A New GRG Formulation: Circle and Area from Binary Radial Modules

Jun 16, 2026

A GRG Formulation: Circle and Area from a Binary Radial Framework

Generative Radial Geometry describes circular measures through a finite binary radial framework. Its working quantities are a radius, a full Turn, a binary resolution, and local chord–tangent constructions.

About the Current GRG Formulation

This page presents the documented binary radial formulation, with geometric definitions and an explicit verification of its measurement kernels. Earlier approaches, including Original Projection and previous Turns formulations, remain part of the research history.

The geometric construction, kernel accuracy, and computational efficiency are separate subjects. The analysis below establishes accuracy for regular symmetric sectors. It does not establish a performance advantage or prescribe an implementation.

The weighted area proposal is documented in A Symmetric GRG Kernel for Area. The geometric ratio notation and proofs below are an analytical clarification of the documented triangles and kernels.

Binary Turns and Resolution

A full rotation is one Turn. Resolve it into equal sectors. The nondegenerate chord–tangent formulas on this page require at least four sectors.

$$
N=2^b,\qquad b\ge2,\qquad R>0.
$$

$$
t_b=\frac1N,\qquad \text{half-sector width}=\frac1{2N}.
$$

One Symmetric Radial Sector

Let a be the distance from the centre to the internal chord, h half that chord, and e half the external tangent side. The tangent is perpendicular to the central radius. The two boundary rays intersect both the circle and this tangent.

$$
a^2+h^2=R^2,\qquad e=R\frac ha.
$$

Figure 1 — Symmetric sector: internal chord, central radius, and external tangent side.

Define dimensionless ratios from these constructed lengths:

$$
c_b=\frac aR,\qquad s_b=\frac hR,\qquad q_b=\frac ha.
$$

$$
c_b^2+s_b^2=1,\qquad q_b=\frac{s_b}{c_b},\qquad c_b=\frac1{\sqrt{1+q_b^2}}.
$$

These symbols denote geometric ratios. They do not specify a trigonometric software routine.

Circumference Kernels

$$
L_i=2h,\qquad L_t=2e.
$$

The arithmetic balance and the classical Snell–Huygens weighted balance give:

$$
\begin{aligned}C_m&=\frac N2(L_i+L_t)=NR(s_b+q_b),\\C_{SH}&=\frac N3(2L_i+L_t)=\frac{2NR}{3}(2s_b+q_b).\end{aligned}
$$

The weighting is a known local arc approximation. This formulation uses it as a kernel within the binary radial framework; the weighting itself is not claimed as an original GRG result.

Area Elements and Kernels

The internal triangle has base 2h and height a. The external triangle has base 2e and height R.

$$
\begin{aligned}I_b&=ah=R^2c_bs_b=R^2\frac{q_b}{1+q_b^2},\\E_b&=Re=R^2q_b.\end{aligned}
$$

$$
A_{\mathrm{in}}=NI_b,\qquad A_{\mathrm{out}}=NE_b.
$$

The arithmetic area estimate and the proposed internal–external weighting are:

$$
\begin{aligned}A_m&=\frac N2(I_b+E_b),\\A_w&=\frac N3(I_b+2E_b).\end{aligned}
$$

$$
A_w=\frac{NR^2}{3}\left(\frac{q_b}{1+q_b^2}+2q_b\right).
$$

Geometric Refinement Bound

The following identity is used to verify regular equal-sector bisection. It is a geometric consequence, not an attribution of a particular numerical algorithm to GRG.

At four sectors the half-sector triangle has equal legs, so q₂ = 1. Bisecting its direction gives the following ratio, obtained by normalising the boundary direction and adding it to the central unit direction:

$$
q_{b+1}=\frac{q_b}{1+\sqrt{1+q_b^2}}.
$$

The denominator is at least two. Induction therefore gives:

$$
0\lt q_b\le 2^{2-b}=\frac4N.
$$

This proves the scale bound needed for the global error estimates. It uses the regular bisection geometry; it does not require a precomputed full-Turn length or area.

Kernel Accuracy: Analytical Verification

Define the local integral below solely as an analytical reference for the exact sector measures. It is not an input to the measurement kernels.

$$
F(q)=\int_0^q\frac{dt}{1+t^2}.
$$

Parameterising the circular boundary by its slope gives exact sector arc length 2RF(q) and exact sector area R²F(q). Thus the fixed full-Turn reference measures satisfy:

$$
C_T(R)=2NRF(q_b),\qquad A_T(R)=NR^2F(q_b).
$$

$$
F(q)=q-\frac{q^3}{3}+\frac{q^5}{5}+O(q^7).
$$

Substitution into the circumference kernels yields:

$$
\begin{aligned}C_m-C_T&=NR\left(\frac{q_b^3}{6}-\frac{q_b^5}{40}+O(q_b^7)\right),\\C_{SH}-C_T&=NR\left(\frac{q_b^5}{10}+O(q_b^7)\right).\end{aligned}
$$

For area:

$$
\begin{aligned}A_m-A_T&=NR^2\left(-\frac{q_b^3}{6}+\frac{3q_b^5}{10}+O(q_b^7)\right),\\A_w-A_T&=NR^2\left(\frac{2q_b^5}{15}+O(q_b^7)\right).\end{aligned}
$$

The area weighting cancels the cubic local residual. Combining these expansions with the geometric scale bound proves, for fixed radius and exact arithmetic:

$$
\begin{aligned}C_m-C_T&=O(N^{-2}),& C_{SH}-C_T&=O(N^{-4}),\\A_m-A_T&=O(N^{-2}),& A_w-A_T&=O(N^{-4}).\end{aligned}
$$

These are asymptotic error statements, not claims of exact equality at finite resolution or guarantees about floating-point performance.

Area Bounds

The geometric inner and outer triangles bound each exact sector. The two balanced area estimates provide a tighter bracket as well:

$$
A_{\mathrm{in}}\le A_m\le A_T\le A_w\le A_{\mathrm{out}}.
$$

For a direct verification, write the normalised kernels as M(q) and W(q). Both differences vanish at zero, and:

$$
\begin{aligned}\frac{d}{dq}\bigl(F(q)-M(q)\bigr)&=\frac{q^2(1-q^2)}{2(1+q^2)^2},\\\frac{d}{dq}\bigl(W(q)-F(q)\bigr)&=\frac{2q^4}{3(1+q^2)^2}.\end{aligned}
$$

Both derivatives are nonnegative for 0 ≤ q ≤ 1, which covers the sectors considered here. Consequently the bracket follows for every b ≥ 2, rather than only for tested resolutions.

Refinement and Quarter-Step Sampling

$$
N_{b+1}=2N_b,\qquad t_{b+1}=\frac{t_b}{2},\qquad \Delta u=\frac{t_b}{4}=\frac1{4N}.
$$

The related area article specifies quarter-step sampling within a module. The verification above concerns the complete sector triangles. An implementation using internal samples must specify how those samples evaluate the same elements; the sampling parameter alone does not define an additional area kernel.

Figure 2 — Binary sector refinement at N = 4, 8, and 16.

Computational Efficiency and Research Scope

Binary resolution is compatible with binary angular indices and power-of-two scaling. A performance advantage for the complete calculation must nevertheless be measured.

A reproducible comparison must use equal accuracy targets, include generation or retrieval of the local measures, account for precomputation and memory, and specify arithmetic precision, rounding, hardware, and execution time. Stability in fixed-point arithmetic requires a separate error analysis.

The results established here concern the documented regular chord–tangent geometry and its kernels. Historical originality and implementation efficiency are separate investigations.

Constructive Figures

Jan 7, 2026

Constructive Figures in GRG: Dyadic Quarter, Mirrored Circle, and Sector Area

This page collects three minimal, constructive drawings used throughout GRG: a dyadic quarter fan, a fully mirrored circle built from four quarters, and a sector-based area accumulator. All figures are π-free inside the loop; conversions happen only at setup or I/O.

1) Dyadic quarter fan (turns-first sweep)

Angle advances in turns. Refinement uses powers of two so that a quarter turn has N = 2^b rays with a step \Delta u = 2^{-b-2}. This aligns with binary hardware.

$$ N = 2^{\,b}, \qquad \Delta u = 2^{-\,b-2} \quad \text{(quarter sampling)} $$

Refinement level b: b = 9 → N = 512 rays per full turn (128 per quarter)

Quarter sweep with N = 2^b rays. Mirror to complete the circle.

2) Full circle via mirrored quarters (total symmetry)

Build one quarter, then mirror/rotate it three times to assemble the full circle with exact bilateral and rotational symmetry.

Four identical quarters enforce perfect symmetry and cache-friendly construction.

3) Sector-based area accumulation (π-free loop; classical limit)

Area is the sum of radial sectors. The loop aggregates in turn units; the classical value appears in the limit.

$$ \Delta A = \tfrac12 r^2\,\Delta\theta \qquad \text{with}\quad \Delta\theta = 2\pi\,\Delta u $$ $$ A \;=\; \sum \Delta A \;\xrightarrow[\Delta u \to 0]{}\; \pi r^2 $$

Sum of ½ r² Δθ over dyadic steps converges to the classical area.

Notes on determinism and implementation

π outside the loop: the runtime loop uses recurrences; π appears only in calibration or reporting.
Bit-shift counts: N = 1 << b replaces divisions with a native shift.
Mirrored quarters: memory and symmetry benefits; identical update sequences aid reproducibility.

Measuring Shape

Oct 12, 2025

Generative Radial Geometry (GRG): measuring shape to change perspective

This page presents concise formulas and proofs showing how GRG replaces constants like $\pi$ with shape measures (circles, hemispheres, ellipses, polygons) and with unit-sum quadratures. The goal is twofold: measurement precision and operational advantage (stability, variance/error reduction, traceability), even in asymmetric scenarios.

1) Principle

In GRG, many occurrences of $2\pi$, $4\pi$ or $\pi$ are not “mysteries” but the area/volume of a shape in the natural domain (e.g., directions, frequencies). By changing the measure on that domain (circular units: turn, hemi, sphere) the constant is absorbed and the integral becomes a normalized average.

2) Core theorems

Theorem 1 — Circular measure equivalence

Let $S^1$ be the unit circle of directions with arc-length measure $\lambda$ so that $\lambda(S^1)=2\pi$. Define the normalized measure $\mu=\lambda/(2\pi)$ with $\mu(S^1)=1$. Then for all $g\in L^1(S^1)$:

$$ \int_{S^1} g \, d\lambda \;=\; 2\pi \int_{S^1} g \, d\mu . $$

Proof. By definition of $\mu$. For $S^2$ the same holds with $\lambda(S^2)=4\pi$ and for the hemisphere with $2\pi$.

Theorem 2 — Anisotropic measure (no symmetry required)

Let $\rho(\theta)\ge 0$ with $\int \rho(\theta)\,d\theta = 1$ be an angular density (possibly non-uniform). Define the weighted measure $d\nu(\theta):=\rho(\theta)\,d\theta$. For all $g\in L^1$:

$$ \int g(\theta)\,\rho(\theta)\,d\theta \;=\; \mathbb{E}_{\theta\sim \rho}\!\big[g(\theta)\big]. $$

Interpretation. The “shape” of the domain is deformed according to $\rho$. Precision comes from encoding asymmetry in the measure, not from any pre-existing symmetry.

Theorem 3 — Radial quadratures with unit-sum weights

There exist sets of directions $\{\hat u_k\}_{k=1}^{N}\subset S^2$ and weights $w_k>0$ with $\sum_k w_k = 1$ such that, for every $g$ in a target family (e.g., polynomials in $\mu=\cos\theta$ up to degree $t$, or spherical harmonics up to degree $t$):

$$ \int_{S^2} g(\hat u)\, d\Omega \;=\; \sum_{k=1}^{N} w_k\, g(\hat u_k). $$

Note. Such cubatures (e.g., spherical t-designs or refined symmetric polyhedra) give controllable accuracy with $N=O(t^2)$.

3) Operational propositions: “removing $\pi$”

Proposition A — Fourier pair in turns (Hz), no $2\pi$

With frequency in cycles/s (Hz) and phase in turns, the Fourier pair

$$ X(f)=\int_{-\infty}^{\infty} x(t)\,e^{-i\,2 f t}\,dt , $$

$$ x(t)=\int_{-\infty}^{\infty} X(f)\,e^{+i\,2 f t}\,df . $$

is equivalent to the classical pair with $\omega=2\pi f$. Proof. Substitute $\omega=2\pi f$ and $d\omega=2\pi\,df$.

Proposition B — RC integral: rad/s vs Hz

$$ \int_{0}^{\infty} \frac{d\omega}{1+(\omega\tau)^2} \;=\; \frac{\pi}{2\,\tau} $$

$$ \Longleftrightarrow\quad \int_{0}^{\infty} \frac{df}{1+(2\pi f \tau)^2} \;=\; \frac{1}{4\,\tau} . $$

Proof. Use $\omega=2\pi f$. The constant $\pi$ is a unit-conversion factor.

4) Demonstrative examples (non–“pure geometry”, explicit shapes)

4.1 Radiometry (Lambert): the number is the area of a disk

For a perfectly diffuse surface,

$$ E \;=\; L \int_{\text{hemisphere}} \cos\theta \, d\Omega . $$

In spherical coordinates $d\Omega=\sin\theta\, d\theta\, d\varphi$, $\varphi\in[0,2\pi)$, $\theta\in[0,\tfrac{\pi}{2}]$. Then

$$ \int_{\text{hemisphere}} \cos\theta \, d\Omega \;=\; \int_0^{2\pi}\! d\varphi \int_0^{\pi/2}\! \cos\theta \sin\theta\, d\theta \;=\; 2\pi \cdot \tfrac{1}{2} \;=\; \pi . $$

Hence $E=\pi L$. With the “hemi” measure (hemisphere $=1$) the average is $\langle \cos\theta\rangle_{\text{hemi}} = \tfrac{1}{2}$ and, expressed per-hemi, $E_{\text{hemi}} = L \cdot \tfrac{1}{2}$. With the projected “Lambert” measure $d\Sigma = \tfrac{\cos\theta}{\pi}\, d\Omega$ (unit disk area), $\int d\Sigma=1$ and therefore $E = L$. In all cases the constant is a shape measure.

4.2 Signals/Control (RC): angle as shape

With the substitution $\omega\tau = \tan\theta$, the integral

$$ \int_0^\infty \frac{d\omega}{1+(\omega\tau)^2} $$

becomes a measure of an angle on the unit circle (arc from $0$ to $\tfrac{\pi}{2}$), which explains $\tfrac{\pi}{2}$. In “turn/Hz” units that angle is $\tfrac{1}{4}$, giving $\tfrac{1}{4\tau}$. Same content, different shape and measure.

4.3 Statistics (Cauchy as a projection of the circle)

If $\Theta \sim \operatorname{Unif}(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ and $X=\gamma \tan\Theta$, then

$$ f_X(x) \;=\; \frac{1}{\pi \gamma}\,\frac{1}{1+(x/\gamma)^2} . $$

Interpretation: $X$ is the tangent projection of a point uniformly distributed on a circle. In “turn” units, let $\Phi \sim \operatorname{Unif}(-\tfrac{1}{4},\tfrac{1}{4})$ and $\Theta = 2\pi \Phi$; with respect to the turn-normalized angular measure (total length $=1$) the normalization constant is absorbed. Converting back to radians restores $1/(\pi\gamma)$.

5) Asymmetry: elliptical metrics and weighted measures

For anisotropic phenomena introduce an elliptical metric $M\succ 0$ and define $r_M(x)=\sqrt{x^\top M x}$. The “balls” of $r_M$ are ellipses, and many constants are absorbed by switching to $M$-radial coordinates. Any directional density $\rho(\theta)$ defines $d\nu=\rho\,d\theta$: integration becomes a unit-sum average without invoking symmetry.

6) Precise discrete re-determination (quadratures)

Design sets $\{(\hat u_k,w_k)\}$ with $\sum_k w_k=1$ that integrate exactly a useful family (e.g., $\{1,\cos\theta,\sin\theta,\cos2\theta,\sin2\theta\}$ for low-order anisotropic BRDFs, or spherical harmonics up to degree $t$). This yields a precise discrete scheme: constants like “$4\pi$” or “$2\pi$” become simply “sum of weights = 1”.

7) Practical conversions (GRG ↔ classical form)

Domain	Classical form	GRG form	Observation
Radiometry (Lambert)	$E=\pi L$	$E=L$ with unit projected measure	$\pi$ = area of the projected disk
Signals (RC)	$\displaystyle \int \frac{d\omega}{1+(\omega\tau)^2}=\frac{\pi}{2\tau}$	$\displaystyle \int \frac{df}{1+(2\pi f\tau)^2}=\frac{1}{4\tau}$	$\pi$ = rad↔turn conversion
Statistics (Cauchy)	$\frac{1}{\pi\gamma}\frac{1}{1+(x/\gamma)^2}$	$\frac{1}{\gamma}\frac{1}{1+(x/\gamma)^2}$ with angle in turns	$\pi$ = length of the angular domain
Fields/Fluxes	$4\pi$ factors on $S^2$	Average over sphere with $\sum w_k=1$	$4\pi$ = area of the unit sphere

8) Measurable benefits

Numerical stability: fewer fixed large/small factors, better conditioning.
Variance reduction: with anisotropic $\rho$, MC error shrinks where it matters.
Traceability: results come pre-normalized (averages on 1); conversion back to standards is multiplication by $2\pi$ or $4\pi$.

9) Validation protocol (A/B)

Select 3 cases (Lambert, RC, Cauchy/Maxwell).
Define the GRG measure (turn/hemi/sphere or weighted $\rho$) and the quadratures (degree $t$).
Compare to the classical baseline at equal budget (samples/time).
Metrics: relative error, variance, conservation (energy/flux), stability along pipelines.
Success threshold: ≥50% lower cost at equal error or ≤½ the error at equal cost in ≥2 out of 3 cases.

10) Conclusion

“Removing $\pi$” is not rhetoric: it is the outcome of measuring the domain in the right shape. GRG provides a measurement language (circular units and unit-sum quadratures) that yields cleaner formulas and, crucially, quantifiable benefits. This is a change of perspective that delivers progress: measurable improvement, not stubbornness.

Wedge Visualization (Interactive)

Sep 21, 2025

GRG — Wedge Visualization: Interactive Uniform vs Adaptive Discretization

This mini-demo provides concrete, reproducible numbers for wedge-stratified GRG. Adjust the controls to compare a uniform wedge count against an adaptive profile on the outer annulus, and see how certified corridor half-widths contract roughly like 1/\sqrt{W_k} for comparable ring budgets.

How the Demo Works (plain language)

Dyadic edges. Rings follow turns-first dyadic refinement with b∈ℕ≥0, N=2^b, and quarter sampling Δu=2^{-b-2}.
Corridor half-width. Each sector’s estimate has a certified interval [E^- , E^+]. The half-width is (E^+−E^-)/2; decisions stop when this width ≤ tolerance τ.
Field model. We synthesize an anisotropic field with mixed angular orders up to fourth order to stress the wedges; values are computed in-page so the numbers are concrete and repeatable.

$$ F(\rho,\theta)=A(\rho)\Big[1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\Big], $$

$$ \textbf{Stop when}\quad \widehat{E}_b^{+}-\widehat{E}_b^{-}\le \tau,\qquad \text{with}\ \ \Delta u=2^{-b-2}. $$

Controls

Rings K: Uniform W: Adaptive outer W_k: Tolerance τ:

Results — Per-Sector Estimates on the Outer Annulus

Hover or tap a bar to read exact values (estimate and corridor half-width). The polar plots show the directional pattern and corridor bands.

Uniform W — outer annulus sectors (bars = estimate, whiskers = corridor half-width).

Adaptive outer W_k — tighter corridors where anisotropy is richer.

Polar — Uniform W.

Polar — Adaptive W_k.

Concrete Numbers (auto-computed)

The table aggregates sector values (outer annulus) for the current settings. Replace later with CSVs from your scripts; the structure is the same.

Notes & Practical Guidance

When to prefer GRG. As angular richness grows or is localized, adaptive W_k exposes structure with linear O(K·W) cost and certified decisions via τ.
When low-order bases suffice. Near-radial scenes with mild 2-fold variation are well captured by a short angular basis; choose GRG when auditability (corridors) matters.
Next steps. Publish side-by-side benchmarks (Zernike, B-splines, Fourier–Bessel) with the same corridor-aware stopping rule and public scripts/datasets.

GRG — Wedge Visualization

Sep 21, 2025

Wedge-Stratified GRG — Visualizing Per-Sector Results

This section visualizes the anisotropic, wedge-stratified example discussed earlier. We compare a uniform W=8 discretization against an adaptive scheme on the outer annulus \rho\in[0.7,1), where angular content is richer. The plots show per-sector estimates with certified corridor half-widths.

Setup Recap

Field with mixed angular orders up to fourth order:

$$ F(\rho,\theta)=A(\rho)\,\Big[1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\Big], $$

Rings and wedges:

$$ \text{Rings }k=1..K,\ \text{wedges }w=1..W_k,\quad I_{k,w}=\int_{\Theta_w}\!\int_{u_{k-1}}^{u_k} F(\rho,\theta)\,\rho\,d\rho\,d\theta. $$

Rings: K=32 (dyadic edges, quarter sampling).
Uniform: W=8 for all rings.
Adaptive: inner W_k=8, mid W_k=12, outer W_k=16 (used for plots below).

Per-Sector Results on the Outer Annulus

The numbers are the same illustrative values used in the write-up; replace them with your measured outputs when available. Error bars display corridor half-widths.

Left: uniform W=8. Right: adaptive outer ring with W_k=16. Hover/tap a bar to read exact values.

Polar View (Directional Pattern)

A simple polar plot places each sector’s estimate radially. Shaded rings visualize the corridor range.

Polar plot of per-sector estimates. Left: W=8. Right: W_k=16.

Reading the Plots

Corridor effect: with higher angular resolution, corridor half-widths shrink (≈ 1/\sqrt{W_k} for comparable ring budgets), revealing higher-order structure.
Decision making: choose the smallest dyadic level b such that the corridor width is below the task tolerance \tau.

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

GRG — Residual Critiques

Sep 20, 2025

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 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 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 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 w	Estimate	Corridor ±
1	0.124	±0.0060
2	0.131	±0.0063
3	0.137	±0.0065
4	0.141	±0.0066
5	0.139	±0.0066
6	0.133	±0.0064
7	0.127	±0.0061
8	0.123	±0.0060

W = 16

Sector w	Estimate	Corridor ±
1	0.062	±0.0035
2	0.066	±0.0036
3	0.070	±0.0037
4	0.072	±0.0038
…	…	…
16	0.061	±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

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.