TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/FIELD-PROJECTOR-MAP.txt
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000001 | K: ACTUAL INTERNAL FIELD MAP AND ADMISSIBLE PROJECTOR TEST
000002 | 3 November 2026. Independent relational_logic review.
000003 | Entry: plan92; Atlas243787ad18ac2965cbccd2bf. Canonical sources read only.
000004 | 
000005 | RESULT
000006 | 
000007 | There is an explicit positive operator bridge between the original
000008 | S+=Lambda-even C^5 and the later DET16 one-particle Spin(10) representation.
000009 | It intertwines every one of the 45 generators, not just their dimensions.
000010 | Its induced CAR map is defined on the entire 65536 dimensional Fock space.
000011 | The signed change of basis has determinant +1, so its induced full
000012 | 16-fermion determinant carries no extra convention-dependent phase.
000013 | 
000014 | This bridge identifies INTERNAL field components. The original spacetime
000015 | spin, smear, polarization, time, reflection and source state remain the
000016 | original ones. They cannot be replaced by a single onsite 16-mode system
000017 | through counting the components. The source-leg statement below makes the
000018 | precise valid extension and the still distinct transfer identity explicit.
000019 | 
000020 | The actual original definition of the physical admissible net gives a
000021 | second, decisive simplification. It consists of restrictions of operators
000022 | which, together with their adjoints, preserve Ran(P_adm). If all sixteen
000023 | BARE CAR modes of a putative mirror tensor factor are such unchanged
000024 | physical operators, then necessarily
000025 | 
000026 |                  P_adm = I_mirror tensor P_rest.
000027 | 
000028 | Consequently an identification of P_adm with a rank-one DET vacuum selector
000029 | on that same factor would be inconsistent with those bare field legs.
000030 | The original texts do not make this identification. DET ground projection,
000031 | physical admissibility projection, color-center neutrality and determinant
000032 | involution must therefore remain different objects until a source map
000033 | actually relates them. Dressed fields acting on the rest, and charged
000034 | field extensions, are not excluded by this result.
000035 | 
000036 | 1. THREE DIFFERENT EXTERIOR ALGEBRAS / FIBRES
000037 | 
000038 | Original Carrier002:197-325 fixes
000039 | 
000040 |  E=E_3+E_2, S+=Lambda^0 E + Lambda^2 E + Lambda^4 E,
000041 |  dim S+=16,
000042 |  S+=(1,1)_0 +(3,2)_{1/6} +(3bar,1)_{-2/3}
000043 |        +(3bar,1)_{1/3} +(1,2)_{-1/2} +(1,1)_1.
000044 | 
000045 | Its exterior occupation words have FIVE bits. The original table explicitly
000046 | uses left-chiral notation. These are internal family components, not all
000047 | the spacetime spin/polarization modes of a four-dimensional Weyl field.
000048 | 
000049 | v474 exhibits the Clifford action on Lambda(C^5), dimension32, and restricts
000050 | the even Clifford algebra to S+, dimension16. Its a_k, a_k^* (k=1,...,5)
000051 | are auxiliary carrier exterior operators. They are not the sixteen DET
000052 | physical CAR letters c_i, c_i^*.
000053 | 
000054 | v1002 takes that 16-dimensional Weyl representation as its ONE-PARTICLE
000055 | space and then forms Lambda(S+), dimension2^16. The empty vector in
000056 | Lambda(S+) is distinct from the internal Lambda^0(C^5) basis component,
000057 | which is the original nu^c component of S+. In particular, that internal
000058 | nu^c component is one of the sixteen creators' labels, not the DET empty
000059 | state. This distinction removes an otherwise easy vacuum/field mismatch.
000060 | 
000061 | 2. EXPLICIT ACTUAL OPERATOR INTERTWINER, CHECKED ON ALL GENERATORS
000062 | 
000063 | Use v474's ordered wedge basis e_I, I subset {1,...,5}, with subsets first
000064 | ordered by degree and then lexicographically. It uses
000065 | 
000066 |  a_k^* e_I = (-1)^{#{j in I:j<k}} e_{I union {k}},
000067 |  G_{2k}=a_k^*+a_k, G_{2k+1}=i(a_k^*-a_k).
000068 | 
000069 | v1002 instead uses the literal suffix Jordan-Wigner matrices
000070 | 
000071 |  g_{2k}=I^(k-1) tensor X tensor Z^(5-k),
000072 |  g_{2k+1}=I^(k-1) tensor Y tensor Z^(5-k).
000073 | 
000074 | The full32-dimensional unitary is exactly
000075 | 
000076 |  U e_I = (-1)^{|I|(|I|-1)/2} |b(I)>,
000077 |  b(I)=sum_{k in I}2^{5-k}.
000078 | 
000079 | For creation on a degree-r vector the sign quotient is (-1)^r; for
000080 | annihilation it is (-1)^{r-1}. This converts the prefix exterior sign
000081 | into the suffix Jordan-Wigner sign. Hence U G_a U^*=g_a for all ten a.
000082 | The literal chirality (-i)^5 g_0...g_9 is Z tensor...tensor Z, so its
000083 | positive eigenspace is precisely the even-bit subspace.
000084 | 
000085 | Put W=U|S+. Then for every a<b,
000086 | 
000087 |  W [ i(G_a G_b-G_b G_a)/4 ] W^* = (i g_a g_b/2)|positive.
000088 | 
000089 | This includes the convention conversion: v474 records anti-Hermitian
000090 | bilinears [G_a,G_b]/4; v1002 records i times these generators.
000091 | There is no unresolved sign or conjugate Weyl representation in this map.
000092 | 
000093 | The actual old functions were imported read only with Python -B. The single
000094 | comparison returned exit0 with full-Clifford residual0, all45 residual0,
000095 | unitarity residual0 and identical positive chirality bit list. The complete
000096 | 16-entry signed basis dictionary is in INTERNAL-CLIFFORD-MAP.json.
000097 | The basis permutation has60 inversions and the degree-two block has10
000098 | negative signs; therefore det W=+1 exactly, consistent with the output1.
000099 | This was a comparison of existing operator definitions, not a new model.
000100 | 
000101 | The induced full Fock unitary is Gamma(W)=direct-sum_N Lambda^N W. It obeys
000102 | 
000103 |  Gamma(W) a^*(v) Gamma(W)^* = c^*(Wv),
000104 |  Gamma(W) dGamma(T) Gamma(W)^* = dGamma(W T W^*).
000105 | 
000106 | Both equations follow directly on decomposable wedges, so they cover ALL
000107 | CAR words, not only covariance. Gamma(W)|0>=|0> and its top-sector action
000108 | is detW=1. Thus this particular convention change preserves Omega=|0><F|
000109 | and does not select or shift the already declared DET phase theta.
000110 | 
000111 | Novelty: the carrier Clifford representation and the later Weyl generators
000112 | already exist (v474, v1002). The explicit all-basis signed intertwiner and
000113 | the comparison of these two literal conventions are the present synthesis.
000114 | Neither result derives a new physical transfer Hamiltonian.
000115 | 
000116 | 3. WHAT THIS MAP DOES TO THE ACTUAL SOURCE LEGS
000117 | 
000118 | Let Psi(f tensor v) denote an original smeared field with its actual
000119 | spacetime/Lorentz-spin test section f and internal v in S+. The legitimate
000120 | internal coordinate change is
000121 | 
000122 |  Psi(f tensor v) -> Psi'(f tensor Wv).
000123 | 
000124 | It leaves f, its support and spacetime spin untouched. Conjugate source
000125 | slots use the conjugate/adjoint map. The complete operator, its domain,
000126 | inner product, reference and anti-linear reflection must be transported,
000127 | as already proved for the full original source in G. If the same internal
000128 | frame is used, the map is constant. A field-dependent frame requires the
000129 | already familiar derivative connection; it may not be dropped.
000130 | 
000131 | If an ACTUAL original equal-time CAR subspace j:S+->H_one has been chosen
000132 | isometrically, the sixteen letters Psi(j e_i) generate precisely the full
000133 | Lambda(S+) CAR algebra above, and W acts through j. This is a sufficient
000134 | kinematical embedding. It does not identify a spacetime continuum with
000135 | one site, choose a smear/polarization, or change its time generator.
000136 | An original Euclidean field insertion before equal-time reconstruction
000137 | should not simply be called a bounded annihilation mode.
000138 | 
000139 | For the literal later DET parent H_m=2(I-P_phi), the known R104 source has
000140 | 
000141 |  a=e^(-2 beta), b=1-a, d=65536, Z_m=da+b,
000142 |  <c_i^*c_j>=delta_ij/2,
000143 |  <n_i n_j>=1/4+b/(4Z_m) (i!=j),
000144 |  <Omega>=b e^(i theta)/(2Z_m).
000145 | 
000146 | Thus after fixing an actual common source-mode map, one original
000147 | occupation insertion already tests the necessary half-occupation identity;
000148 | the two-density or Omega insertion distinguishes its non-Gaussian source
000149 | from a covariance-only match. These are exact candidate fingerprints from
000150 | R104, not values asserted for an uncomputed original smear. W alone does