"""Reproduce the HC3–HC9 proof diagram.""" from pathlib import Path SVG='\nExact corner repair, an increasing matrix chain, and diagonal indexAn approximate corner projection q is repaired to r with trace tau(p), then conjugated onto p. Full dyadic approximations build a nested chain whose trace-preserving Hilbert unitary gives pRp isomorphic to R. Isomorphic complementary corners give index one over t times one minus t.\n\nEvery positive corner trace, including irrational traces\nR = weak closure of ⋃ M₂⊗k; τₚ = τ / τ(p); 0 < τ(p) ≤ 1\nRepair the corner exactly\nq = 1[½,1](E_D(p))τ(r) = τ(p), uru* = p‖u − 1‖₂ ≤ 2‖r − p‖₂uD₀u* ⊂ pRp, unit pHC3–HC4; (C5)–(C9)\nBuild nested full matrices\nBₙ ≅ M₂ᵏⁿ, kₙ ≥ nBₙ ⊂ Bₙ₊₁ ⊂ pRpdist₂(xⱼ, Bₙ) < 2⁻ⁿAll xⱼ with j ≤ nHC5–HC6; (C10)–(C12)\nIdentify both closures\nU x̂ = φ(x)̂U L(pRp) U* = L(R)φ preserves τₚ and τpRp ≅ R for every p ≠ 0HC7–HC8; (C13)–(C14)\n\n\nComplementary corners realize every finite real d ≥ 4\nτ(p) = t, q = 1 − p; θ : pRp → qRq; τ(θ(x)) = (1 − t)τ(x)/t\nN = {x + θ(x)} ⊂ R; [R:N] = 1/t + 1/(1 − t) = 1/[t(1 − t)]\nt = (1 − √(1 − 4/d))/2 ∈ (0, ½]; d = 4 gives t = ½\nHC9; (C15)–(C17). The corner proof itself does not use the index formula.\nProof mechanism; boxes are schematic. No finite-stage membership of p is assumed.\n\n' Path(__file__).with_suffix(".svg").write_text(SVG,encoding="utf-8")