ITERATIVE GEOMETRIC THEORY (IGT) One-Page Summary Erik Northmen · Independent Researcher · 2026 The Core Idea IGT inverts 2,300 years of polygon construction. Instead of inscribing a polygon inside a pre-existing circle, the Recursive Isosceles Chain Construction (RICC) builds regular polygons outward from a baseline through local compass-and-straightedge iteration, governed by the construction ratio CR(n) = 2cos(π/n)/(2cos(π/n) − 1). Two operations — torsion extension and construction network examination — generate a 2×2 classification of regular and star structures in two and three dimensions. Central Result: The Diagonal Reciprocity Theorem The heptagon is the unique regular polygon whose two innermost diagonal excesses are multiplicative inverses: (D₂ − 1)(D₃ − 1) = 1. The proof reduces this to the supplementary diagonal identity D₄ = D₃ via the Chebyshev product D₂ · D₃ = D₄ + D₂, showing that the coincidence CR(7) = α(7) ≈ 2.247 is structurally determined: n = 3 + 4 = 7, forced by the diagonal orders the framework operates on. No other polygon satisfies this identity. Further Results (8 Theorems, 3 Propositions, 4 Lemmas) 3D Theory: The Torsion Compression Theorem gives the exact star torsion as a closed-form function of n, k, and base torsion, derived from the cylindrical structure of alternating-torsion skew polygons. The Helical Non-Closure Theorem proves constant-torsion chains cannot close in ℝ³. Star skew polygon regularity is proved via symmetry group analysis. Puckering Framework: A unified Fourier-mode classification explains why even-n rings admit alternating torsion (Nyquist degeneracy) while odd-n rings require pseudorotation. The linearized torsion-puckering constant C(n) unifies both cases. CNU Cascades: Iterated construction networks are exactly self-similar with scaling factor D₂ = 2cos(π/n). For odd n, D₂ acts as an integer circulant on ℤⁿ (proved algebraically). The n = 5 cascade recovers the Penrose inflation hierarchy. The n = 7 cascade predicts heptagonal quasicrystalline order with numerically verified sharp Bragg peaks in a 7-dimensional embedding — a class of aperiodic order not yet observed experimentally. Polyhedral Geometry: Platonic dihedral angles expressed as sin²(δ/2) = (α(f)+1)/(3−α(n)), with curvature classification α(n)+α(f) ≲ 2. The coprimality condition governs star structure at three levels: face, chain, and polyhedron. Molecular Correspondence TICC closure solutions match cyclohexane bond angles to 0.08° and glucose to 0.02° with zero free parameters. The golden ratio appears at two DNA scales: α(5) = φ (deoxyribose ring) and α(10) = φ² (helical repeat), connected by φ² = φ + 1. These are geometric correspondences, not physical predictions. Status The monograph (14 sections, 10 figures, 7 tables) has passed three rounds of adversarial review with zero algebraic errors. All proofs are self-contained. Novelty claims are explicitly separated from connections to known results. Six of eight original open directions are resolved or partially resolved. The remaining open questions are: exact odd-n torsion formula, CNU matching rules for space-filling tilings, polyhedral CNU in 3D, and experimental molecular validation.