GRG — Unified Update: Examples, Validation, Mini-App, and Anisotropic Extension

This unified page consolidates the latest GRG status: cross-domain numerical examples (optics and chemistry), an external validation plan against standard bases (e.g., Zernike), public Mini-App notes, and a practical recipe for anisotropic, non-radial systems via wedge-stratified GRG. All formulas use turns-first dyadic refinement with proposed (not independently certified) corridors for auditable error control.

Key Takeaways

π-free numerics. GRG avoids \pi in accumulation, without establishing an improvement in numerical precision or performance.
proposed (not independently certified) corridors. The proposed workflow seeks upper/lower bounds; validity must be established for each integrand; stopping rules are explicit.
Interdisciplinary reach. Geometry, optics, chemistry, and beyond—all under a single corridor framework.
Open path to validation. Public scripts, datasets, and side-by-side benchmarks with standard methods.

Numerical Example A — Optics (MTF at a Target Frequency)

Goal. Maximize on-axis contrast at target spatial frequency using corridor-aware sampling. We set unit pupil and dyadic refinement with b \in \mathbb{N}_{\ge 0}, N = 2^b, and \Delta u = 2^{-b-2}.

$$ \mathrm{MTF}(f) \;=\; \frac{1}{\pi}\! \int_{0}^{2\pi}\!\!\int_{0}^{1} P(\rho,\theta)\,\cos\!\bigl(2\pi f\,\rho\cos\theta\bigr)\,\rho\,d\rho\,d\theta $$

Illustrative setup. Clear pupil (P=1). Apodizer family A_\alpha(\rho)=\rho^\alpha. Target frequency f^\star is scaled to the unit pupil. For each b, GRG produces an estimate and a proposed (not independently certified) corridor.

MTF at f^\star with GRG corridors (illustrative values).

b	N=2^b	Δu	MTF est.	Corridor ±
4	16	2^{-6}=0.015625	0.412	±0.020
5	32	2^{-7}=0.0078125	0.417	±0.012
6	64	2^{-8}=0.00390625	0.419	±0.007
7	128	2^{-9}=0.001953125	0.420	±0.004

Practical stop. Choose the smallest b such that the corridor half-width is below a task tolerance \tau.

$$ \text{Stop when}\quad \widehat{E}^{+}_b - \widehat{E}^{-}_b \le \tau. $$

Numerical Example B — Chemistry (Radial Distribution Function)

Goal. Estimate a smooth radial distribution in a dense cluster (e.g., ligand shell) using dyadic radii and corridor-aware accumulation. After mild smoothing with a compact kernel, corridors certify stability across small samples.

$$ g(r) \;=\; \frac{1}{4\pi r^2 \Delta r}\,\frac{1}{N} \sum_{i=1}^{N} \#\bigl\{\, j\neq i:\ r \le \|x_j-x_i\| < r+\Delta r \,\bigr\} $$

Radius r	Δu	g(r) estimate	Corridor ±
0.20	2^{-8}	1.38	±0.06
0.40	2^{-8}	0.97	±0.05
0.60	2^{-8}	0.81	±0.04
0.80	2^{-8}	0.76	±0.04

Interpretation. Corridor-aware RDF avoids over-reading small oscillations and makes cross-dataset comparisons fair and reproducible.

Validation Plan — Public, Scriptable, Comparable

We will release side-by-side benchmarks with open scripts and datasets. Figures and tables will include corridors so decisions are auditable.

$$ \text{Compare}\quad \bigl\|\,T_{\text{GRG}} - T_{\text{ref}}\,\bigr\|_{2} \ \ \text{vs.}\ \ b,\; \text{runtime},\; \text{memory}, $$

Tasks: PSF/MTF fitting; surface reconstruction; denoising with corridor stops; radial distributions.
Baselines: Zernike, polar B-splines, Fourier-Bessel, and pixel discretizations.
Artifacts: One-command scripts; dataset DOIs; fixed seeds/parameters; CSV outputs for independent plotting.

Mini-App — Visibility & Usability

The public demo exposes corridor overlays, a b slider, CSV/PNG export, and presets (optics/chemistry). A short screencast walks through an optimization and a corridor-based stopping rule.

Anisotropy — Wedge-Stratified GRG for Non-Radial Systems

Idea. Partition each ring into W angular wedges. Corridors apply per wedge; proposed (not independently certified) aggregation preserves guarantees. Increase W where angular gradients are steep.

$$ \text{For rings }k=1..K,\ \text{wedges }w=1..W:\quad I_{k,w}\;=\;\int_{\theta\in\Theta_w}\!\int_{\rho\in[u_{k-1},u_k)} F(\rho,\theta)\,\rho\,d\rho\,d\theta. $$

Per-wedge certification enables anisotropic handling without losing corridor guarantees.

FAQ — What’s New Here?

Concrete cross-domain demos: optics (MTF) and chemistry (RDF) with corridor-aware tables/plots.
External validation: transparent plan, open artifacts, standard baselines.
Mini-App: public demo with overlays, dyadic control, CSV/PNG export, and a short screencast.
Non-radial systems: wedge-stratified recipe with simple heuristics for W allocation.