Emergent Geometry from Relational Constraints: The Bounded-Lattice Dual-Deltoid Engine Summary This work presents a relational geometric framework in which classical Euclidean structures arise as consequences of bonded constraints rather than as primitive assumptions. The construction begins with the Elastic Immovable Mesh Model (EIMM), a variational system defined on a globally coupled relational lattice whose configuration is determined by a single bonded potential enforcing radius invariance, antipodal pairing, covariance isotropy, and mesh stiffness. At the centre of the construction lies a primary focus surrounded by six sub-focus nodes forming a fully coupled microcell; their relational bonds establish a star-plus-complete mesh in which every node constrains every other node, preventing local deformation modes and enforcing global covariance across the lattice. The minimising configuration of this system produces a structured axis triple whose support envelope forms a stacked dual-deltoid geometry. From this internal envelope a square emerges through a staged boundary-closure functional and subsequently promotes to a cube through a transdimensional extrusion governed by the mesh radius. In the generative layer of the framework no trigonometric functions, angular coordinates, or explicit appearance of π are introduced; classical angular descriptions arise only as external representations of the equilibrium configuration. A structural bridge identifies the support-function deltoid produced by the relational mesh with the rolling-circle deltoid of classical kinematic geometry, establishing a shared geometric object linking the variational and dynamical regimes and revealing a universal proportional constant connecting both descriptions. Building on this identification, the Bounded-Lattice Dual-Deltoid Engine (BLDDE) introduces a gear-natural dynamical architecture whose phase structure forms a bounded lattice governed by a twelve-well potential. Within this system, curvature transport, areal kernels, and thermal activity functions arise directly from the deltoid geometry generated in the relational layer, producing a dynamical architecture whose phase organisation remains confined within the lattice potential. The resulting architecture admits structural closure through a sequence of theorems establishing phase-space compressibility, bounded attractor formation, well-index invariance under bounded perturbations, and long-cycle stability within the lattice potential. Because the underlying construction derives geometric objects purely from relational bond constraints, the framework provides a structural lens through which classical coordinate geometry, circular parametrisations, and angular measures may be interpreted as emergent consequences of deeper constraint-driven systems rather than as axiomatic premises. Within this perspective the dual-deltoid lattice also functions as a geometric focusing structure capable of resolving focal symmetries and curvature organisation in complex relational networks. The six-point focus configuration behaves as a resolution lattice centred on a primary focus, providing a natural mechanism for structural disambiguation within highly coupled systems. This focusing property suggests interpretive applications for phenomena that require high-precision structural resolution, including lattice-based dynamical systems, attractor-driven architectures, particle-interaction networks, cosmological structure modelling, and curvature-based focusing phenomena such as gravitational lensing. The document is presented as a 77-page technical monograph and is intended for readers with advanced multidisciplinary training in the specific domains required to interpret the model: mathematical physics, variational calculus, differential and computational geometry, geometric topology, graph theory and mesh topology, dynamical systems theory, cosmological structure modelling, gravitational lensing interpretation frameworks, and particle-scale physical modelling. Within these domains the work demonstrates how ordered geometric forms and bounded dynamical architectures can emerge from invariant relational constraints without introducing those forms axiomatically.
Lance Thomas Davidson (Fri,) studied this question.