GRG (2D/Circle): A Generative, π-Free Reconstruction with Error Bounds — Reply to Common Objections

Abstract. Generative Radial Geometry (GRG) reconstructs a circle’s circumference and area from the radius alone using finite steps and exact border points (x²+y²=R² by construction). Angles are measured in turns \(u\in[0,1)\) with \(\Delta u=1/N\). The radial track (full turn) yields perimeter and radial area via inscribed/circumscribed polygons; the Cartesian track (quarter) yields area via Δx-trapezoids with a certified corridor. π is not assumed; it emerges as a limit.

GRG construction — three coordinated modes

• Radial perimeter (full turn): sample turns \(u_j=j/N\); build border points \(P_j=(x_j,y_j)\) with \(x_j^2+y_j^2=R^2\) via a GRG recurrence (no trig); close the loop with \(P_N:=P_0\). Sum chords $$ L_{\text{in}}=\sum_{j=0}^{N-1}\big\|P_{j+1}-P_j\big\| \quad\text{(inscribed, lower).} $$ Tangent intersections give a circumscribed polygon with length \(L_{\text{out}}\) (upper). Hence \(L_{\text{in}}\le C\le L_{\text{out}}\).
• Radial area (full turn): use the same \(P_j\) (counter-clockwise). Inscribed polygon area (shoelace): $$ A_{\text{in}}=\tfrac12\sum_{j=0}^{N-1}\!\big(x_j y_{j+1}-x_{j+1} y_j\big) \quad\text{(lower).} $$ Circumscribed polygon from tangent intersections \(T_j\) (see below) gives \(A_{\text{out}}\) (upper): $$ A_{\text{out}}=\tfrac12\sum_{j=0}^{N-1}\!\big(T_j^x T_{j+1}^y – T_{j+1}^x T_j^y\big). $$ Hence \(A_{\text{in}}\le A\le A_{\text{out}}\).
• Cartesian area (quarter only): abscissas \(x_i=i\,\Delta x\), \(\Delta x=R/N\); heights \(y_i=\sqrt{R^2-x_i^2}\). Right/midpoint/trapezoid give a Projection Corridor on the quarter; multiply by 4 for the disk.

Orientation. Assumi l’ordine antiorario così che \(\det_j>0\); se i punti sono orari, adegua i segni.

Error control (π-free statements)

• Perimeter corridor: \(L_{\text{in}}(N)\nearrow C \le L_{\text{out}}(N)\searrow\). Il gap si riduce con \(N\).
• Radial area corridor: \(A_{\text{in}}(N)\nearrow A \le A_{\text{out}}(N)\searrow\).
• Cartesian corridor (quarter): \(A_{\text{right,q}}\le A_{\text{mid,q}}\le A_{\text{quarter}}\le A_{\text{trap,q}}\) (arco concavo); disco via \(4\times\).