Results

Results

Sep 13, 2025

Results and Verification

This page separates analytical results from numerical illustrations and performance measurements.

Analytical Results

The current binary radial formulation defines regular chord–tangent sectors and gives exact-arithmetic proofs of second-order error for arithmetic kernels and fourth-order error for the weighted circumference and area kernels. It also proves an area bracket for the defined regular sectors.

These statements have specific geometric hypotheses. They do not establish numerical exactness at finite resolution, hardware stability, originality, or superiority over other methods.

Numerical Checks

Numerical evaluations may check consistency with the analytical results. A publishable record must include the generating procedure, radius, sector count, precision, reference definition, raw output, and executable code.

Performance Results

The former gallery mixed example cards with timing figures and incomplete dataset records. Those figures are not retained as validated performance results. No measured efficiency advantage is established by this page.

Performance results will be added when complete reproducible evidence is available under the comparison protocol.

Methods

Methods

Sep 12, 2025

Methods and Their Scope

Current Binary Radial Kernels

The current formulation defines regular equal sectors, local chord–tangent lengths, triangular areas, and weighted kernels. Its proofs concern those definitions.

Earlier Methods

Algorithm Specifications

A runnable method must identify initial values, refinement or direct evaluation, arithmetic format, outputs, and stopping conditions. A geometry identity alone is not a runtime benchmark.

Extensions

Adaptive sampling, general curves, multiple seeds, and three-dimensional constructions require their own definitions and validation. No error order or performance advantage is transferred automatically from the regular-circle proof.

Overview

Overview

Sep 12, 2025

GRG Overview

Generative Radial Geometry studies finite radial constructions and their measurement kernels. The current two-dimensional reference uses equal binary sectors, internal chord triangles, and external tangent triangles.

Current Reference

Definitions, kernel formulas, bounds, and exact-arithmetic error proofs.

Research History

Original Projection, Cartesian trapezoidal methods, and earlier Turns formulations retain their own definitions and scope. They are not interchangeable specifications.

Evidence

Geometric accuracy, floating-point stability, computational performance, and originality are separate questions. See Results and the benchmark protocol for their status.