TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/QUARTER-STRESS-CURRENT-RECOVERY.txt
Original file SHA-256: 5861c8516e38bd1ed303622e019c7bef182d9312a25142bc754456beac573141
Inclusive original lines: 33-115
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000033 | Put p=alpha-1/2. On the same CAR finite-energy core,
000034 | 
000035 |   L_n^alpha=L_n^(1/2)+p J_n+(p^2/2) delta_(n,0).
000036 | 
000037 | This includes the original relative charge Q=J0. At the two quarter values,
000038 | p_plus=1/4 and p_minus=-1/4, the quadratic constants agree. Therefore
000039 | 
000040 |   J_n=2(L_n^(3/4)-L_n^(1/4))                           (1)
000041 | 
000042 | for EVERY integer n, including n=0 and nonzero charge sectors.
000043 | The exact coefficient search imposes equality of the original CAR matrix
000044 | units for symbolic r,n. It finds the unique solution (2,-2). A stochastic
000045 | genetic search is unnecessary for this affine identity: exact elimination
000046 | decides the entire declared coefficient class at once.
000047 | 
000048 | Equation (1) initially compares two families on the SAME CAR generators.
000049 | It does not permit subtraction of unrelated operators merely because both
000050 | represent c=1 Virasoro algebras with lowest weight1/32.
000051 | 
000052 | 3. The actual winding supplies that comparison on the old full cover
000053 | --------------------------------------------------------------------
000054 | Write the already given source as Hcov=H0 direct-sum H1, with H0=W0K and
000055 | H1=W1K. On K direct-sum K the source winding is
000056 | 
000057 |   U=u_cov=[[0,u_base],[I,0]],       u_base e_m=e_(m+1).
000058 | 
000059 | The polarization is C=C0 direct-sum C1, with both Cj the sea m<0.
000060 | Only the wrap arrow contains u_base, so [C,U] has rank one. The already
000061 | used CAR implementability theorem gives a Fock unitary V=Gamma(U), unique
000062 | up to phase. Ad(V) is independent of that phase.
000063 | 
000064 | On the FIRST arrow, H0 to H1, U is exactly I in the common K chart.
000065 | Cover modes 2m go to2m+1. Their occupation signs agree for every m:
000066 | 
000067 |   2m<0 iff 2m+1<0.
000068 | 
000069 | Thus pulling the second factor's CAR matrix units back by Ad(V)^(-1)
000070 | preserves their Wick subtraction. It also preserves the vacuum state on
000071 | that factor. Consequently the literal full-cover identity is
000072 | 
000073 |   J_0(f)=2(Ad(V)^(-1)(T_(3/4)(f))-T_(1/4)(f)),          (2)
000074 | 
000075 | where subscript0 on J labels H0, not a Fourier zero mode.
000076 | 
000077 | The second arrow goes from2m+1 to2m+2. At m=-1 it crosses the sea boundary.
000078 | That rank-one charge carry is retained. In particular V does NOT preserve
000079 | the full cover vacuum. Its state-preserving restriction used in (2) and
000080 | its global charged action are different statements and are compatible.
000081 | 
000082 | 4. The local algebra, with its real support
000083 | ---------------------------------------------
000084 | Let I be a proper interval of the base circle x=2theta. The fields of each
000085 | character factor over I are supported on p^(-1)(I), the two preimage arcs
000086 | of the cover. U=e^(i theta) multiplies pointwise and does not enlarge support.
000087 | 
000088 | Let M_pair(I) be the von Neumann algebra generated by both stress unitary
000089 | groups in (2), for smooth real f supported in I. On their common core the
000090 | difference is J0(f). For the existing current representation, J(f) and the
000091 | Sugawara stress are essentially self-adjoint on the finite-energy core.
000092 | The same holds for their real shifted stress families. These are the
000093 | standard current-net facts stated in Tanimoto, arXiv:1807.11723v2, sections
000094 | 3.1 and3.2. Local current conjugation gives the shifted stress within the
000095 | same current net, so M_pair(I) is contained in A_J0(I).
000096 | 
000097 | Conversely, use the unitary product formula on the common core:
000098 | 
000099 |  exp(i t J0(f))
000100 |    =strong-lim_N [ Ad(V)^(-1)(exp(i2t T_(3/4)(f)/N))
000101 |                     exp(-i2t T_(1/4)(f)/N) ]^N.         (3)
000102 | 
000103 | The products are stable with norm1. Their derivative on the core is iJ0(f),
000104 | whose closure is the generator of a unitary group. The Trotter/Chernoff
000105 | criterion applies; this is not a formal BCH truncation. Strong closure
000106 | therefore proves the reverse inclusion:
000107 | 
000108 |   M_pair(I)=A_J0(I).                                   (4)
000109 | 
000110 | This is a local algebra statement in the existing source. It does not
000111 | identify stress control parameters with physical time or claim a performed
000112 | pulse protocol. The later eightfold W8 support must still be transported
000113 | through its existing field dictionary. Equation (4) alone is not a claim
000114 | that a scalar original current subnet already contains every multi-arc
000115 | CAR charge transporter or every folded field.