoping and parity invariants: Strict odd parity of layers, absence of shared prime factors with boundary sides, and finite difference laws (constant (n-1)-th differences equal to n!). - Structural crossroads (muros): How classical number sequences (centered polygonal and star numbers, Cuban primes, Catalan numbers, and Lucas numbers) arise naturally from layer geometry.