"""Reproduce the exact proof schematic and its stated finite checks.""" from pathlib import Path SVG='\nTwo sufficient routes to an exact finite whole-tunnel partition\nA selected finite square can be completed by a positive residual trace certificate or by a small cut and overlap certificate. The unrestricted input from amenability remains unproved. Boxes and arrows encode proof steps, not traces or geometry.\n\n\n\nTwo sufficient routes to a finite whole-tunnel partition\nActual finite-index II₁ inclusion; finite targets Y; ε > 0\n\n76.4: selected whole-tunnel blocks and physical residual f\nP₀, Q₀; old physical supports are orthogonal; target error < ε/2\nEach selected block has its complete actual finite whole-tunnel pair\n\n\n\nif the positive certificate is supplied\n\nif the small-cost certificate is supplied\n\nRoute A: exact positive residual certificate\nτ(f) ∈ Γ₊ ⇔ finite positive trace certificate [FP.9]\nj ≥ m; z ∈ Z^(s_j); ωⱼ·z = 0; vⱼ+z ≥ 0\nAll-trace σ(vⱼ+z) > 0 suffices [FP.10]; no upper test\n\nRoute B: small cut / ordered overlap certificate\nΔⱼ ≤ ℰⱼ ≤ qΔⱼ [FP.12]\nℰ < [ε / (2(1+√2)R)]² [FP.16]\nR = max(1, max of target norms); overlap is ordered\n\n\n\nFP.3: split ranks into finitely many bounded vectors\nFP.5–FP.7: choose physical residual cells fₐ ≤ f\nand place each in its own actual finite tunnel.\nOld whole-stage blocks stay fixed; P₀ ⊂ P*.\nCanonical certificate pieces need not be orthogonal.\n\n80.3–80.4: optimal cuts remove total trace Δⱼ\nEnlarge the physical residual by the removed pieces.\nComplement ranks certify that new residual exactly.\nExtra target error ≤ (1+√2)‖y‖√Δⱼ.\nCut old supports may change; targets remain fixed.\n\n\n\nFinite full support partition: sum of physical supports = 1\nEvery cell has its complete supported actual whole-tunnel pair\nE_N E_{P_*} = E_{P_*} E_N = E_{Q_*}; target error < ε [FP.6–FP.7 / FP.16]\n\nRemaining unrestricted input: not proved here\nAmenability must supply a positive certificate, a sufficiently cheap actual selection,\nor another complete construction of the original finite partition [FP.0–FP.1].\nPositions, areas and colors carry no trace, rank or geometric measurement.\n' Path(__file__).with_suffix(".svg").write_bytes(SVG.encode("utf-8")) from fractions import Fraction from itertools import product import random w = [Fraction(3, 15), Fraction(5, 15), Fraction(7, 15)] T = {sum((w[i]*h[i] for i in range(3)), Fraction()) for h in product(range(2), repeat=3)} t = Fraction(2, 5) assert t not in T and 2*w[0] == t and 3*w[0]+t == 1 assert 6*w[0] == 3*(2*w[0]) rng = random.Random(20261005) cases = 0 for n in range(1, 13): for q in range(1, 8): for _ in range(20): h = [rng.randrange(n+1) for _ in range(q)] excess = max(sum(h)-n, 0) retained, remaining = h[:], excess for i in range(q): cut = min(remaining, retained[i]) retained[i] -= cut remaining -= cut cursor, projections = 0, [] for old_rank, new_rank in zip(h, retained): core = set(range(cursor, cursor+new_rank)) cursor += new_rank outside = sorted(set(range(n))-core) projections.append(core | set(outside[:old_rank-new_rank])) energy = sum(len(projections[i] & projections[j]) for i in range(q) for j in range(q) if i != j) assert excess <= energy <= q*excess cases += 1 assert cases == 1680