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Implement a Python script that sweeps a finer grid of curvature exponents (k) (e.g., [0.1, 0.5] in steps of 0.001) over larger (N_{\max}) (up to 10,000) using the frame-shifted residue (\theta'(n, k) = \phi \cdot ((n \mod \phi)/\phi)^k). Apply the Domain Curvature Transformation to correct for uniform density biases, computing (e_{\max}(k)) as Z = n(Δₙ/Δmax) where Δₙ measures bin density shifts. Output a heatmap of enhancements, proving concept that nonlinear warping reveals prime over-densities around (k^* \approx 0.3), scalable with NumPy for efficiency.

Build a script fitting GMMs (5-10 components) to normalized prime residues (x_p = (\theta'(p, k) \mod \phi)/\phi) at fixed (k=0.3). Use Universal Form Transformer to map variances (\bar{\sigma}(k)) as Z = T(v/c) with v as cluster spread rates and c as interval limit [0,1). Visualize component means and sigmas in 2D plots, demonstrating PoC for isolating prime clusters with scikit-learn, highlighting compactness gains of 10-20% over uniform models.

Create a Python tool approximating prime density (\rho(x)) with Fourier series (order M=10), focusing on sine coefficients (S_b(k)) as measures of asymmetric frame shifts. Frame this via Universal Frame Shift Transformer, where b_m coefficients encode Δₙ deviations from even distributions. Sweep k values, plot (S_b(k)) curves, and prove concept by showing peak asymmetries correlate with density enhancements, using SciPy for least-squares fits and Matplotlib for wave reconstructions.

Extend the main.py helix visualization by parameterizing HELIX_FREQ with golden ratio multiples (e.g., (\phi / 2, \phi)), applying Numberspace transformer as Z = value * (B / C) with B as math.e and C as pi. Script generates interactive 3D scatters (via Plotly) separating primes in helical coordinates, proving PoC that oscillatory frame shifts (sin(pi * freq * n)) amplify prime visibility in logarithmic scales for N up to 10,000.

Script a variant of main.py's logarithmic spiral, incorporating curvature k in angles: angle = n * 0.1 * pi * (n mod phi / phi)^k. Use Domain Curvature Transformation to warp radii as log(n) * (Δₙ/Δmax), plotting primes in polar 3D with Matplotlib. PoC demonstrates enhanced clustering of primes along spiral arms, relating to Beatty sequences, with exportable SVG for geometric insights.

From main.py's torus section, develop a script mapping numbers mod (p,q) where p,q are primes (e.g., 17,23), using toroidal coordinates x=(R+r*cos(theta))*cos(phi). Apply Universal Form Transformer with rate as gcd(n, pq) / max_gcd, coloring by Z = n(Δₙ/Δmax) for residue shifts. Visualize wireframe tori with prime stars, proving PoC for periodicity in multi-modular spaces, scalable to larger moduli via NumPy meshes.

Enhance main.py's zeta plot by overlaying more primes (up to 1,000) on the critical line s=0.5 + i*t, computing log|ζ(s)| with SciPy.special.zeta. Incorporate frame shifts as Z = T(v/c) with v as imag part rates and c as e, surfacing non-trivial zeros approximations. PoC script outputs 3D surfaces, showing prime correlations to zeta dips, as empirical link to Riemann hypothesis via density waves.

Using geometric_projection_utils.py, script a focused triangulator for "twin_primes" in ranges like (1e6, 1e6+10000), chaining cylindrical/hyperbolic projections. Compute triangles with areas <5 as Z = area * (Δ_area / Δmax_area), classifying via gcd and differences. Visualize top triangles in 3D, proving PoC that sequential frame shifts isolate twin pairs with 85th percentile densities, outputting candidate lists.

Adapt the GeometricTriangulator for "mersenne_primes", sampling large ranges (e.g., 2^10 to 2^20 exponents) with spherical projections tuned to PHI rates. Use Universal Frame Shift Transformer to adjust z_values as log2(n+1) * (fs / max_fs), triangulating vertices where is_mersenne holds. PoC generates histograms and scatter plots, forecasting potential Mersennes by density thresholds, verified against known ones like 31, 127.

Extend triangulation to "coprimes", building a graph where triangle vertices connect if gcd=1, using NetworkX for adjacency. Apply chained projections with correction_factor = rate / UNIVERSAL, mapping densities as Z = n(Δ_gcd / Δmax=1). Script visualizes coprime clusters in 2D force layouts overlaid on 3D projections, proving PoC for revealing Euler's totient patterns in geometric spaces, with export to JSON for further analysis.