"""Reproduce the exact actual-corner and controlled-tunnel schematic (CC0).""" from pathlib import Path SVG='\n\nWhole supports, actual full corners, and the remaining bridge\nAll areas are schematic. Physical targets Y stay fixed. Proof locators refer to the adjacent note.\n\n\nWhole-stage input\nF = Nₘ, p ∈ F′ ∩ N, t = τ(p), u = ρ_F(p)\nFp ⊂ pNp has index Iₚ = dᵐtu\nActual old pair: pAₘp / pBₘp [OT.2; GTB.4]\nActual skipped Jones triple\nT = Nₘ₊₂ₕ ⊂ S = Nₘ₊ₕ ⊂ F\nqₕFqₕ = Tqₕ, E_S(qₕ) = d⁻ʰ1\nqₕ is an actual projection [14.8; GTB.5]\n\n\n\n\n\nProved full corner: r = pqₕ [GTB.2]\nτ(r) = td⁻ʰ ρ_T(r) = ud⁻ʰ support product = tu d⁻²ʰ\n[rNr : Tr] = dᵐtu = Iₚ, unchanged; [rMr : rNr] = d\nThe old operators r(pAₘp)r are retained in the full supported late cell.\nEach canonical tower keeps its own cups, expectations and traces, at its own index.\n\n\n\n\nFinite actual orthogonal refinement\nL = ⌊dʰ⌋ copies inside F; individual continuations\n0 ≤ discarded physical mass < td⁻ʰ\nEach new compressed local index is still Iₚ.\nWhole-stage supports persist. [GTB.3; GTB.10–11]\nAdditional actual corner-chain input\npMp ⊃ pNp ⊃ ⋯ ⊃ Sₖ ⊂ G′ ∩ pNp\nRequires actual cups, expectations and generation.\n⇔ whole tail p ∈ Tₖ retaining G physically\nNumerical index equality is insufficient. [GTB.4]\n\n\n\nSufficient route to the full finite partition\nActual corner chains for the finite near-cover\n→ tail family → OT.1 → full actual cells summing to1\nBoth expectations, A₀ and strict target error retained.\nOriginal residual is finitely realized here. [GTB.5]\n\nGenuine Jones test: d =9, rank-two p\n(t,u) = (2/3,2/3) → (2/27,2/27)\nIₚ =4 before and after the full-corner cup\n4 ≠9ᵏ for every integer k ≥0. [GTB.14–19]\nThis tests this route; it does not refute the theorem.\n\nRemaining unrestricted implication\nGeneral amenability ⇒ controlled candidates/chains, different cells, or another full finite construction.\nExact retention of every selected block can fail at every finite length: GTB.6 below.\nGTB.8 sufficient budget: ε₀ + √Σδᵢ² < ε/2, τ(f) < ε²/(16R²). General input remains open.\nHuman source: S. Popa, Acta Mathematica172(1994),1.3.2 p.178 and4.4 pp.220–222. Independent schematic.\n\n\nSame rank-two example, every length\npNp = Mat₂ ⊗ F ⊂ pMp = Mat₃ ⊗ Mat₂ ⊗ F\nEvery actual smaller stage: Mat₃ʲ [GTB.21]\nExact G retention ⇒ unital Mat₂ inside Mat₃ʲ.\nTwo equal integer ranks cannot sum to odd 3ʲ.\nFull smooth-representation amenability: GTB.6a.\nControlled actual matrix completion\nL = 3ʲ, τ_F(z) = 1 − 1/L, q = (L − 1)/2\nL − 1 old units + equal-trace residual → Mat_L\nRotate the tunnel; fixed targets stay in place.\n36 two-leg targets: error² = 1/(6L).\nAt j = 3: error = 1/√162 < 1/10. [GTB.23–24]\n\nActual L = 27 completion in Mat₂ ⊗ Mat₂₇ [GTB.7 / Exercise GTB.2]\nEach colored diagonal has normalized D-trace 1/27 and ordinary rank two.\nE₁₁ row\nE₂₂ row\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n\nresidual\ntrace 1/27\nrank two\nConnect all 27 equal-trace diagonals by actual partial isometries in the fixed physical D.\nThe 26 colored units retain E_ab ⊗ z. The gray support carries exactly the error E_ab ⊗ (1 − z).\nGTB.6 obstructs exact selected-block retention. GTB.7 proves controlled retention in this actual example.\nNo universal centered-core construction or full original partition theorem is claimed from this figure.\n' Path(__file__).with_suffix(".svg").write_bytes(SVG.encode("utf-8"))