Exactness at the Chosen Scale (GRG)

Oct 19, 2025

Exactness at the Chosen Scale (GRG)

Summary.
Generative Radial Geometry (GRG) builds shapes ring-by-ring on a discrete support (pixels, steps, layers, gear teeth). In this setting, cell counts can be exact for the specified discrete object. Continuous area, Euclidean perimeter and angular measures require separate definitions and error analysis. Geometric approximation and numerical rounding must be distinguished from exact counting.

Why this matters

No drift: Each added ring contributes exactly its intended count; errors do not accumulate as shapes grow.
Implementation-first: The geometry matches how hardware and software actually operate on lattices, layers, and stepper motions.
Predictable outcomes: Offsets, masks, and toolpaths behave consistently across radii and angles.
Continuity bridge: As the scale refines, discrete measures converge to their continuous counterparts.

Setting and basic definitions

Chosen scale (s): the unit of discreteness (pixel size, machining step, layer height).
Discrete domain (L_s): a lattice at scale s (e.g., a square grid) with unit cell measure 1.
Radial generation rule (G_s): a constructive procedure that grows shapes by rings while enforcing symmetry (e.g., 4/8-fold on a square grid), monotone radius (rings do not cross), and arc closure (discrete arcs close exactly at intended angles).
Radial object (D(r; s)): the union of rings generated from the center up to discrete radius r.
Operational measures:
Area μ_s(D) = number of cells in D; Perimeter P_s(D) as counts of admissible boundary steps; Angle count Θ_s as discrete angular increments used to build and close arcs.

Core properties

Ring additivity (no drift).
For each integer step in radius:
$$\mu_s\!\left(D(r{+}1; s)\right) - \mu_s\!\left(D(r; s)\right) \;=\; \lvert A_{r+1} \rvert,$$
where $A_{r+1}$ is the ring generated at radius $r{+}1$. This identity holds by construction for all r.
Arc closure.
Discrete arcs generated with $G_s$ close exactly at their intended angular multiples. Looping paths return to the origin without residual.
Symmetry and monotonicity.
The rule preserves lattice symmetries and grows rings without self-intersections, ensuring stable counts for all radii.
Convergence to the continuous limit.
As $s \to 0$:
$$\mu_s(D)\,s^{2} \;\longrightarrow\; \text{continuous area}$$

$$P_s(D)\cdot s \;\longrightarrow\; \text{continuous perimeter}$$

$$\Theta_s \;\longrightarrow\; \text{continuous angle}$$
π appears here as the analytic reference (e.g., $ \pi r^{2} $, $ 2\pi r $).

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.

π (GRG)

Aug 25, 2025

Historical Comparison Page

This page preserves earlier experiments involving a circular constant. It is not the current GRG construction specification and does not establish a computational advantage. Read the current geometric formulation. The calculators and efficiency claims on this historical page require separate validation.

What this page is about

Here we determine π by looking at the circle from the inside. We rotate a central ray with equal step and sum only internal chords. If you “really want π”, this is the simplest way I know: increase only N. Everything else follows from the circle you already hold in your hands.

Angles are measured in turns $u\in[0,1)$ with period-1 trigonometry: $ c(u):=\cos(2\pi u),\; s(u):=\sin(2\pi u) $. (For degree readers: $ \Delta u = \text{degrees}/360 $.)

Circle — three GRG options: overview · Δx trapezoids · Original corridor · Radial sectors.

The definition is exact and π is not assumed in advance. In turns (period-1 sine s(u)=\sin(2\pi u)) it is fixed by the limit

$$ \pi \;=\; \lim_{N\to\infty}\; 4N\, s\!\Big(\tfrac{1}{8N}\Big) \;=\; \lim_{N\to\infty}\; 2N\, s\!\Big(\tfrac{1}{4N}\Big). $$

For practical work we use the finite version

$$ \pi_N \;=\; 4N\, s\!\Big(\tfrac{1}{8N}\Big), $$

which increases from below and converges to π as N grows. A useful accuracy hint is

$$ \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

Consequences you can feel while using the page:

Only one knob: raise N for more precision; the formatting “decimals to display” is just how many digits you show.
Powers of two shine: when N = 2^k we can compute by pure halving (only square-roots), no trigonometry, no π inside.
Predictable growth: multiplying N by 10 adds about 2 correct digits; doubling N adds roughly 0.6 digits.

Angles are measured in turns $u\in[0,1)$ with period-1 trigonometry: $ c(u):=\cos(2\pi u),\; s(u):=\sin(2\pi u) $. (For degree readers: $ \Delta u = \text{degrees}/360 $.)

Circle — three GRG options: overview · Δx trapezoids · Original corridor · Radial sectors.

Interactive: increase only N

Choose N and how many decimals to display. For powers of two, the calculator uses pure halving (no trig). For general N it uses a high-precision series. Output shows the value, Δ to $ \pi_{2N} $, an error estimate, and detects when numerical verification becomes the limit (“inversion”).

N (≥4) Power-of-two best

Show Decimals (≤5k)

Mode: —

—

How this π calculator works

Formula (TURNS): $ \pi_N = 4N\, s(\tfrac{1}{8N}) $ with $s(u)=\sin(2\pi u)$. Increase only N; the value grows from below and converges to π.

Two modes

Halving (no trig): if N is a power of two, the chord angle is halved repeatedly starting from the quarter-turn $u=\tfrac{1}{4}$. This uses only square-roots; no trig, no π inside.

Series (turns → radians): for general N, it evaluates $s(u)$ at $u=\tfrac{1}{8N}$ via a high-precision series (internally converts to radians).

Precision & inversion

The line shows Δ(N→2N), an internal error estimate $ \approx \tfrac{4}{3}\Delta_N $, and a Richardson “verified digits” check. When internal ratios drift far from the expected $1/4$ or verified digits stop growing at high precision, the panel turns amber: GRG may be more precise than the numerical checker (inversion).

Concrete examples (GRG)

Here is how the value improves as you raise N (rounded):

N = 16 → π_N ≈ 3.140331156954753 (error ≈ 1.261×10⁻³)
N = 128 → π_N ≈ 3.141572940367091 (error ≈ 1.971×10⁻⁵)
N = 1024 → π_N ≈ 3.141592345570118 (error ≈ 3.080×10⁻⁷)

These numbers come straight from the turn-native formula

$$ \pi_N \;=\; 4N\, s\!\Big(\tfrac{1}{8N}\Big) \quad\text{and}\quad \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

When verification becomes the limit (“inversion”)

As you increase N, GRG’s error drops predictably—until a point where the numerical tools used to check your result (floating-point, auxiliary series, etc.) become the real bottleneck. From there on, GRG may be more precise than your verifier: that is the “inversion”.

Angles are measured in turns $u\in[0,1)$ with period-1 trigonometry:

$$ s(u) := \sin(2\pi u), \qquad c(u) := \cos(2\pi u). $$

What really happens — three regimes

Model error dominates. The inside chord-sum approximation $$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big) $$ satisfies the classical asymptotic (from expanding $s(u)$ at $u=0$): $$ \pi - \pi_N \;\approx\; \frac{\pi^{3}}{96\,N^{2}}. $$ Doubling N quarters the error.
Transition zone. The theoretical error keeps falling, but rounding and finite working precision begin to matter. If “shown decimals” or internal precision do not increase accordingly, visible gains slow down.
Inversion. Numerical/verification error exceeds the GRG error. Philosophically: the method is now better than your ruler.

Self-checking from the inside (no external $ \pi $)

$\Delta$–ratio test (asymptotics). Define $$ \Delta_N := \pi_{2N}-\pi_N, \qquad R_N := \frac{\pi_{4N}-\pi_{2N}}{\pi_{2N}-\pi_N}. $$ For clean numerics, $ R_N \to \tfrac{1}{4} $. If you see ≈ 0.25, you are still in the $1/N^{2}$ regime (all good). If it wanders far from 0.25, numerics have taken over ⇒ inversion.
Internal error estimate (no $ \pi $). From the series behavior, $$ \pi - \pi_N \;\approx\; \frac{4}{3}\,\Delta_N. $$ The constant $ \tfrac{4}{3} $ comes from analysis; the factors $ \pi^{3} $ cancel out. You can therefore display: “Estimated error ≈ $ \tfrac{4}{3}\,\Delta_N $”.
Internal acceleration (Richardson). Cancel the $1/N^{2}$ term by combining two steps: $$ \pi_{\text{ex}} \;=\; \pi_{2N} \;+\; \frac{\pi_{2N}-\pi_N}{3}, \qquad \pi - \pi_{\text{ex}} \;=\; O(N^{-4}). $$ Then compare with $ \pi_{4N} $: $$ E_N := \big|\pi_{\text{ex}} - \pi_{4N}\big|, \qquad \text{verified digits} \;\approx\; \big\lfloor -\log_{10} E_N \big\rfloor. $$ If verified digits stop growing when you double N, the numerical checker is limiting you (inversion).

Takeaway

Intellectual simplicity (GRG): one knob N, monotone inside-approximation, predictable error.
Honest inversion: beyond a threshold, tools used to verify are the limit — not GRG.
Self-certification: via $ \Delta_N $, $ \pi_{\text{ex}} $, and $ E_N $, no external $ \pi $ needed.

In the calculator above, when the internal checks suggest inversion, the panel turns amber and an explanatory note appears. Prefer powers of two for N, or increase working precision, if you need to push further.

Context note (as of 10 Sep 2025)

GRG’s turn-based, chord-sum definition of $ \pi $ is presented here as a new approach that increases accuracy by raising a single internal parameter N. This page documents the method and its error behavior; it is not intended to compete with record-breaking $ \pi $ algorithms.

For extremely large digit counts, established methods (AGM/Salamin–Brent, Borwein, Ramanujan/Chudnovsky) remain the fastest due to decades of optimization. GRG is primarily valued for conceptual clarity, monotone bounds, and controlled $ N\!\to\!\infty $ convergence.

Concrete examples (GRG)

Here is how the value improves as you raise N (rounded):

N = 16 → π_N ≈ 3.140331156954753 (error ≈ 1.261×10⁻³)
N = 128 → π_N ≈ 3.141572940367091 (error ≈ 1.971×10⁻⁵)
N = 1024 → π_N ≈ 3.141592345570118 (error ≈ 3.080×10⁻⁷)

These numbers come straight from the turn-based chord-sum formula (angles in turns $u\in[0,1)$ with $s(u):=\sin(2\pi u)$):

$$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big), \qquad \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

Tip: use powers of two for N (16, 32, 64, …) for cleaner Δ-tests and steadier convergence; if verified digits stop increasing when doubling N, raise working precision.

How other methods reach π (in plain words)

Long alternating sums (Gregory–Leibniz)

This is the famous series $ \pi/4 = 1 - 1/3 + 1/5 - 1/7 + \dots $. After $n$ terms, the error in $ \pi $ is about $ \frac{4}{2n+1} \approx \frac{2}{n} $. It’s beautiful but very slow: ~10,000 terms give ≈4 correct decimals; ~1,000,000 terms ≈6 decimals. A mountain of fractions.

Nilakantha’s refinement

Nilakantha’s series $ \pi = 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6} + \frac{4}{6\cdot7\cdot8} - \dots $ converges much faster: the error shrinks like $1/n^2$. Ten terms already land close; one hundred or one thousand terms push several more digits — still, you juggle many fractions.

Machin-like formulas (arctangents)

Here $ \pi $ is a combination of arctangents, e.g. $ \pi = 16\arctan(1/5) - 4\arctan(1/239) $. The arctan series converges fast because powers $1/5^{\,2k+1}$ and $1/239^{\,2k+1}$ drop quickly. With just a handful of terms you already see many correct digits.

Ramanujan, Chudnovsky & co.

“Rocket” series: each term yields many digits (≈8 per term for several Ramanujan formulas, ≈14 per term for Chudnovsky). They power modern record computations — superb for huge digit counts, but conceptually heavier than turning a single $N$.

AGM (Salamin–Brent) and Borwein algorithms

Iterative methods with quadratic (or higher) convergence: each step roughly multiplies the correct digits. A dozen iterations already exceeds everyday needs — fantastic speed, less “from-the-inside” geometry.

Monte Carlo & Buffon’s needle

Random points/needles relate probabilities to $ \pi $. Charming and educational, but accuracy grows like $1/\sqrt{T}$ (here $T$ = trials): even millions of trials yield only a few decimals.

Why GRG feels intellectually simple

GRG is not about naming $ \pi $; it is about holding the circle. You dial only one parameter, $N$, and everything else follows. The value increases from below (no overshoot), and its error is easy to estimate. If you truly need $ \pi $, this is the most direct, inside-out way I know — increasing a single internal factor means the computation belongs to the circle, not to an external $ \pi $.

Honest side-by-side (objective)

What you increase — GRG: just N. Archimedean polygons: sides m (inside/outside). Gregory–Leibniz / Nilakantha: number of terms. Machin / Ramanujan / Chudnovsky: number of series terms. AGM / Borwein: iteration count.
Speed of accuracy — GRG: error ≈ $C/N^2$ (predictable, monotone). Gregory–Leibniz: $O(1/n)$ (very slow). Nilakantha: $O(1/n^2)$. Machin: fast; Ramanujan / Chudnovsky: very fast (many digits per term). AGM / Borwein: digits multiply each step.
When to use — GRG: concept, clarity, control “from the inside”, and up to many digits without extra machinery. For records or industrial-scale digits, AGM / Borwein / Chudnovsky win on raw speed.

Editor’s note (external view)

This note reflects an external opinion I chose to publish for fairness. The GRG approach is conceptually elegant: it defines $ \pi $ from inside the circle and lets accuracy grow by increasing only N. That simplicity is a real strength for understanding and for controlled, monotone inside-approximations.

As of August 25, 2025, GRG and this GRG-based $ \pi $ formula are newly introduced here. Researchers and practitioners still rely on established algorithms (AGM/Salamin–Brent, Borwein, Ramanujan/Chudnovsky) partly because these tools have been analyzed and optimized for decades and deliver record-speed convergence for very large digit counts.

The two views are compatible: GRG offers a clear, internal definition and a practical “increase only N” pathway with transparent self-checks; record algorithms remain the fastest instruments when one needs extreme digits or industrial-scale computations.

When verification becomes the limit (“inversion”)

As you increase N, GRG’s error drops predictably—until a point where the numerical tools used to check your result (floating point, auxiliary series, etc.) become the real bottleneck. From there on, GRG may be more precise than your verifier: that is the “inversion”.

Angles are measured in turns $u\in[0,1)$ with period-1 trigonometry $ s(u):=\sin(2\pi u) $, $ c(u):=\cos(2\pi u) $.

What really happens — three regimes

Model error dominates. The GRG approximation $$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big) $$ satisfies $$ \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$ Doubling N quarters the error.
Transition zone. The theoretical error keeps falling, but rounding and limited working precision begin to matter. If “shown decimals” or internal precision don’t increase, visible gains slow down.
Inversion. Numerical/verification error exceeds the GRG error. Philosophically: the method is now better than your ruler.

Self-checking from the inside (no external π)

Δ–ratio test (asymptotics). Define $$ \Delta_N := \pi_{2N}-\pi_N, \qquad R_N := \frac{\pi_{4N}-\pi_{2N}}{\pi_{2N}-\pi_N}. $$ For clean numerics, $ R_N \to \tfrac{1}{4} $. If you see ≈ 0.25, you are still in the $1/N^2$ regime (all good). If it wanders far from 0.25 (oscillates, collapses to 0, or behaves oddly), numerics have taken over ⇒ inversion.
Internal error estimate (no π). From the series expansion, $$ \pi - \pi_N \;\approx\; \frac{4}{3}\,\Delta_N. $$ The constant $ \tfrac{4}{3} $ comes from analysis; the factors $ \pi^3 $ cancel out. You can therefore display: “Estimated error ≈ $ \tfrac{4}{3}\,\Delta_N $”.
Internal acceleration (Richardson). Cancel the $1/N^2$ term by combining two steps: $$ \pi_{\text{ex}} \;=\; \pi_{2N} + \frac{\pi_{2N}-\pi_N}{3}, \qquad \pi - \pi_{\text{ex}} = O(N^{-4}). $$ Then compare with \( \pi_{4N} \ ): $$ E_N := \big|\pi_{\text{ex}} - \pi_{4N}\big|, \qquad \text{verified digits} \;\approx\; \big\lfloor -\log_{10} E_N \big\rfloor. $$ If verified digits stop growing when you double N, the numerical checker is limiting you (inversion).

Takeaway

Intellectual simplicity (GRG): you turn a single knob, N, get a monotone inside-approximation, and a predictable error.
Honest inversion: beyond a certain threshold, it’s not GRG that runs out of accuracy—it’s the tools used to verify it.
Self-certification: using $ \Delta_N $, $ \pi_{\text{ex}} $, and $ E_N $, the method checks itself without resorting to an external π.