GRG — Residual Critiques

Sep 20, 2025

Scientific correction — 13 October 2026

The application intervals and charts on this historical page are proposals or illustrations, not certified bounds for arbitrary fields or validated performance results. Certification requires assumptions on each integrand, a proved quadrature residual, rounding control, and propagation through subsequent transformations. Regular circular-sector bounds do not automatically bound optical, chemical, or anisotropic quantities.

Current reference: Circle and Area from Binary Radial Modules.

GRG — Addressing Residual Critiques: Demo Access, Benchmarks, and a Concrete Wedge-Stratified Example

This page responds to the latest external feedback by: (i) exposing how the public Mini-App demo and screencast will be surfaced, (ii) detailing the publication plan for side-by-side benchmarks, and (iii) adding a fully worked anisotropic, wedge-stratified example with concrete W, K, per-sector corridor bounds, and a clear complexity model.

1) Mini-App Demo & Screencast — Access Plan

  • Public demo URL: will be placed in the site header under Community → Mini-App and reiterated in the hero of all demo-related posts (single click from the homepage).
  • Screencast: a short walkthrough (≈2–3 min) showing corridor overlays, the b slider, CSV/PNG export, and a corridor-based stopping rule.
  • Repro artifacts: each preset (Optics/MTF, Chemistry/RDF) includes a “Download CSV” button containing the estimate and its proposed (not independently certified) corridor.

Status marker: until the public URL is visible, the hero of this page shows a “Demo forthcoming” badge so the absence is obvious—not hidden.

2) Benchmarks — What Will Be Published

We will release side-by-side comparisons versus Zernike, polar B-splines, Fourier–Bessel, and pixel discretizations. Each figure carries proposed (not independently certified) corridors so decisions are auditable.

$$ \text{For each task } \mathcal{T} \in \{\text{PSF/MTF fit},\ \text{surface recon.},\ \text{denoise},\ \text{RDF}\}: \quad \begin{aligned} &\text{Report}\ \ \|T_{\text{GRG}} - T_{\text{ref}}\|_2,\ \text{runtime},\ \text{memory} \\ &\text{as functions of}\ b\ (\text{dyadic level}),\ N\ (=2^b),\ \text{and problem size.} \end{aligned} $$
  • Code & data: single-command scripts, dataset DOIs, fixed seeds, CSV outputs for independent plotting.
  • Decision rule: declare a winner only when corridor half-width < task tolerance \tau and confidence intervals do not overlap.

3) Concrete Anisotropic Example — Wedge-Stratified GRG

Scenario. A weakly freeform pupil with sagittal/meridional asymmetry (e.g., elliptical sag). We integrate a radial-angular field F(\rho,\theta) over dyadic rings and angular wedges, with proposed (not independently certified) corridors per sector.

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

Setup

  • K=32 dyadic rings (b=5 for ring edges; quarter sampling uses \Delta u=2^{-b-2}).
  • Two angular granularities: W=8 (coarse) and W=16 (refined).
  • Field: F(\rho,\theta)=A(\rho)\,\bigl(1+\epsilon\cos 2(\theta-\theta_0)\bigr) with A(\rho)=\rho^{0.6}, \epsilon=0.15, \theta_0=20^\circ.

Per-sector corridors (illustrative values)

We report the aggregate over outer half of the pupil \rho\in[0.5,1). Numbers show the estimate and the symmetric corridor half-width.

W = 8

Sector wEstimateCorridor ±
10.124±0.0060
20.131±0.0063
30.137±0.0065
40.141±0.0066
50.139±0.0066
60.133±0.0064
70.127±0.0061
80.123±0.0060

W = 16

Sector wEstimateCorridor ±
10.062±0.0035
20.066±0.0036
30.070±0.0037
40.072±0.0038
………
160.061±0.0035

Observation. Doubling W tightens per-sector corridors roughly as 1/\sqrt{W} (for comparable sampling budgets), while revealing the expected \cos 2(\theta-\theta_0) anisotropy.

Complexity & Scaling

$$ \text{Work} \ \sim\ O(K\cdot W) \quad\text{and}\quad \text{Memory} \ \sim\ O(K\cdot W) $$

With K=32 and W=\{8,16\}, the refined case doubles angular bins, hence ~2× arithmetic and storage. In practice, ring accumulation is cache-friendly; wall-clock growth is close to linear until compute becomes memory-bound. Corridor contraction with b remains unchanged; W mainly dictates angular resolution.

Visualization — Wedge-Stratified Layout

Wedge-stratified rings with per-sector corridors Annulus split into K rings and W wedges; thicker arcs indicate corridor aggregation on the outer half.
Per-sector certification on the outer half of the pupil; increase W where angular gradients are steep.

4) Practical Stopping with Corridors

Independently of W, choose the smallest dyadic level b such that corridor half-width is below the tolerance relevant to the task.

$$ \textbf{Stop when}\quad \widehat{E}^{+}_{b} - \widehat{E}^{-}_{b} \le \tau. $$

What This Page Adds

  • Clear placement and visibility of the public Mini-App and screencast.
  • A concrete wedge-stratified example (values for W, K, per-sector corridors).
  • Simple complexity model O(K\cdot W) with practical remarks on scaling.
  • Benchmark publication plan with auditable corridor-aware comparisons.