A Symmetric GRG Kernel for Area

A Symmetric GRG Kernel for Area

Jun 17, 2026

A Symmetric GRG Kernel for Area

The binary radial formulation combines complementary internal and external sector triangles. The area kernel uses one internal contribution and two external contributions, unlike the circumference kernel’s two-to-one chord–tangent balance.

Geometric Elements

Resolve one Turn into N equal symmetric sectors. Let a be the internal triangle’s height, h half its chord, and e half the external tangent side. Define q = h/a.

$$
N=2^b,\quad b\ge2,\quad R>0,\qquad a^2+h^2=R^2,\quad e=R\frac ha.
$$
$$
I_b=ah=R^2\frac{q_b}{1+q_b^2},\qquad E_b=Re=R^2q_b.
$$

Two Area Kernels

$$
A_m=\frac N2(I_b+E_b),\qquad A_w=\frac N3(I_b+2E_b).
$$

Accuracy for the Defined Triangles

The current formulation gives an analytical verification using a local integral as an exact sector reference, not as a kernel input.

$$
F(q)=\int_0^q\frac{dt}{1+t^2},\qquad A_T(R)=NR^2F(q_b).
$$
$$
\begin{aligned}A_m-A_T&=NR^2\left(-\frac{q_b^3}{6}+\frac{3q_b^5}{10}+O(q_b^7)\right),\\A_w-A_T&=NR^2\left(\frac{2q_b^5}{15}+O(q_b^7)\right).\end{aligned}
$$

Regular geometric bisection yields a scale bound. Together with the expansion, it proves second-order total error for the arithmetic kernel and fourth-order total error for the weighted kernel in exact arithmetic.

$$
0\lt q_b\le\frac4N,\qquad A_m-A_T=O(N^{-2}),\qquad A_w-A_T=O(N^{-4}).
$$

The same analysis proves the following bracket for these sectors:

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

Sampling and Scope

$$
\Delta u=\frac1{4N}.
$$

The earlier version introduced this quarter-step sampling parameter inside each module. The proof here concerns the complete sector triangles defined above. A different sample-based area construction requires its own explicit definition and proof.

GRG length and area kernel inversion A clean schematic comparison between the GRG circumference kernel, based on two internal chord contributions and one external tangent contribution, and the proposed GRG area kernel, based on one internal area contribution and two external area contributions. Length kernel 2 chord + 1 tangent Chord Chord Tangent Area kernel 1 internal + 2 external Internal External External weight inversion
Internal–external weights are reversed between the circumference and area kernels. This diagram is schematic; the triangle definitions determine the calculation.

Research Status

The cancellation is established for the regular triangular elements above. This does not establish original authorship of the weighting, numerical stability of an implementation, or a computational advantage. Those questions require separate literature and implementation comparisons.

Earlier revisions of this page described fourth-order behaviour as a candidate. The current analytical clarification identifies the geometric assumptions under which that behaviour follows.

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

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.

$$
a^2+h^2=R^2,\qquad e=R\frac ha.
$$
Symmetric binary radial sector with chord and tangent side A circle with a symmetric radial sector. The internal chord lies between two points on the circle. The external tangent side lies on the tangent line perpendicular to the central radius. O R tᵦ tangent line external tangent side upper sector boundary lower sector boundary internal chord
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.

Binary refinement of a radial circle Three circles divided into increasing binary sector counts, showing refinement from coarse radial modules to a smoother circular construction. b = 2, N = 4 b = 3, N = 8 b = 4, N = 16
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.

Lens (ERG Volume)

Lens (ERG Volume)

Oct 18, 2025

GRG Lens — Points • Dual Surface • ERG Volume • Smooth (C²)

Abstract. A lens is the envelope of two surfaces (front $S_1$, back $S_2$) with a shared rim. At resolution $\varepsilon$, the local step obeys:

$$ h(u,v) \;\le\; 2\,\sqrt{\frac{\varepsilon}{\kappa_{\max}(u,v)}}, $$

We build the closed solid and compute the ERG volume, then apply Smooth$_\varepsilon$ with $C^{2}$ continuity to avoid optical edges.

1) Definitions (curvatures & adaptive step)

  • Principal curvatures $(\kappa_1,\kappa_2)$; $\kappa_{\max}=\max(|\kappa_1|,|\kappa_2|)$.
  • Local step $h(u,v)$: imposed by $2\sqrt{\varepsilon/\kappa_{\max}}$.
  • Perfect at scale $\varepsilon$: geometric deviation $\le \varepsilon$ on the whole lens.

Axisymmetric case $z=f(r)$

$$ \kappa_{\text{mer}}=\frac{f''(r)}{\bigl(1+f'(r)^{2}\bigr)^{3/2}}, \qquad \kappa_{\text{circ}}=\frac{f'(r)}{r\,\sqrt{1+f'(r)^{2}}}, \qquad \kappa_{\max}=\max\!\bigl(|\kappa_{\text{mer}}|,|\kappa_{\text{circ}}|\bigr). $$

2) Points → Dual surface → ERG volume

2.1 Points (adaptive radial–angular grid)

Sample $S_1$ and $S_2$ with $\Delta r$, $\Delta \theta$ such that local geodesic spacing never exceeds $h(u,v)$. Density increases where $\kappa_{\max}$ is higher (small radii or high-curvature regions).

2.2 Surfaces and closure

Build two coherent meshes $S_1, S_2$ and close with a thin lateral ring (cylindrical rim) to obtain a closed, oriented solid.

2.3 ERG volume

Tetrahedra method (as for the sphere) on the closed mesh:

$$ V_{\mathrm{ERG}}=\frac{1}{6}\sum_{\triangle ijk}\bigl((p_i\times p_j)\cdot p_k\bigr). $$

If axisymmetric, cylindrical shells give a consistent Riemann sum:

$$ V_{\mathrm{ERG}}\;\approx\; 2\pi\sum_{k}\; t(r_k)\, r_k\,\Delta r, \qquad t(r)=z_1(r)-z_2(r), $$

which matches the tetrahedra estimate with $O(h^{2})$ error when the sampling bound is enforced.

3) Smooth$_{\varepsilon}$ for optics (C² continuity)

Goal: remove micro-faceting and guarantee surface and derivative continuity ($C^{2}$) for stable normals and curvature.

  1. Local MLS/RBF fit (degree $\ge 2$) on a radius $\rho\in[h,2h]$ for each face.
  2. Resample vertices on the smoothed patches; deviation $\le \varepsilon/2$.
  3. Normal filtering along geodesic neighborhoods, bandwidth $\propto \varepsilon$.
  4. Optical check: angular variation of normals consistent with expected faceting and no edges (no $C^{1}$ discontinuities).
  5. Topology check: no self-intersections after smoothing.

4) Algorithm (concise)

input: surfaces S1, S2 (or z1=f1(r), z2=f2(r)), ε
for each face S∈{S1, S2}:
    compute κ_max(u,v)  # or, for revolution, κ_mer(r), κ_circ(r)
    set h(u,v) ← 2*sqrt(ε/κ_max(u,v))
    generate adaptive grid (Δr, Δθ) satisfying h

T ← triangulate(S1) ∪ triangulate(S2) ∪ cylindrical_rim
orient(T)
A ← sum_areas(T)
V_ERG ← tetra_volume(T)

if optical requirements:
    T' ← smooth_MLS_RBF_C2(T) with deviation ≤ ε/2
    check_normals(T')          # continuity & angular variation
    check_self_intersections(T')
output: (closed mesh T', A, V_ERG, normals)

5) Checks & validation

  • Hausdorff (ideal ↔ mesh) $\le \varepsilon$.
  • $C^{2}$ continuity after smoothing (tests on numerical derivatives of patches).
  • Triangle quality and no self-intersections.
  • (Optional) qualitative optical test: no “steps”.

6) Why it matters

Lenses require $C^{2}$ continuity: even tiny edges degrade refraction. GRG adaptive sampling densifies where needed; Smooth$_\varepsilon$ controls deviation. The result is a lens “perfect at the chosen resolution”, with validated ERG volume.

7) Figures (inline SVG)

Meridional section: z1(r), z2(r) and cylindrical rim Two smooth profiles forming a biconvex lens; the rim is closed with a short cylinder to make a watertight solid. z r z₁ z₂ rim
L2 — Meridional section (profiles $z_1$, $z_2$) and rim closure.
Adaptive step h(u,v) from κmax Radial density sketch: smaller step near high curvature regions; circles represent iso-radius samples with varying spacing.
L1 — Density map driven by $\kappa_{\max}$: smaller $h$ where curvature is higher.

8) FAQ

How do I choose $\varepsilon$ for optics?
Start from the required form tolerance (e.g., $\lambda/10$ or microns) and translate it into a geometric deviation limit.

Does smoothing alter dioptric power?
No, if capped at $\varepsilon/2$ and validated with deviation maps and derivative checks.

What about the rim?
Close with a coherent ring: no sharp or inverted edges; enforce correct orientation.

ERG grid icon Concentric arcs with dyadic ticks representing ERG refinement.

This compact tool computes the quarter-sampling step Δu and the grid size K from the dyadic level b, previews the uk values, and exports the grid to CSV.

Inputs

  • Dyadic level b ∈ ℕ≥0 (recommended range: 0–18)
  • Optional: recent estimates V2b-1 and V2b to check the dyadic stopping rule
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
\[ \text{Stop if}\quad \bigl|V_{2^{b}} - V_{2^{b-1}}\bigr| \le \varepsilon. \]

Calculator

Results

  • Δu = — (exact: —)
  • K = — (i.e., number of grid points)
  • N = — (turns-first level, N = 2b)
Preview of u-grid uk = k · Δu

First 8: —

Last 8: —

—

Practical checklist

  • Use dyadic refinement: increase b until the dyadic difference falls below ε.
  • Quarter-sampling is assumed: Δu = 2-(b+2).
  • Grid is deterministic; CSV can be versioned for reproducibility.
ERG volume

ERG volume

Oct 18, 2025

GRG Sphere — Points • Surface • ERG Volume • Smooth

Abstract. A sphere of radius $R$ is generated as a quasi-uniform point set. With sampling step:

$$ h_{\max}(\kappa,\varepsilon)\;=\;2\sqrt{\frac{\varepsilon}{\kappa}}, \qquad \kappa=\frac{1}{R}, $$

the reconstructed surface is perfect at scale $\varepsilon$ (geometric error $\le \varepsilon$); area and ERG volume converge with $O(h^2)$ error.

1) Definitions

  • Resolution $\varepsilon$: admissible indistinguishability threshold.
  • Step $h$: distance between nearby surface samples.
  • Curvature $\kappa$: for a sphere, $\kappa = 1/R$.
  • Perfect at scale $\varepsilon$: Hausdorff distance (ideal ↔ reconstructed) $\le \varepsilon$.

Geometric reminder (sagitta of an arc of radius $R$):

$$ s \;\approx\; \frac{h^{2}}{8R} \;=\; \frac{\kappa h^{2}}{8} $$
$$ \Rightarrow\quad h_{\max} \;=\; 2\sqrt{\frac{\varepsilon}{\kappa}} \;=\; 2\sqrt{\varepsilon R}. $$

2) Points → Surface → ERG volume

2.1 Points (quasi-uniform sampling)

  • Spherical Fibonacci lattice: simple and isotropic.
  • Icosphere (subdivided icosahedron): regular connectivity.

Point count (triangles near-equilateral, area $(\sqrt{3}/2)\,h^2$):

$$ N \;\approx\; \frac{4\pi R^{2}}{(\sqrt{3}/2)\,h^{2}} \;=\; \frac{2\pi}{\sqrt{3}}\,\frac{R}{\varepsilon}. $$

It grows linearly with $R/\varepsilon$.

2.2 Surface (closed, oriented mesh)

Triangulate coherently on the spherical manifold. Face normal: $\mathbf{n}_f \propto (p_j-p_i)\times(p_k-p_i)$.

Discrete area:

$$ A \;=\; \sum_{\triangle ijk} \frac{1}{2}\,\big\|\, (p_j-p_i)\times(p_k-p_i)\, \big\|. $$

2.3 ERG volume (from closed surface)

With an oriented closed mesh (tetrahedra w.r.t. the origin):

$$ V_{\mathrm{ERG}} \;=\; \frac{1}{6}\sum_{\triangle ijk}\big((p_i\times p_j)\cdot p_k\big). $$

Convergence: $A$ and $V_{\mathrm{ERG}}$ approach the ideal values with $O(h^2)$ error.

3) Smooth$_{\varepsilon}$ (when and how)

Goal: stabilize normals and reflections without exceeding $\varepsilon$.

  1. Local Taubin/MLS filter (forward/backward pass to avoid shrinkage).
  2. Projection onto the sphere ($\lVert v\rVert \to R$).
  3. Check: Hausdorff (pre/post) $\le \varepsilon/2$.
  4. Smoothed normals from local MLS fit ($\ge C^{1}$ continuity).

When it’s not needed: if you only measure area/ERG volume and faceting is already below the $\varepsilon$-scale threshold.

4) Algorithm (concise)

input: R, ε
κ ← 1/R
h_max ← 2*sqrt(ε/κ)       # = 2*sqrt(εR)
N    ← round((2π/√3) * R/ε)

P ← fibonacci_sphere(N)    # or icosphere(h_max)
T ← triangulate_on_sphere(P)
orient(T)                  # outward normals

A ← sum_areas(T)
V_ERG ← tetra_volume(T)

if need_smooth_normals:
    P' ← taubin_mls(P, T)  # controlled smoothing
    P' ← project_to_sphere(P', R)
    assert hausdorff(P, P') ≤ ε/2
    recompute_normals(P', T)
output: (P, T, A, V_ERG, normals)

5) Checks & validation

  • Hausdorff distance (ideal ↔ mesh) $\le \varepsilon$.
  • Triangle quality (avoid extreme angles).
  • No self-intersections.
  • $O(h^{2})$ convergence for area and ERG volume.

6) Why it matters

The sphere is an ideal testbed: constant curvature and closed-form $A,V$. The chain points → surface → ERG volume → smooth at scale $\varepsilon$ sets the general GRG/ERG procedure for solids.

7) Figures (inline SVG)

Sagitta geometry: s ≈ h^2/(8R) Arc of radius R, chord of length h, sagitta s; all lengths in the same units. R O s | ⟂ h
F3 — Sagitta relation used to bound $h_{\max}$.
Quasi-uniform points on a sphere Wireframe sphere with dots to suggest Fibonacci vs icosphere distributions.
F1 — Point distribution (Fibonacci vs icosphere idea) at equal $h$.

8) FAQ

Is “perfect” absolute?
No: it is perfect at scale $\varepsilon$. Details $<\,\varepsilon$ are not guaranteed.

Do we always need smoothing?
No: needed for normals/optics stability; otherwise optional.

Fibonacci or icosphere?
Both work: Fibonacci is simple and isotropic; icosphere offers regular connectivity.

How do I choose $\varepsilon$?
Based on use: measurement accuracy, rendering, fabrication, etc.

ERG grid icon Concentric arcs with dyadic ticks representing ERG refinement.

This compact tool computes the quarter-sampling step Δu and the grid size K from the dyadic level b, previews the uk values, and exports the grid to CSV.

Inputs

  • Dyadic level b ∈ ℕ≥0 (recommended range: 0–18)
  • Optional: recent estimates V2b-1 and V2b to check the dyadic stopping rule
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
\[ \text{Stop if}\quad \bigl|V_{2^{b}} - V_{2^{b-1}}\bigr| \le \varepsilon. \]

Calculator

Results

  • Δu = — (exact: —)
  • K = — (i.e., number of grid points)
  • N = — (turns-first level, N = 2b)
Preview of u-grid uk = k · Δu

First 8: —

Last 8: —

—

Practical checklist

  • Use dyadic refinement: increase b until the dyadic difference falls below ε.
  • Quarter-sampling is assumed: Δu = 2-(b+2).
  • Grid is deterministic; CSV can be versioned for reproducibility.
Complexity

Complexity

Sep 12, 2025

Computational Complexity

Complexity must identify the task, algorithm, arithmetic precision, and output representation. This page distinguishes costs rather than assigning a single complexity to all GRG methods.

Scalar Kernel Evaluation

Given local geometric ratios, the current circle kernels require a fixed number of arithmetic operations. Obtaining the ratios has its own cost. A binary-bisection evaluation performs a number of refinement stages proportional to the binary depth; square-root cost depends on precision and implementation.

Explicit Geometry

Generating and storing all sector nodes has a cost proportional to the output count. Streaming may reduce storage, but does not remove output work.

Raster and Volume Tasks

Boundary drawing, disk filling, surface generation, and voxel filling have different output sizes. Their cost statements must specify whether every output element is generated.

Comparisons

Report setup, precomputation, precision, memory, and runtime at equal achieved accuracy under the benchmark protocol. Earlier repeated cost models have been consolidated here.

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.