TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/check_gauge_locality_completion.py
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Inclusive original lines: 1-197
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000001 | """Exact control of a CONDITIONAL locality change, not its P1 derivation.
000002 | 
000003 | Question: can a fermion-parity Gauss completion make the existing half flux
000004 | endpoint-local while preserving the full even physical Hilbert space and
000005 | the original register Hamiltonian? The proof is all-size; this small exact
000006 | CAR control checks its signs with the actual v1033 hopping coefficients.
000007 | The first new premise is the Gauss/local-observable rule, not a new energy.
000008 | """
000009 | from pathlib import Path
000010 | import hashlib
000011 | import importlib
000012 | import json
000013 | import sys
000014 | import numpy as np
000015 | import sympy as s
000016 | 
000017 | [OMITTED: non-scientific conversation or execution metadata]
000018 | sys.path.insert(0,str(ROOT/'verification'))
000019 | v1033=importlib.import_module('v1033_charged_disorder')
000020 | 
000021 | 
000022 | def need(value,label):
000023 |     if not value:raise ValueError(label)
000024 | 
000025 | 
000026 | def zero(matrix):
000027 |     return all(s.expand(x)==0 for x in matrix.todok().values())
000028 | 
000029 | 
000030 | def annihilator(m,k):
000031 |     out=s.MutableSparseMatrix(2**m,2**m,{})
000032 |     for bits in range(2**m):
000033 |         if bits&(1<<k):
000034 |             out[bits^(1<<k),bits]=(-1)**((bits&((1<<k)-1)).bit_count())
000035 |     return out
000036 | 
000037 | 
000038 | def exact_matrix(np_matrix):
000039 |     return s.SparseMatrix([[s.nsimplify(x) for x in row] for row in np_matrix])
000040 | 
000041 | 
000042 | def second_quantize(h,ops):
000043 |     out=s.zeros(ops[0].rows)
000044 |     for j in range(h.rows):
000045 |         for k in range(h.cols):
000046 |             if h[j,k]:out+=h[j,k]*ops[j].T*ops[k]
000047 |     return s.SparseMatrix(out)
000048 | 
000049 | 
000050 | # Three columns, one transverse site, both actual QWZ orbital components.
000051 | # This is solely an exact group/CAR/sign control, not the physical width8
000052 | # source or a new source model. The analytic isometry works for arbitrary
000053 | # column Hilbert spaces, onsite terms and original hopping coefficients.
000054 | L,ny=(int(sys.argv[1]) if len(sys.argv)>1 else 3),1
000055 | m=2*L*ny
000056 | dim=2**m
000057 | ops=[annihilator(m,k) for k in range(m)]
000058 | even=[n for n in range(dim) if n.bit_count()%2==0]
000059 | states=[(r,n) for r in range(4) for n in even]
000060 | pos={a:i for i,a in enumerate(states)}
000061 | 
000062 | 
000063 | def col_parity(n,j):
000064 |     mask=((1<<(2*ny))-1)<<(2*ny*j)
000065 |     return (n&mask).bit_count()%2
000066 | 
000067 | 
000068 | def ext_index(b,z,n):
000069 |     return (b*2**L+z)*dim+n
000070 | 
000071 | 
000072 | extended=2*2**L*dim
000073 | E=s.MutableSparseMatrix(extended,len(states),{})
000074 | for (r,n),col in pos.items():
000075 |     b,w=r%2,r//2
000076 |     for tbits in range(2**(L-1)):
000077 |         t=[0]+[(tbits>>(j-1))&1 for j in range(1,L)]
000078 |         z=sum((((w if j==L-1 else 0)+t[j]+t[(j+1)%L])%2)<<j for j in range(L))
000079 |         phase=(-1)**sum(t[j]*col_parity(n,j) for j in range(1,L))
000080 |         E[ext_index(b,z,n),col]=phase/s.sqrt(2**(L-1))
000081 | E=s.SparseMatrix(E)
000082 | need(E.T*E==s.eye(len(states)),'isometry on FULL even register-matter carrier')
000083 | 
000084 | gauss=[]
000085 | electric=[]
000086 | for j in range(L):
000087 |     G=s.MutableSparseMatrix(extended,extended,{})
000088 |     X=s.MutableSparseMatrix(extended,extended,{})
000089 |     for b in range(2):
000090 |         for z in range(2**L):
000091 |             for n in range(dim):
000092 |                 col=ext_index(b,z,n)
000093 |                 G[ext_index(b,z^(1<<j)^(1<<((j-1)%L)),n),col]=(-1)**col_parity(n,j)
000094 |                 X[ext_index(b,z^(1<<j),n),col]=1
000095 |     G,X=s.SparseMatrix(G),s.SparseMatrix(X)
000096 |     need(G*E==E,'all Gauss constraints, including anchor, on even carrier')
000097 |     gauss.append(G);electric.append(X)
000098 | 
000099 | # Assemble the exact original dGamma(h_r) from the current builder.
000100 | Hblocks=[second_quantize(exact_matrix(v1033.qwz_cylinder(L,ny,1.,r)),ops) for r in range(4)]
000101 | H_old=s.MutableSparseMatrix(len(states),len(states),{})
000102 | for r in range(4):
000103 |     for n in even:
000104 |         for nprime in even:
000105 |             value=Hblocks[r][nprime,n]
000106 |             if value:H_old[pos[r,nprime],pos[r,n]]=value
000107 | H_old=s.SparseMatrix(H_old)
000108 | 
000109 | # The gauged hopping is obtained from the SAME coefficients: link signs z_j
000110 | # multiply the forward QWZ hopping at every j; i^b remains at the seam.
000111 | H_new=s.MutableSparseMatrix(extended,extended,{})
000112 | TX=exact_matrix(v1033.TX)
000113 | for b in range(2):
000114 |     for z in range(2**L):
000115 |         h=exact_matrix(v1033.qwz_cylinder(L,ny,1.,b))
000116 |         for x in range(L):
000117 |             amp=(s.I**b if x==L-1 else 1)
000118 |             change=((-1)**((z>>x)&1)-1)*amp
000119 |             if change:
000120 |                 for a in range(2):
000121 |                     for c in range(2):
000122 |                         i,j=2*((x+1)%L)+a,2*x+c
000123 |                         h[i,j]+=change*TX[a,c]
000124 |                         h[j,i]+=s.conjugate(change*TX[a,c])
000125 |         block=second_quantize(h,ops)
000126 |         for n in range(dim):
000127 |             for nprime in range(dim):
000128 |                 if block[nprime,n]:H_new[ext_index(b,z,nprime),ext_index(b,z,n)]=block[nprime,n]
000129 | H_new=s.SparseMatrix(H_new)
000130 | need(zero(H_new*E-E*H_old),'EXACT original physical Hamiltonian and all time words')
000131 | need(all(H_new*G==G*H_new for G in gauss),'local Gauss constraints commute with Hamiltonian')
000132 | 
000133 | W=s.diag(*[s.I**b*(-1)**z.bit_count() for b in range(2) for z in range(2**L) for n in range(dim)])
000134 | Zold=s.diag(*[s.I**r for r,n in states])
000135 | need(W*E==E*Zold,'old local register Z becomes global Wilson loop')
000136 | 
000137 | for endpoint in range(L):
000138 |     F=s.MutableSparseMatrix(len(states),len(states),{})
000139 |     for (r,n),col in pos.items():
000140 |         parity=sum(col_parity(n,x) for x in range(endpoint+1))%2
000141 |         F[pos[(r+2)%4,n],col]=(-1)**parity
000142 |     need(electric[endpoint]*E==E*F,'existing half string becomes LOCAL electric endpoint')
000143 | 
000144 | # Any nontrivial register-grade q continues to obstruct old local inclusion.
000145 | X4=s.zeros(4)
000146 | for r in range(4):X4[(r+1)%4,r]=1
000147 | Z4=s.diag(1,s.I,-1,-s.I)
000148 | need(all(Z4*X4**q*Z4.conjugate().T==s.I**q*X4**q for q in range(4)),
000149 |      'all register grades survive arbitrary matter dressing and homogeneous words')
000150 | 
000151 | # Independently check the one-particle gauge covariance at the ACTUAL width8.
000152 | # All values are binary rationals times powers of i, so these identities are
000153 | # exact in floating representation, not tolerance-based spectral checks.
000154 | def signed_cylinder(nx,ny,b,z):
000155 |     h=v1033.qwz_cylinder(nx,ny,1.,b).copy()
000156 |     for x in range(nx):
000157 |         if (z>>x)&1:
000158 |             change=-2*(1j**b if x==nx-1 else 1)
000159 |             for y in range(ny):
000160 |                 i=2*((x+1)%nx*ny+y)
000161 |                 j=2*(x*ny+y)
000162 |                 h[i:i+2,j:j+2]+=change*v1033.TX
000163 |                 h[j:j+2,i:i+2]+=np.conj(change)*v1033.TX.conj().T
000164 |     return h
000165 | 
000166 | width8_cases=0
000167 | for b in range(2):
000168 |     for z in range(8):
000169 |         h=signed_cylinder(3,8,b,z)
000170 |         for x in range(3):
000171 |             u=np.ones(48)
000172 |             u[16*x:16*(x+1)]=-1
000173 |             znew=z^(1<<x)^(1<<((x-1)%3))
000174 |             need(np.array_equal(u[:,None]*h*u[None,:],signed_cylinder(3,8,b,znew)),
000175 |                  'actual width8 QWZ gauge covariance with SAME coefficients')
000176 |             width8_cases+=1
000177 | 
000178 | result={
000179 |     'date':'2026-10-22',
000180 |     'verdict':'EXACT_GAUSS_LOCALITY_COMPLETION_CONDITIONAL_NOT_P1_DERIVED',
000181 |     'control_columns':L,'control_transverse_sites':ny,
000182 |     'original_even_physical_dimension':len(states),
000183 |     'extended_redundant_dimension':extended,
000184 |     'isometry_and_all_Gauss_constraints_exact':True,
000185 |     'same_original_QWZ_hopping_coefficients_and_physical_H_exact':True,
000186 |     'half_flux_maps_to_local_electric_link_operator_exact':True,
000187 |     'old_register_Z_maps_to_GLOBAL_Wilson_loop_exact':True,
000188 |     'actual_width8_one_body_gauge_covariance_cases':width8_cases,
000189 |     'no_new_physical_dimension_or_energy_coefficient':True,
000190 |     'first_new_assumption':'Local Gauss constraints and gauge-invariant local observable assignment; not supplied by original P1.',
000191 |     'full_Dirac_chiral_measure_selected':False,
000192 |     'second_edge_removed':False,
000193 |     'E8_spinor_origin_derived':False,
000194 |     'source_sha256':hashlib.sha256((ROOT/'verification/v1033_charged_disorder.py').read_bytes()).hexdigest(),
000195 | }
000196 | Path(__file__).with_name('gauge-locality-completion-results.json').write_text(json.dumps(result,indent=2)+'\n')
000197 | print(json.dumps(result,indent=2))