TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/NEUTRAL-DEFECT-PAIRS-AND-VECTOR-WARDS.txt
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000076 | 4. Complete-vector Ward test: the next genuine obstruction
000077 | 
000078 | This paragraph uses the ORIGINAL FULL PRODUCT DIRAC/CAR SOURCE, not an
000079 | unproved identification of that source with the QWZ strip. On one boundary
000080 | face its two DIRAC-GRADING components have covariances C and1-C. They are
000081 | NOT the two strip edges and NOT the two collar faces.
000082 | 
000083 | For implementable smooth commuting pointwise vector profiles H_i h and fI,
000084 | the complete Lagrangian Cauchy source satisfies
000085 |   N C_full N*=1-C_full, [N,H_i h]=[N,fI]=0.
000086 | The original normal-complement lemma gives opposite cocycles for the two
000087 | grading components, so the vector current obeys
000088 |   [J_i,vec(h), J_vec(f)] = 0.                         (4)
000089 | The noncentral term is zero because the pointwise profiles commute.
000090 | For their smooth one-parameter phase implementations on the common
000091 | analytic domain (and hence their bounded Weyl forms),
000092 |   [J_i,vec(h), V_vec(f)] = 0,
000093 |   V_vec(f)=exp(i J_vec(f)).                            (5)
000094 | This is an infinitesimal/equal-time Ward assertion; zero curvature alone
000095 | is not being promoted to a global holonomy theorem for arbitrary loops.
000096 | 
000097 | In the separately given chiral current normalization
000098 |   [J_i,+(h),J_+(f)] = i/(2pi) integral h f',
000099 | while the opposite grading component has the negative form. For a smooth
000100 | approximation to f=pi 1_(x,y), the endpoint coefficient tends to
000101 |   (+/-) (h(x)-h(y))/2.                                (6)
000102 | The sign of the corresponding commutator with exp(iJ(f)) depends on the
000103 | chosen implementer convention; the two component residues are opposite
000104 | and their vector sum is EXACTLY ZERO. This is the decisive convention-
000105 | independent content.
000106 | 
000107 | A target spinor component has every Cartan charge lambda_i=+/-1/2.
000108 | An endpoint pair S_lambda(x) S_lambda*(y) must have Ward
000109 |   lambda_i (h(x)-h(y)) times the pair amplitude.        (7)
000110 | The complete vector action(5) cannot have(7) with nonzero pair amplitude.
000111 | All eight Cartan tests are necessary: testing only total U1 would miss
000112 | spinor weights whose total charge is zero.
000113 | 
000114 | Scalar renormalization, smearing and positive joint word limits cannot
000115 | change the exact zero in(5) IF the same vector Ward insertions/current
000116 | action are retained. Thus these COMPLETE VECTOR phase histories cannot
000117 | automatically be identified as the desired charged spinor pair of those
000118 | same vector currents. This does NOT exclude charged spinor fields relative
000119 | to a CHIRAL current subnet, a shifted sector representation, other
000120 | originally derived operations, or a different physical measure.
000121 | 
000122 | For a finite CAR carrier there is no bosonization assumption in the phase
000123 | operator identity: Gamma(exp(if))=exp(i dGamma(f)), and
000124 | J_C(f)=dGamma(f)-Tr(Cf)I changes it only by a scalar. In particular f=pi E
000125 | gives actual parity. For the full smooth Dirac representation use its
000126 | specified normal-ordered implementers. No equality of an UNCONSTRUCTED
000127 | singular continuum limit of the lattice parity with a chiral vertex is
000128 | inferred from these identities.
000129 | 
000130 | 5. Finite complementary determinant: exact amplitude control
000131 | 
000132 | For ANY finite Hermitian covariance 0<=C<=1 and unitary g define
000133 |   chi_C(g)=det(I-C+Cg).
000134 | Let A=I-C+gC. Sylvester's determinant identity gives det A=chi_C(g).
000135 | Since
000136 |   A* g=(I-C)g+C,
000137 | taking determinants proves
000138 |   chi_(1-C)(g)=det(g) conjugate(chi_C(g)).              (8)
000139 | No commutation of C and g is required. If g=exp(if), the two normal-order
000140 | phases multiply to exp(-i Tr f). Consequently
000141 |   exp(-i Tr(Cf)) chi_C(g)
000142 |   times exp(-i Tr((1-C)f)) chi_(1-C)(g)
000143 |     = |chi_C(g)|^2.                                   (9)
000144 | This applies to the TWO GRADING components on ONE Dirac boundary face.
000145 | 
000146 | Equation(9) is exact and finite, including nonprojective C=n(A). It is NOT
000147 | an unregulated determinant formula for a typeIII local algebra or for
000148 | arbitrary mixed temporal words. A continuum extension needs its own trace/
000149 | renormalization and domain control. The analytic Ward result(4)-(7) uses
000150 | the original current theorem, independently of that extension.
000151 | 
000152 | The checker uses both original quarter A spectra and the actual face
000153 | occupations n(A) at L=2 log2, making them rational. A finite unitary Cayley
000154 | control has a nonzero individual component phase, which cancels exactly
000155 | in(8)-(9). It is a finite algebra control, not a new P1 boundary choice.
000156 | Klich, arXiv:cond-mat/0209642, provides the standard finite Fock-trace/
000157 | single-particle determinant background; (8) is proved directly above.