TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/check_higgs_mixture_wick.py
Original file SHA-256: d4a4921905bfe2d3984c3017909c8d59864ed4b9facff8ce1d1b594fbcb01b5a
Inclusive original lines: 1-120
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | """Original full96 Higgs source versus covariance-only modular reconstruction.
000002 | 
000003 | Analytic proof supplies all positive Higgs weights. Numerical constructor
000004 | controls reuse v252's original seed252; rational Fock control is diagnostic.
000005 | No physical Yukawa, boson measure, geometry or vacuum is selected here.
000006 | """
000007 | from pathlib import Path
000008 | import sys, json
000009 | import numpy as np
000010 | import sympy as s
000011 | ROOT=Path(__file__).resolve().parents[4]
000012 | sys.path.insert(0,str(ROOT/'verification'))
000013 | import v252_full_finite_triple as v
000014 | 
000015 | checks={}; details={}
000016 | def need(name,ok):
000017 |  checks[name]=bool(ok)
000018 |  if not ok:raise AssertionError(name)
000019 | 
000020 | # The ACTUAL original constructor inputs and real/selfadjoint one-form.
000021 | rng=np.random.default_rng(252)
000022 | Ys=[rng.normal(size=(3,3))+1j*rng.normal(size=(3,3)) for _ in range(4)]
000023 | mr=rng.normal(size=(3,3));MR=mr+mr.T
000024 | z3=np.zeros((3,3),complex);z2=np.zeros((2,2),complex)
000025 | DY=v.full_D(*Ys,z3);DM=v.full_D(z3,z3,z3,z3,MR)
000026 | K=v.Kswap();P=v.rep(1,z2,z3);gamma=v.grading()
000027 | p=.25;ell=1.0 # retained quarter mode; ell=1 is a diagnostic, not a scale fit
000028 | def phi(H):
000029 |  h1,h2=H;q=np.array([[np.conj(h2),h1],[-np.conj(h1),h2]])
000030 |  a=v.rep(0,q,z3);B=a@(DY@P-P@DY);B=B+B.conj().T
000031 |  return B+K@B.conj()@K+DM
000032 | def cov(H):
000033 |  ph=phi(H);h=p*gamma-1j*gamma@ph
000034 |  need('full96_hermitian_'+str(H),np.max(abs(h-h.conj().T))<1e-12)
000035 |  ev,U=np.linalg.eigh(h)
000036 |  return (U*(1/(1+np.exp(2*ell*ev))))@U.conj().T
000037 | left=[[v.IDX[(name,'L',col,g)] for name in ['u','d'] for g in range(3)] for col in range(3)]
000038 | sing=np.r_[np.linalg.svd(Ys[0],compute_uv=False),np.linalg.svd(Ys[1],compute_uv=False)]
000039 | def mean_formula(u):
000040 |  E=np.sqrt(p*p+u*sing*sing)
000041 |  return 3-p/2*np.sum(np.tanh(ell*E)/E)
000042 | Hs=[(0j,1+0j),(.6+0j,.8+0j),(.6j,.8+0j),(0j,2+0j)]
000043 | Cs=[cov(H) for H in Hs]
000044 | means=[]
000045 | for k,C in enumerate(Cs):
000046 |  blocks=[C[np.ix_(ii,ii)] for ii in left]
000047 |  need('three_actual_color_blocks_identical_'+str(k),all(np.max(abs(A-blocks[0]))<2e-13 for A in blocks))
000048 |  need('actual_cross_color_covariance_zero_'+str(k),all(np.max(abs(C[np.ix_(left[a],left[b])]))<2e-13 for a in range(3) for b in range(3) if a!=b))
000049 |  m=float(np.trace(blocks[0]).real);means.append(m)
000050 |  u=sum(abs(z)**2 for z in Hs[k])
000051 |  need('original96_mean_matches_six_singular_values_'+str(k),abs(m-mean_formula(u))<3e-13)
000052 | need('whole_H_direction_does_not_change_color_mean',max(abs(means[k]-means[0]) for k in [1,2])<3e-13)
000053 | need('actual_radial_H_change_is_detected',means[3]>means[0])
000054 | details['constructor_means']=means
000055 | 
000056 | # Direct finite CAR matrices, independent of Wick formula / covariance log.
000057 | # Two internal modes per color, all 64 Fock states, exact integer operators.
000058 | n=6;dim=1<<n;ann=[]
000059 | for j in range(n):
000060 |  a=np.zeros((dim,dim),dtype=np.int64)
000061 |  for b in range(dim):
000062 |   if b&(1<<j):a[b^(1<<j),b]=(-1)**((b&((1<<j)-1)).bit_count())
000063 |  ann.append(a)
000064 | E=[[sum((ann[2*c+i].T@ann[2*d+i] for i in range(2)),np.zeros((dim,dim),dtype=np.int64)) for d in range(3)] for c in range(3)]
000065 | N=sum((E[c][c] for c in range(3)),np.zeros((dim,dim),dtype=np.int64))
000066 | Q=sum((E[c][d]@E[d][c] for c in range(3) for d in range(3)),np.zeros((dim,dim),dtype=np.int64))
000067 | O=3*N@N-Q
000068 | need('exact_Fock_CAR',all(np.array_equal(ann[i]@ann[j].T+ann[j].T@ann[i],np.eye(dim,dtype=np.int64) if i==j else np.zeros((dim,dim),dtype=np.int64)) for i in range(n) for j in range(n)))
000069 | need('exact_color_singlet_O',all(np.array_equal(O@E[c][d],E[c][d]@O) for c in range(3) for d in range(3)))
000070 | T=[[sum((ann[2*c+i].T@ann[2*c+j] for c in range(3)),np.zeros((dim,dim),dtype=np.int64)) for j in range(2)] for i in range(2)]
000071 | need('exact_internal_basis_invariance',all(np.array_equal(O@T[i][j],T[i][j]@O) for i in range(2) for j in range(2)))
000072 | quartic=np.zeros((dim,dim),dtype=np.int64)
000073 | for a in range(n):
000074 |  for b in range(n):quartic+=3*ann[a].T@ann[b].T@ann[b]@ann[a]
000075 | for c in range(3):
000076 |  for d in range(3):
000077 |   for i in range(2):
000078 |    for j in range(2):quartic+=ann[2*c+i].T@ann[2*d+j].T@ann[2*d+i]@ann[2*c+j]
000079 | need('exact_pure_quartic_normal_ordering',np.array_equal(O,quartic))
000080 | 
000081 | def density2(A):
000082 |  d=s.factor(A.det());r=s.zeros(4);r[0,0]=s.factor((s.eye(2)-A).det());r[3,3]=d
000083 |  r[1:3,1:3]=A-d*s.eye(2)
000084 |  return r
000085 | def fullrho(A):
000086 |  d=density2(A)
000087 |  return s.kronecker_product(d,d,d)
000088 | def expect(rho,M):
000089 |  ii,jj=np.nonzero(M)
000090 |  return s.simplify(sum(rho[int(j),int(i)]*int(M[i,j]) for i,j in zip(ii,jj)))
000091 | A0=s.Matrix([[s.Rational(1,3),s.Rational(1,6)],[s.Rational(1,6),s.Rational(1,2)]])
000092 | A1=s.Matrix([[s.Rational(2,3),-s.I/8],[s.I/8,s.Rational(1,3)]])
000093 | wt=s.Rational(1,3);Abar=wt*A0+(1-wt)*A1
000094 | for i,A in enumerate([A0,A1,Abar]):
000095 |  need('strict_positive_Fock_control_covariance_'+str(i),A.det()>0 and (s.eye(2)-A).det()>0 and A[0,0]>0 and 1-A[0,0]>0)
000096 |  rho=fullrho(A);m=s.trace(A);tau=s.trace(A-A*A)
000097 |  need('exact_normalized_density_'+str(i),s.trace(rho)==1)
000098 |  need('direct_Fock_N2_'+str(i),s.simplify(expect(rho,N@N)-9*m*m-3*tau)==0)
000099 |  need('direct_Fock_color_Casimir_'+str(i),s.simplify(expect(rho,Q)-3*m*m-9*tau)==0)
000100 |  need('direct_Fock_quartic_factor24_'+str(i),s.simplify(expect(rho,O)-24*m*m)==0)
000101 | mixed=wt*fullrho(A0)+(1-wt)*fullrho(A1);gauss=fullrho(Abar)
000102 | var=s.factor(wt*(1-wt)*(s.trace(A0)-s.trace(A1))**2)
000103 | defect=s.factor(expect(mixed-gauss,O))
000104 | need('same_full_two_point_but_nonzero_invariant_four_point',all(expect(mixed-gauss,ann[j].T@ann[i])==0 for i in range(n) for j in range(n)) and defect==24*var and defect>0)
000105 | details.update(exact_Fock_defect=str(defect),exact_Fock_variance=str(var),Fock_dimension=dim)
000106 | 
000107 | # Original96 constructor control with two positive diagnostic H weights.
000108 | # This does not choose the original physical H distribution.
000109 | cl=[C[np.ix_(left[0],left[0])] for C in [Cs[0],Cs[3]]]
000110 | m0,m1=means[0],means[3]
000111 | w0=.5;bar=w0*cl[0]+(1-w0)*cl[1]
000112 | full4=24*(w0*m0*m0+(1-w0)*m1*m1)
000113 | gauss4=24*np.trace(bar).real**2
000114 | need('original96_color_invariant_Wick_defect',abs((full4-gauss4)-24*w0*(1-w0)*(m0-m1)**2)<2e-12 and full4>gauss4)
000115 | details['original96_diagnostic_defect']=float(full4-gauss4)
000116 | need('constant_H_norm_control',abs(means[0]-means[1])<3e-13)
000117 | 
000118 | out=dict(all_passed=all(checks.values()),checks=checks,details=details,physical_H_weight_selected=False,new_lean=False)
000119 | Path(__file__).with_name('HIGGS-MIXTURE-WICK.json').write_text(json.dumps(out,indent=2)+'\n')
000120 | print(json.dumps(dict(all_passed=out['all_passed'],checks=len(checks),details=details)))