"""Reproduce the schematic and exact actual tensor-Jones identities.""" from pathlib import Path SVG='\n\nTail cuts retain operators; whole supports change the compressed index\nExact actual objects and proof locators. The original general finite partition remains unresolved.\n\nStronger actual tail-family input: OT.1\np_i in Q_i = N_mi^(i); P_i = p_i A_mi^(i) p_i\nold error < eps/2; tau(f) < eps^2/(16 R^2)\nPopa4.4 pp.220-221 uses this tail membership.\nUnrestricted amenability has not supplied it here.\n\nAvailable general whole support: OT.2\np in B_m = Q\' cap N; Q = N_m; D = d^m\nQp is an actual II1 factor inside pNp.\n(Qp)\' cap pMp = p A_m p: old operators retained.\nThis is the membership supplied by general76.2.\n\n\n\nFinite cups give exact compatible budgets\nsum c_i = 1; 0 < a_i < tau(p_i) for good cells\nChoose r_i ≤ p_i inside Q_i with tau(r_i) = a_i.\nu_i in Q_i fixes A_mi^(i), and places a copied cup.\nr_i enters actual B_(mi+J+1); residual trace is a_0.\nOT.3-OT.8: no near-identity unitary assumption\n\nThe literal corner-tunnel match is blocked\n[pNp : Qp] = D tau(p) rho_Q(p) < D\nB_m cap Q = C1; a proper p is outside every later tail.\nLater compressed index = d^(m+l) tau(p) rho_Q(p).\nA depth shift needs tau(p) rho_Q(p) = d^(-a).\nOT.13-OT.16: necessary, not sufficient\n\n\n\nExact finite whole pairs; fixed physical y\nc_y = sum r_i b_y r_i in P*; b_y = E_P0(y)\nerror < eps/2 + eps/(2 sqrt(2)) < eps\nOT.9-OT.10; both expectation orders are exact.\n\nActual tensor Jones example\nQ = 1 tensor 1 tensor R; N = M_2 tensor 1 tensor R\nM = M_2 tensor M_2 tensor R; d = 4\ntau(p) = rho_Q(p) = 1/2; local index = 1\nOT.17-18: constructed operators; no theorem counterexample.\n\n\nOriginal unrestricted implication still required\nGeneral amenability → actual finite operator transfer, exact residual certificate, or cheap actual overlap.\nNo common stage, prefix or generating theorem is inferred from these individual-cell results.\nAll positions and areas are schematic; no trace geometry is inferred. Exact proof locators OT.1-OT.18.\nSources: Popa(1994), pp.220-222; TakesakiIII XIX.2.6-2.8, pp.422-423. Dashed arrow marks the open input.\n' Path(__file__).with_suffix(".svg").write_bytes(SVG.encode("utf-8")) from fractions import Fraction as F import itertools,json operator_checks=0;generation_checks=0 for n in [2,3,4]: e={(a*n+a,b*n+b):F(1,n) for a in range(n) for b in range(n)} def addmul(a,b): c={} for (i,k),v in a.items(): for (l,j),w in b.items(): if k==l:c[i,j]=c.get((i,j),F(0))+v*w return {k:v for k,v in c.items() if v} def scale(a,t):return {k:v*t for k,v in a.items() if v*t} def first(a,b):return {(a*n+k,b*n+k):F(1) for k in range(n)} assert addmul(e,e)==e assert {(j,i):v for (i,j),v in e.items()}==e trace=sum(e.get((i,i),F(0)) for i in range(n*n))/F(n*n) assert trace==F(1,n*n) partial={(a,c):sum(e.get((a*n+b,c*n+b),F(0)) for b in range(n))/n for a in range(n) for c in range(n)} assert partial=={(a,c):F(1,n*n) if a==c else F(0) for a in range(n) for c in range(n)} for a,b in itertools.product(range(n),repeat=2): assert addmul(addmul(e,first(a,b)),e)==scale(e,F(1,n) if a==b else F(0)) operator_checks+=1 for u,v,a,b in itertools.product(range(n),repeat=4): generated=scale(addmul(addmul(first(u,a),e),first(b,v)),n) assert generated=={(u*n+a,v*n+b):F(1)} generation_checks+=1 rank_left_p=sum(1 for a,b in itertools.product(range(n),repeat=2) if a==0) tau_p=F(1,n);rho_p=F(rank_left_p,n*n) assert n*n*tau_p*rho_p==1 operator_checks+=5 constant_checks=0 for q in range(1,10): eps=F(1,3);R=2;f=eps**2/(32*R**2);old=(1-f)/q eta=min(old/8,eps**2/(64*q*R**2)) assert 00 constant_checks+=1 print(json.dumps({'passed':True,'actual_tensor_Jones_sizes':[2,3,4],'generation_checks':generation_checks,'projection_expectation_module_checks':operator_checks,'strict_transfer_constant_checks':constant_checks,'scope':'Finite exact operator identities verify the explicitly constructed tensor Jones example; scalar checks verify bounds only, with no unrealized Jones array claim.'})) from fractions import Fraction as F from itertools import product import json def mul(a,b): c={} for (i,k),x in a.items(): for (k2,j),y in b.items(): if k==k2: c[i,j]=c.get((i,j),F(0))+x*y return {k:v for k,v in c.items() if v} def scale(a,s): return {k:v*s for k,v in a.items() if v*s} def first(n,u,v): return {(u*n+a,v*n+a):F(1) for a in range(n)} generation=0; scalar_expectation=0; module=0 for n in [2,3,4]: e={(a*n+a,b*n+b):F(1,n) for a,b in product(range(n),repeat=2)} assert mul(e,e)==e assert all(e.get((j,i),F(0))==v for (i,j),v in e.items()) assert sum(e.get((i,i),F(0)) for i in range(n*n))/F(n*n)==F(1,n*n) en={(u,v):sum(e.get((u*n+a,v*n+a),F(0)) for a in range(n))/n for u,v in product(range(n),repeat=2)} assert en=={(u,v):(F(1,n*n) if u==v else F(0)) for u,v in product(range(n),repeat=2)} for u,v in product(range(n),repeat=2): assert mul(mul(e,first(n,u,v)),e)==(scale(e,F(1,n)) if u==v else {}) scalar_expectation+=1 for u,a,b,v in product(range(n),repeat=4): assert scale(mul(mul(first(n,u,a),e),first(n,b,v)),n)=={(u*n+a,v*n+b):F(1)} generation+=1 assert F(n,n*n)==F(1,n) assert n*n*F(1,n)*F(n,n*n)==1 module+=1 constants=0 for q in range(1,10): eps=F(1,3); R=F(2); f=eps*eps/(32*R*R); old=(1-f)/q eta=min(old/8,eps*eps/(64*q*R*R)) assert eta0 C2=F(25**q-1,6)+F((5**q-1)**2,4) assert C2>0 constants+=1 assert F(1,8)