# 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. """ Strong sector: QCD-1 (confinement) and QCD-2 (the SU(3) rank-three frame). QCD-1. Confinement is the area law of the Wilson loop in the strong-coupling (saturated) regime of the COMPACT finite gauge group. We compute the strong-coupling string tension sigma(beta) for compact U(1) and for Z_N and exhibit the linear potential V(r)=sigma*r, with the crossover to the weak-coupling Coulomb phase. The finite gauge group is compact, so the confined phase exists; non-abelian asymptotic freedom puts the IR there. QCD-2. SU(3) is the special unitary group of a Hermitian 3-form over the quadratic extension K=F_{q^2} (the spinor channel) with Frobenius conjugation. Built exactly for q=2 (F_4): order 216, centre Z_3 (colour triality), controlled by the cube-root structure 3 | q+1. """ import numpy as np import itertools from fractions import Fraction as Fr import math # ===================================================================== # QCD-1 : strong-coupling string tension and the linear potential # ===================================================================== def besselratio(beta, kmax=60): """I1(beta)/I0(beta) by series -- the U(1) single-plaquette character coeff.""" I0 = sum((beta/2)**(2*k) / (math.factorial(k)**2) for k in range(kmax)) I1 = sum((beta/2)**(2*k+1) / (math.factorial(k)*math.factorial(k+1)) for k in range(kmax)) return I1 / I0 def sigma_U1(beta): # string tension, strong-coupling leading order return -math.log(besselratio(beta)) def sigma_ZN(beta, N): """Z_N gauge: single-plaquette character coeff c1 = /<...>.""" ths = [2*math.pi*k/N for k in range(N)] Z = sum(math.exp(beta*math.cos(t)) for t in ths) c1 = sum(math.cos(t)*math.exp(beta*math.cos(t)) for t in ths) / Z return -math.log(c1) def qcd1(): print("QCD-1 confinement = area law in the saturated (strong-coupling) regime") print(" string tension sigma(beta) = -ln(plaquette character coeff):") print(" beta sigma_U(1) sigma_Z2 sigma_Z3 phase") for beta in (0.2, 0.5, 1.0, 2.0, 4.0): sU = sigma_U1(beta); s2 = sigma_ZN(beta, 2); s3 = sigma_ZN(beta, 3) phase = "confined (linear V)" if sU > 0.3 else "-> Coulomb (deconfined)" print(f" {beta:4.1f} {sU:8.4f} {s2:8.4f} {s3:8.4f} {phase}") # linear potential: V(r) = -lim_T (1/T) ln = sigma * r (area law) beta = 0.5; s = sigma_U1(beta) print(f"\n linear potential at beta={beta} (sigma={s:.4f}): V(r) from area law") print(" r : " + " ".join(f"{r}" for r in range(1, 7))) print(" V(r): " + " ".join(f"{s*r:.3f}" for r in range(1, 7)) + " (V/r constant => linear => confining)") print(" capacity link: small beta = low channel capacity = saturated;") print(" sigma ~ ln(1/beta) is the string tension; sqrt(sigma) sets Lambda_QCD.\n") # ===================================================================== # QCD-2 : SU(3, F_q) as the special unitary group of a Hermitian 3-form # over K = F_{q^2}. Exact construction for q=2 (K=F_4). # ===================================================================== # F_4 = {0,1,2=w,3=w^2}, char 2. Frobenius conjugation x -> x^2. 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]) # x -> x^q = x^2 (Frobenius over F_2) def f4_matmul(A, B): """batched 3x3 matmul over F_4. A,B: (...,3,3) int arrays.""" C = np.zeros(A.shape[:-2] + (3, 3), dtype=np.int64) for i in range(3): for j in range(3): acc = np.zeros(A.shape[:-2], dtype=np.int64) for k in range(3): acc = ADD[acc, MUL[A[..., i, k], B[..., k, j]]] C[..., i, j] = acc return C def f4_det(M): a,b,c = M[...,0,0],M[...,0,1],M[...,0,2] d,e,f = M[...,1,0],M[...,1,1],M[...,1,2] g,h,i = M[...,2,0],M[...,2,1],M[...,2,2] def m(x,y): return MUL[x,y] t1 = m(a, ADD[m(e,i), m(f,h)]) t2 = m(b, ADD[m(d,i), m(f,g)]) t3 = m(c, ADD[m(d,h), m(e,g)]) return ADD[ADD[t1, t2], t3] def qcd2(): print("QCD-2 SU(3,2): special unitary group of a Hermitian 3-form over F_4") # all 3x3 matrices over F_4 N = 4**9 digits = np.array(np.unravel_index(np.arange(N), (4,)*9)).T # (N,9) M = digits.reshape(N, 3, 3) # conjugate-transpose M^dagger = conj(M)^T Mdag = CONJ[M].transpose(0, 2, 1) # Hermitian form H = identity: unitary <=> M^dag M = I prod = f4_matmul(Mdag, M) I3 = np.eye(3, dtype=np.int64) unitary = np.all(prod == I3, axis=(1, 2)) det1 = (f4_det(M) == 1) su3 = unitary & det1 order = int(su3.sum()) formula = 2**3 * (2**2 - 1) * (2**3 + 1) # q^3 (q^2-1)(q^3+1) print(f" |SU(3,2)| enumerated = {order} formula q^3(q^2-1)(q^3+1) = {formula}" f" match: {order == formula}") # centre: scalar matrices xI with x^3=1 and norm(x)=1 centre = [] for x in range(4): S = (x * I3) # not F_4 scalar mult; build xI by hand S = np.zeros((3,3), dtype=np.int64) for j in range(3): S[j,j] = x Sd = CONJ[S].T if np.all(f4_matmul(Sd[None], S[None])[0] == I3) and f4_det(S[None])[0] == 1: centre.append(x) print(f" centre = scalar cube-roots of unity {{x I : x in {centre}}} -> Z_{len(centre)}" f" (colour triality)") print(f" cube roots of unity in F_4^x = C_3 present (3 | q^2-1); centre Z_3 since 3 | q+1.") print(f" the defining 3-module over K is the colour triplet.\n") return order, formula, len(centre) if __name__ == "__main__": print("=" * 70) print("STRONG SECTOR (QCD-1 confinement, QCD-2 the SU(3) rank-three frame)") print("=" * 70 + "\n") qcd1() qcd2() print("=" * 70)