"""Reproduce the exact invariant-cost and conditional-variance schematic.""" from pathlib import Path SVG='\nThe actual invariant cost and its common-center conditional variance\nThe original transfer T equals Q P(w times h). In the same state completion it equals Q P alpha(ell times h). A positive lower bound compares T with corrected alternating expectations and proves its fixed algebra is the intersection of the represented centers. The remaining cost has zero spectral lower endpoint exactly when its common-center conditional variance has zero lower endpoint. General amenability has not supplied this endpoint.\n\nThe fixed algebra is the common center; the original endpoint remains\nOriginal a = b̂; b = d² √(1−4/d) Q(|E_D0(f−E_C(f))|). Full centers retained. (IE.1)\n\nThe same physical transfer, with a proved state-completion comparison. (IE.2–IE.3)\nOriginal Q preserves α; original P preserves γ = θα. New P_α(x) = P(θ⁻¹x).\nT(h) = Q P(wh) = Q P_α(ℓh); ℓ ≥ r = p/q > 0; Qℓ = P_αℓ = 1.\nK(h) = Q P_α(h); L = (T−rK)/(1−r) is positive, unital and α-preserving.\nThese are maps in the same GNS completion. No original P = P_α or Pℓ = 1 is asserted.\n\nA quantitative energy comparison identifies the exact fixed algebra. (IE.4–IE.7)\nFor selfadjoint h: α(h(h−Th)) ≥ r ||h−P_αh||²₂.\nFor original bounded h∈U: actual total basis energy ≥ 2dr ||h−P_αh||²₂.\nFix(T) = F = U_GNS ∩ V_GNS. Both Kⁿ and Cesàro(T) converge in state L² to Π.\nE_F = ΠQ is the α-preserving expectation on the full joint algebra; E_F P_α = E_F.\n\nThe remaining endpoint is an explicit conditional variance. (IE.8–IE.10)\nθ = ℓ/w; v = E_F((θ−1)²) = E_F(P(θ)−1), using P_αθ = 1.\nā = Πb = d E_F(w|θ−1|); (dr/(H−1)) v ≤ ā ≤ d R* √v.\nmin sp_F(ā) = 0 ⇔ min sp_F(v) = 0. Their kernel projections coincide.\nSmall variance cuts in F return by LC.3 to actual compatible M-central states on B.\n\nA positive genuine minimum would require uniform conditional variance. (IE.11)\nAt a true minimizing ψ₀ with cost m: ā = m1; (m/(dR*))² ≤ v ≤ ((H−1)/(dr)) m.\nThe general amenability-to-zero-endpoint implication remains unproved.\nFull BF, infinite-capacity localization, full partitions, alignment and generation stay assigned.\nExact maps and complete proofs: IE.1–IE.11. Schematic panels; no trace areas encoded.\nHuman context: Popa1994, Definitions3.1.1–3.1.2 and Theorem4.2.2. Original diagram: CC0.\n\n' Path(__file__).with_suffix(".svg").write_bytes(SVG.encode("utf-8"))