# framed-rational status: MIXED -- exact framed-rational core; continuum constructs are confined to clearly-labelled [approx] / degenerate-idealisation comparisons (Tier 2/3/5 or measured-data), never to an exact claim. """ D6b -- the colour string tension from the finite strong-coupling expansion. ================================================================================ Closes the one *open* (closeable, not Omega-hard) residue of the strong sector: the string tension sigma = -ln c_1(beta) had been computed only for the abelian gauge groups U(1) and Z_N (qcd.py). Here we give the NON-ABELIAN tension. The area law itself (sigma>0) is a corollary of the positivity of S_rho (Prop. corrarea, correspondence.py block F). What was "still to be done" is the *value* of the single-plaquette coefficient c_1(beta) for the colour group, in finite units. We supply it in three exact pieces and one resolved-window piece: T1 finite-group area law: for ANY finite gauge group, the rectangular Wilson loop tiles to c_1^Area with c_1 = <(1/N) chi_f>_w an EXACT finite sum (finite-group character orthogonality; no Haar measure, no Omega->infty). T2 SU(2,F_3)=2T (the corpus's exact non-abelian instance, 2T < SU(2,C)): c_1(beta) in closed form; leading sigma = ln(4/beta), reproducing the continuum SU(2) tension EXACTLY through order beta^3. T3 colour SU(3): resolved-window single-link expansion c_1 = beta/18 + beta^2/216 + ... = (beta/18)(1 + beta/12 + ...), positive and confining; the finite SU(3,F_q) reproduces it (Thm corr). S structural float-free checks on SU(3,F_2): |G|=216, centre Z_3. Everything exact is done over Q / closed form; the continuum Haar integrals are the resolved-window (Omega->large) reference the finite group converges to. Conventions match correspondence.py block F: action S_rho(U) = 1 - (1/N) Re chi_f(U) (positive class function), weight w(U) = e^{-beta S_rho} ~ e^{beta (1/N) Re chi_f(U)} (const cancels), coeff c_1(beta) = < (1/N) chi_f >_w , sigma = -ln c_1 . """ import numpy as np import math, itertools from fractions import Fraction as Fr import sympy as sp PASS = [] def check(name, ok): PASS.append(bool(ok)) print(f" [{'PASS' if ok else 'FAIL'}] {name}") # ===================================================================== print("="*72) print("CHECK 0 normalisation matches the corpus (U(1): sigma(0.5)=1.4168)") # ===================================================================== def Iv(v, beta, k=35): return sum((beta/2)**(2*j+v)/(math.factorial(j)*math.factorial(j+v)) for j in range(k)) c1_U1 = lambda beta: Iv(1, beta)/Iv(0, beta) # I1/I0 check("U(1) c_1=I1/I0, sigma(0.5)=1.4168", abs(-math.log(c1_U1(0.5)) - 1.4168) < 1e-3) # ===================================================================== print("="*72) print("CHECK T1 finite-group area law: orthogonality gives c_1 as an EXACT sum") # ===================================================================== # The two identities the tiling uses, on a finite group G (uniform average # (1/|G|) sum), verified on the symmetric group S_3 as a generic finite group: # (i) = (1/|G|) sum_g chi_r(g) = 0 for any nontrivial irrep r # (ii) (1/|G|) sum_g chi_r(a g) chi_s(g^{-1} b) = delta_{rs} chi_r(ab)/d_r # (i) is exactly what makes the fundamental link integrate to zero (no tadpole), # so the loop tiles its minimal surface; both are finite identities, no measure. # Character table of S_3: classes e,(12),(123) sizes 1,3,2; irreps triv,sgn,std(2). classes = [('e',1),('t',3),('c',2)] # name, size chars = {'triv':{'e':1,'t':1,'c':1}, 'sgn' :{'e':1,'t':-1,'c':1}, 'std' :{'e':2,'t':0,'c':-1}} order = 6 for r in ('sgn','std'): s = sum(sz*chars[r][nm] for nm,sz in classes) check(f"S_3: sum_g chi_{r} = 0 (fundamental has no singlet -> tiling closes)", s == 0) # orthonormality (ii) at a=b=e: (1/|G|) sum |chi_r|^2 = 1 for r in chars: s = sum(sz*chars[r][nm]**2 for nm,sz in classes) check(f"S_3: = 1", Fr(s,order) == 1) # ===================================================================== print("="*72) print("CHECK T2 SU(2,F_3) = 2T < SU(2,C): exact closed-form string tension") # ===================================================================== def quat_to_su2(q): a,b,c,d = q return np.array([[a+1j*b, c+1j*d], [-c+1j*d, a-1j*b]]) G2T = [] for s in ([1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]): for sg in (1,-1): G2T.append([sg*x for x in s]) for signs in itertools.product([0.5,-0.5], repeat=4): G2T.append(list(signs)) mats = [quat_to_su2(q) for q in G2T] check("|2T| = 24 unit quaternions", len(mats) == 24) check("2T < SU(2,C): all det=1, unitary", all(abs(np.linalg.det(M)-1) < 1e-9 and np.allclose(M.conj().T @ M, np.eye(2)) for M in mats)) from collections import Counter trace_mult = Counter(int(round(np.trace(M).real)) for M in mats) check("trace multiset {2:1,-2:1,0:6,1:8,-1:8}", dict(trace_mult) == {2:1,-2:1,0:6,1:8,-1:8}) # sum of the (nontrivial) fundamental character vanishes -> T1 hypothesis holds check("sum_{g in 2T} chi_f(g) = 0 (orthogonality with trivial)", abs(sum(np.trace(M) for M in mats)) < 1e-9) # exact closed form via the trace multiset, c_1 = <(1/2)chi>_w b = sp.symbols('beta', positive=True) mult = {2:1, -2:1, 0:6, 1:8, -1:8} Zsum = sum(m*sp.exp(b*sp.Rational(t,2)) for t,m in mult.items()) Nsum = sum(m*sp.Rational(t,2)*sp.exp(b*sp.Rational(t,2)) for t,m in mult.items()) c1_2T = sp.nsimplify(sp.simplify(Nsum/Zsum)) print(" c_1^{2T}(beta) =", sp.simplify(c1_2T)) ser = sp.series(c1_2T, b, 0, 6).removeO() print(" series =", ser) # leading term beta/4 lead = sp.limit(c1_2T/b, b, 0) check("leading c_1^{2T} = beta/4 (=> sigma = ln(4/beta))", lead == sp.Rational(1,4)) # continuum SU(2) single-link c_1 = I2/I1; compare series c1_SU2_series = sp.series( (sum((b/2)**(2*j+2)/(sp.factorial(j)*sp.factorial(j+2)) for j in range(6))) / (sum((b/2)**(2*j+1)/(sp.factorial(j)*sp.factorial(j+1)) for j in range(6))), b, 0, 6).removeO() # agreement through beta^3: difference starts at beta^5 diff = sp.simplify(ser - c1_SU2_series) low = sp.series(diff, b, 0, 5).removeO() check("2T reproduces continuum SU(2) tension through O(beta^3) (diff starts beta^5)", sp.simplify(low) == 0) f2T = sp.lambdify(b, c1_2T, 'math') def c1_SU2(beta): return Iv(2,beta)/Iv(1,beta) print(" beta c_1^2T c_1^SU2(Haar) sigma^2T reldiff") for beta in (0.2,0.5,1.0,2.0): a, c = f2T(beta), c1_SU2(beta) print(f" {beta:4.1f} {a:.6f} {c:.6f} {-math.log(a):7.4f} {abs(a-c)/c:.2e}") check("sigma^{2T}(beta) > 0 for all tested beta (confining)", all(-math.log(f2T(x)) > 0 for x in (0.2,0.5,1.0,2.0,4.0))) # ===================================================================== print("="*72) print("CHECK T3 colour SU(3): resolved-window single-link expansion") # ===================================================================== def c1_SU3(beta, n=600): p = np.linspace(0, 2*math.pi, n, endpoint=False) P1, P2 = np.meshgrid(p, p); P3 = -(P1+P2) z1, z2, z3 = np.exp(1j*P1), np.exp(1j*P2), np.exp(1j*P3) vdm = np.abs((z1-z2)*(z1-z3)*(z2-z3))**2 # SU(3) Weyl measure chi = z1+z2+z3 w = np.exp(beta*chi.real/3.0) # w = e^{beta (1/3) Re chi} return float(np.sum((chi.real/3.0)*w*vdm)/np.sum(w*vdm)) bs = np.array([0.02,0.04,0.06,0.08,0.10,0.12]) vals = np.array([c1_SU3(x) for x in bs]) coef,*_ = np.linalg.lstsq(np.vstack([bs,bs**2,bs**3]).T, vals, rcond=None) print(f" fit: c_1^SU(3) = {coef[0]:.6f} b + {coef[1]:.6f} b^2 + ...") check("SU(3) leading c_1 = beta/18 = 1/(2N^2)", abs(coef[0]-1/18) < 1e-4) check("SU(3) next coeff = beta^2/216 (= (beta/18)(1+beta/12))", abs(coef[1]-1/216) < 1e-4) print(" beta c_1^SU(3) sigma=-ln c_1") for beta in (0.2,0.5,1.0,2.0,4.0): c = c1_SU3(beta); print(f" {beta:4.1f} {c:.6f} {-math.log(c):7.4f}") check("sigma^{SU(3)}(beta) > 0 for all tested beta (confining), ->0 as beta->inf", all(-math.log(c1_SU3(x)) > 0 for x in (0.2,0.5,1.0,2.0,4.0,8.0))) # ===================================================================== print("="*72) print("CHECK S structural float-free checks on SU(3,F_2) over F_4") # ===================================================================== ADD = np.array([[0,1,2,3],[1,0,3,2],[2,3,0,1],[3,2,1,0]]) MUL = np.array([[0,0,0,0],[0,1,2,3],[0,2,3,1],[0,3,1,2]]) CONJ = np.array([0,1,3,2]) # Frobenius x->x^2 def herm(u,v): s = 0 for i in range(3): s = ADD[s, MUL[CONJ[u[i]], v[i]]] return s def det3(c0,c1,c2): M = [[c0[0],c1[0],c2[0]],[c0[1],c1[1],c2[1]],[c0[2],c1[2],c2[2]]] a,b_,c = M[0]; d,e,f = M[1]; g,h,i = M[2] pos = ADD[ADD[MUL[a,MUL[e,i]],MUL[b_,MUL[f,g]]],MUL[c,MUL[d,h]]] neg = ADD[ADD[MUL[c,MUL[e,g]],MUL[a,MUL[f,h]]],MUL[b_,MUL[d,i]]] return ADD[pos,neg] # char 2: +/- identical vecs = list(itertools.product(range(4), repeat=3)) norm1 = [v for v in vecs if herm(v,v) == 1] cnt = 0; centre = set() for c0 in norm1: for c1 in norm1: if herm(c0,c1) != 0: continue for c2 in norm1: if herm(c0,c2) != 0 or herm(c1,c2) != 0: continue if det3(c0,c1,c2) != 1: continue cnt += 1 if (c0[1]==0 and c0[2]==0 and c1[0]==0 and c1[2]==0 and c2[0]==0 and c2[1]==0 and c0[0]==c1[1]==c2[2]): centre.add(c0[0]) check("|SU(3,2)| = 216 = q^3(q^2-1)(q^3+1) at q=2", cnt == 216 == 2**3*(2**2-1)*(2**3+1)) check("centre = Z_3 (norm-one cube roots of unity; 3 | q+1)", len(centre) == 3 and (2+1) % 3 == 0) # ===================================================================== print("="*72) print(f"RESULT: {sum(PASS)}/{len(PASS)} checks pass") print("D6b: the string tension sigma(beta) is computed in finite units --") print("exact for U(1),Z_N (qcd.py) and 2T (closed form here), resolved-window") print("series for SU(3). The remaining strong-sector residue is the PHYSICAL") print("value of beta (= the cross-scale running fixing Lambda_QCD = sqrt(sigma)),") print("which is D6c, Omega-hard. Clean cut: the function sigma(beta) is T; the") print("point beta_phys on it is Omega.") assert all(PASS), "some checks failed"