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Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/FOCK-DEFECT-REVIEW.txt
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000001 | FOCK DEFECT OF THE RECOVERED CIRCLE SOURCE
000002 | ===========================================
000003 | 
000004 | 25 October 2026. Independent analytic review. This note changes no Atlas,
000005 | index, paper or archived source. It uses the already recovered circle pair and
000006 | the already stated conditional full-CAR/time interface. It does not choose a
000007 | new covariance, twist, one-particle vector or heat operator.
000008 | 
000009 | Verdict
000010 | -------
000011 | 
000012 | For the old circle pair
000013 | 
000014 |   B_a psi_n=(n+a)psi_n,       0<a<1,
000015 |   u psi_n=psi_(n+1),
000016 | 
000017 | put
000018 | 
000019 |   P_+=1_(0,infinity)(B_a),    C=P_-=1_(-infinity,0)(B_a),
000020 |   Pi=|psi_0><psi_0|.
000021 | 
000022 | Then the shifted filled sea has the exact one-particle covariance
000023 | 
000024 |   u C u*=C+Pi.                                           (1)
000025 | 
000026 | Moreover [u,C] has rank one and Hilbert-Schmidt norm one. The Bogoliubov
000027 | automorphism induced by u is therefore implementable in the Fock
000028 | representation of the given sea. With the explicit convention
000029 | 
000030 |   U c_n U*=c_(n+1),                                      (2)
000031 | 
000032 | its implementer is unique up to phase and satisfies
000033 | 
000034 |   U Omega=phase c_0* Omega.                              (3)
000035 | 
000036 | Thus Pi is not a freely inserted rank-one model. It is the exact excess
000037 | two-point covariance of the u-shifted original sea, conditional precisely on
000038 | the already isolated original full-CAR and time identification. This does not
000039 | identify the full later P_adm with P_+, and it does not make U a neutral
000040 | observable.
000041 | 
000042 | The charged implementer itself is not a new result of this note. The actual
000043 | source unit winding W1, its mode action and its Fock implementation were
000044 | already established in
000045 | 
000046 |   experiments/theory-contracts/mixed-current-origin-20261014/
000047 |   ATLAS-KMS-LOCALITY-AND-SOURCE-WINDING.txt:75-164
000048 | 
000049 | and
000050 | 
000051 |   experiments/theory-contracts/seam-joint-ward-state-20261014/
000052 |   PROOF.txt:742-850,1003-1087.
000053 | 
000054 | Those proofs explicitly warned that their SOURCE circle had not been
000055 | identified with the primitive P1 core circle. The contribution checked here
000056 | is the direct operator match with the now recovered fullv21 pair: on the
000057 | common basis exp(i(n+a)theta), its M_z is precisely the one-particle shift
000058 | implemented by the earlier source W1. This is an equality of the displayed
000059 | circle/CAR operators at their stated one-particle and Fock types. A physical map
000060 | from the fullv21 gluing representative into the primitive P1 source remains
000061 | an additional source identification.
000062 | 
000063 | 1. Exact projection and commutator calculation
000064 | -----------------------------------------------
000065 | 
000066 | Since 0<a<1,
000067 | 
000068 |   C H=closed span{psi_n:n<0},
000069 |   P_+ H=closed span{psi_n:n>=0}.
000070 | 
000071 | The range of u C u* is the closed span of psi_m with m<=0. Hence
000072 | 
000073 |   u C u*=C+Pi,       C Pi=Pi C=0.                        (4)
000074 | 
000075 | Since P_+=I-C, the same identity is exactly the previously reconstructed
000076 | Toeplitz defect formula
000077 | 
000078 |   P_+-u P_+ u*=Pi.
000079 | 
000080 | Equivalently, on the complete mode basis,
000081 | 
000082 |   (u C-C u)psi_n = psi_0  if n=-1,
000083 |                     0      otherwise,
000084 | 
000085 | so that
000086 | 
000087 |   [u,C]=|psi_0><psi_(-1)|,
000088 |   rank[u,C]=1,
000089 |   ||[u,C]||_2=1.                                        (5)
000090 | 
000091 | The Shale-Stinespring criterion for CAR therefore applies: the one-particle
000092 | unitary u gives an implementable Bogoliubov automorphism in the pure
000093 | quasifree representation with covariance C. The implementation is projective,
000094 | so its only ambiguity here is an overall phase.
000095 | 
000096 | The reverse shift is genuinely different:
000097 | 
000098 |   u* C u=C-Pi_(-1),       Pi_(-1)=|psi_(-1)><psi_(-1)|,  (6)
000099 |   [u*,C]=-|psi_(-1)><psi_0|.
000100 | 
000101 | It removes one occupied negative mode instead of adding psi_0. Equations
000102 | (4)--(6) also fix the direction of every later charge and energy statement.
000103 | 
000104 | 2. Fock convention, charge and the shifted vacuum
000105 | --------------------------------------------------
000106 | 
000107 | Use the CAR convention of GROUND-STATE-SOURCE-REDUCTION.txt. The filled-sea
000108 | vacuum obeys
000109 | 
000110 |   c_n Omega=0       for n>=0,
000111 |   c_n* Omega=0      for n<0.                            (7)
000112 | 
000113 | Let U satisfy (2), and put Xi=U Omega. From
000114 | 
000115 |   U* c_m U=c_(m-1)
000116 | 
000117 | one obtains
000118 | 
000119 |   c_m Xi=0          for m>=1,
000120 |   c_m* Xi=0         for m<=0.                           (8)
000121 | 
000122 | These are exactly the annihilation relations of the sea in which all n<0
000123 | modes and the additional n=0 mode are occupied. Since the quasifree Fock
000124 | vacuum for a fixed polarization is unique up to phase,
000125 | 
000126 |   Xi=phase c_0* Omega.                                  (9)
000127 | 
000128 | The norm is one because {c_0,c_0*}=1 and c_0 Omega=0.
000129 | Relative to the original sea, Xi has charge +1. With u* and the oppositely
000130 | oriented implementer, the state is the hole
000131 | 
000132 |   phase c_(-1) Omega,
000133 | 
000134 | of relative charge -1. This is the Fock counterpart of the minus sign in
000135 | (6).
000136 | 
000137 | The same result follows directly at the level of two-point functions. With
000138 | 
000139 |   omega_C(c(f)*c(g))=<g,Cf>,
000140 | 
000141 | the vector state of U Omega has
000142 | 
000143 |   <U Omega,c(f)*c(g)U Omega>
000144 |       =<g,u C u* f>
000145 |       =<g,Cf>+<g,Pi f>.                                 (10)
000146 | 
000147 | Thus Pi is an excess one-particle covariance. It is not the rank-one Fock
000148 | projection |Xi><Xi|; the two objects act on different Hilbert spaces.
000149 | 
000150 | 3. Exact positive-energy line, with no heat limit
000151 | -------------------------------------------------
000152 | 
000153 | Assume the already stated source interface: the original positive-energy
000154 | Fock time restricts to
000155 | 
000156 |   alpha_t(c(f))=c(exp(i t B_a)f),
000157 |   alpha_t(A)=exp(i t H) A exp(-i t H),
000158 |   H>=0,       H Omega=0.                                (11)
000159 | 
000160 | This assumption is the existing full-CAR/time bridge. Under it the ground
000161 | state is forced to have C=P_-; no second quasifree state is chosen here.
000162 | 
000163 | Since B_a psi_0=a psi_0, anti-linearity of c and (11) give
000164 | 
000165 |   exp(i t H)c_0*Omega=exp(i t a)c_0*Omega,
000166 |   H c_0*Omega=a c_0*Omega.                              (12)
000167 | 
000168 | Consequently the Euclidean positive semigroup acts exactly by
000169 | 
000170 |   exp(-tH)c_0*Omega=exp(-ta)c_0*Omega.                  (13)
000171 | 
000172 | If
000173 | 
000174 |   Pi_F=|c_0*Omega><c_0*Omega|,
000175 | 
000176 | then
000177 | 
000178 |   Pi_F exp(-tH) Pi_F=exp(-ta)Pi_F.                      (14)
000179 | 
000180 | This is an exact reduced transition line, not an infrared or large-t heat
000181 | limit. For the reversed shift the hole c_(-1)Omega has energy
000182 | 
000183 |   -(a-1)=1-a.                                           (15)
000184 | 
000185 | Equations (12)--(15) use the Fock Hamiltonian H and its original time. They
000186 | do not identify the one-particle B_a with H, and they do not identify the
000187 | later electromagnetic response operator with Pi_F or Pi.
000188 | 
000189 | 4. Relative Fredholm determinant and its orientation
000190 | -----------------------------------------------------
000191 | 
000192 | The ordinary determinants det(I-zC) and det(I-z(C+Pi)) are not defined as
000193 | Fredholm determinants: C has infinite rank, so the individual perturbations
000194 | of the identity are not trace class. Their directed quotient is well-defined.
000195 | For z different from 1, set
000196 | 
000197 |   A(z)=I-zC,
000198 |   A_u(z)=I-z(C+Pi).
000199 | 
000200 | Using C Pi=0 and Pi^2=Pi gives