GRG in One Page: π-Free Corridors, Benchmarks, and a Lean Mini-App

This single page addresses reviewer questions with concrete numbers, one minimal code box, and a practical validation plan. All constructions are turns-first and dyadic: b ∈ ℕ≥0, N = 2^b, step Δu = 2^{-(b+2)} = 1/(4N). We use only sums and square roots.

1) Worked example (r = 100) with certified circumference/area corridors

Let c_N be the unit-radius chord spanning an arc of 1/N turns. Starting from the square, c_4 = √2, we refine dyadically by a π-free half-angle step:

With radius r, certified inscribed/circumscribed perimeters and areas are

Hence a certified corridor for π:

Numeric results (r = 100; dyadic N)

2) One minimal code box (Python) that reproduces the table

Drop-in, π-free implementation. Only square roots and sums via half-angle edge doubling.

# GRG: π-free corridors for circumference and area (dyadic N=2^b)
# r: radius; N: number of edges (dyadic)
import math

def c_dyadic(N):
    # start from square (N=4): c_4 = sqrt(2)
    if N & (N-1):
        raise ValueError("N must be a power of 2")
    if N < 4:
        raise ValueError("N must be >= 4")
    c = math.sqrt(2.0)
    n = 4
    while n < N:
        c = math.sqrt(2.0 - 2.0*math.sqrt(1.0 - 0.25*c*c))
        n *= 2
    return c  # unit radius

def corridors(r=100.0, b_list=(3,4,5,6)):
    out = []
    for b in b_list:
        N = 2**b
        c = c_dyadic(N)
        t = c / math.sqrt(1.0 - (0.5*c)**2)  # computed π-free
        C_minus = N * r * c
        C_plus  = N * r * t
        apothem = r * math.sqrt(1.0 - (0.5*c)**2)
        A_minus = 0.5 * C_minus * apothem
        A_plus  = 0.5 * C_plus  * r
        pi_lo, pi_hi = C_minus/(2*r), C_plus/(2*r)
        out.append(dict(N=N, C_minus=C_minus, C_plus=C_plus,
                        A_minus=A_minus, A_plus=A_plus,
                        pi_low=pi_lo, pi_high=pi_hi,
                        width=pi_hi-pi_lo, mid=0.5*(pi_hi+pi_lo)))
    return out

if __name__ == "__main__":
    for row in corridors():
        print(row)

Why this matters: Reviewers asked for “real code.” The snippet above is reference-grade and numerically stable for dyadic N in double precision.

3) Mini-calculator (in-page): corridor and area, π-free

Try different radii and resolutions. The calculator only uses the recursion above (no trigonometric calls).

4) External validation: baselines, datasets, metrics

4.1 Baselines

• Geometry: regular polygon perimeter/area via trig (reference), rasterization by angular stepping, standard kernel approximations.
• Optics: Zernike-based phase optimization (asphere/phase plate) and uniform samplers; compare against GRG dyadic turns.
• Chemistry: radial distribution (isotropic) vs. anisotropic stress tests; compare GRG dyadic shells to uniform/log binning.

4.2 Datasets and tasks

• Circles/Spheres: circumference/area/volume corridors at fixed sample budgets.
• Optical toy problem: paraxial lenslet: maximize on-axis MTF at a target spatial frequency under a fixed simulator budget.
• Chemical toy problem: radial distribution functions with/without anisotropy; report L_2 error vs. a high-precision reference.

4.3 Metrics and stats

• Error vs budget: corridor width or absolute error vs. number of samples.
• Time vs accuracy: seconds to reach a target error.
• Robustness: worst-case error across seeds; bootstrap CI.

5) Scalability and complexity

Cost: computing c_N is O(log N) half-angle steps; producing C_N^{±}, A_N^{±} is O(1).

Beyond circles/spheres: For radially defined profiles, GRG’s dyadic turns yield nested certified bounds. For anisotropy, use a wedge-stratified scheme: split [0,1) turns into dyadic wedges and apply GRG within each wedge; corridors add up across wedges.

6) GRG Mini-App: current status and UI notes

• Scope: live computation of C_N^{±}, A_N^{±}, π corridor, and dyadic overlays (“inscribed polygon preview”, “row layout”).
• Roadmap: public demo + short tutorial video; CSV export of computed corridors; copy-paste of the Python above.
• Accessibility: keyboard-operable sliders for b, ARIA live regions on numeric outputs (as in the mini-calculator).

7) What to expect next

• Open-source drop: the single Python file (MIT/BSD-3) + tiny unit tests.
• Benchmarks: publish results vs. Zernike and uniform samplers on the toy problems; scripts and seeds included.
• Papers & talks: π-free corridor proofs, dyadic error scaling, wedge-stratified anisotropy experiments.

8) Optics: toy MTF setup (micro-section)

This toy problem shows how GRG guides sampling while any preferred field solver evaluates performance. We optimize a rotationally symmetric height profile under a fixed evaluation budget.

8.1 Objective

Constraints: height range (manufacturability), smoothness (bounded second difference), and a hard cap on simulator calls.

8.2 GRG sampling policy

• Choose dyadic level b; sample radii on N=2^b turns (rings). For anisotropic stress tests, split the pupil into dyadic wedges and apply the same policy per wedge.
• At each refinement, reuse prior evaluations (nested design), cutting waste in flat regions.

8.3 Budget-fair comparison

• Compare against equal-angle and equal-arc samplers at the same number of simulator calls.
• Report \Delta \mathrm{MTF} = \mathrm{MTF}_{\text{GRG}} - \mathrm{MTF}_{\text{baseline}} and time-to-target MTF.

8.4 Reproducibility hook

The π-free recursion and nested sampling policy above serve as the public spec. A scalar paraxial PSF/OTF evaluator—or your preferred solver—can be dropped in without altering the sampling design.