TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/SPIN-FIELD-TEST.txt Original file SHA-256: d4d4a30b409b43aa8ea88913cdd52d6213139d800bc65f234c5d624cb58052a2 Inclusive original lines: 1-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | K: SPATIAL SPIN TEST OF THE LITERAL SAME-WEYL-FIELD DET16 VERTEX 000002 | 3 November2026. relational_logic independent analytical test. 000003 | Read-only canonical sources; no new model or numerical execution. 000004 | 000005 | DECISION 000006 | 000007 | The pure internal epsilon16 determinant made from the SAME local 000008 | left-Weyl family fields is a spatial spin8 tensor, not a rotation scalar. 000009 | Consequently a nonzero scalar coefficient times this literal vertex plus 000010 | its adjoint cannot be a rotation-invariant derivative-free local term on 000011 | those field legs. This tests the first physical group action after the 000012 | already successful INTERNAL Spin(10) basis map W. 000013 | 000014 | It does not invalidate the native DET16 lattice construction, whose modes 000015 | were not supplied with this physical Lorentz-spin action. Nor is it an 000016 | exclusion of all sixteen-fermion invariants or any TFPT/IR realization. 000017 | The excluded identification is specified precisely below. 000018 | 000019 | 1. ACTUAL ORIGINAL FIELD TYPE AND THE EXTRA IDENTIFICATION UNDER TEST 000020 | 000021 | Original v45Carrier002:197-292 calls S+=Lambda-even(E_3+E_2) one chiral 000022 | Standard Model family including nu^c, explicitly in left-chiral notation. 000023 | Its table counts 16 INTERNAL family components. The original technical 000024 | companion008:24-29 states explicitly that the packet has SIX Weyl field 000025 | types and 16 chiral states per family; internal color/weak multiplicities 000026 | account for the difference. Thus '16' does not count the two Lorentz-spinor 000027 | components of each Weyl field. 000028 | 000029 | Original QFT001:177-199 separately uses the physical Euclidean kinetic 000030 | Dirac operator and chiral projectors (1-/+gamma5)/2 in the quark sectors. 000031 | These confirm that internal half-spinor and physical spacetime chirality 000032 | are distinct slots. The argument here takes the original physical Weyl 000033 | readout literally; it does not derive a 4D field theory from a count. 000034 | 000035 | On a four-dimensional rotationally covariant branch write one family as 000036 | 000037 | psi_{alpha i}(x), alpha in S=C^2, i in V=S+. 000038 | 000039 | Spatial rotations act on S by the fundamental SU(2) representation and 000040 | on the spacetime argument, while the internal Spin(10)/SM action acts on 000041 | V. This is the usual Weyl transformation law, independently checked in 000042 | the primary lecture source cited below. 000043 | 000044 | The ADDITIONAL map being tested identifies the native sixteen DET letters 000045 | with these same local Weyl fields at one common spatial/orbital location, 000046 | without derivatives, another orbital tensor, a spin-carrying rest 000047 | coefficient, or a fixed directional background compensating their spin. 000048 | The actual DET Hamiltonian has scalar coefficient I_R multiplying Omega 000049 | and Omega^*, as in its original H_R+2(I-P_phi) definition. W fixes only V; 000050 | it does not supply the additional spacetime-spin identification above. 000051 | 000052 | More precisely, V=S+ is the irreducible complex internal half-spinor. 000053 | Schur's lemma gives 000054 | 000055 | Hom_Spin(10)(V,C^2_spin tensor V)=C^2_spin tensor C I_V. 000056 | 000057 | Consequently EVERY Spin(10)-equivariant point-local linear embedding has 000058 | the form j(v)=chi tensor v. For an isometric CAR embedding chi has norm1. 000059 | Its range is chi tensor V; it is not invariant under all spatial SU(2), 000060 | because SU(2) has no invariant line in C^2. This proves the initial 000061 | identification problem before choosing any Hamiltonian. The successful 000062 | internal W and Gamma(W) are retained; a full same-local-Weyl mode map also 000063 | has to say what happens to chi under rotations. This statement assumes 000064 | the declared full internal Spin(10) intertwining and the same local spin 000065 | slot; it does not impose that group as an additional physical gauge law. 000066 | 000067 | 2. EXACT GRASSMANN EXCHANGE TEST 000068 | 000069 | Form the candidate internal determinant tensor at a regulated point: 000070 | 000071 | T_{alpha1...alpha16} 000072 | = sum_{i1,...,i16} epsilon^{i1...i16} 000073 | psi_{alpha1 i1} ... psi_{alpha16 i16}. 000074 | 000075 | Exchange any adjacent alpha slots. Relabel the corresponding two dummy 000076 | internal indices. The epsilon tensor gives one minus sign; restoring the 000077 | original fermion order gives another. Therefore 000078 | 000079 | T_{...alpha beta...}=T_{...beta alpha...}. 000080 | 000081 | Thus the spin tensor lies in Sym^16(S), not Lambda^16(S), even though it 000082 | uses the alternating internal determinant. The statement is a finite 000083 | Grassmann/CAR polynomial identity; no coincident-distribution product is 000084 | assumed without a regulator. A covariant local renormalized version, if 000085 | defined, has the same representation type. Different smears/orbital legs 000086 | can carry additional permutation/spin data and are outside this test. 000087 | 000088 | The tensor is not identically zero: its all-plus component is 000089 | 000090 | T_{+...+}=16! psi_{+1}...psi_{+16}. 000091 | 000092 | It is a nonzero monomial in the regulated independent Weyl letters. 000093 | 000094 | 3. NO ROTATION SCALAR IN THIS ENTIRE TENSOR CLASS 000095 | 000096 | Represent Sym^16(C^2) by homogeneous polynomials in z_+,z_-. Its torus 000097 | weights are m=8,7,...,-8. In the convention 000098 | R_z(theta)=exp(-i theta sigma3/2), the all-plus component transforms as 000099 | 000100 | T_{+...+} -> e^(-i8theta) T_{+...+}. 000101 | 000102 | Changing the active/passive sign convention conjugates this phase and 000103 | does not change the conclusion. Repeated lowering generates all17 000104 | independent symmetric spin components: this is precisely spin8. 000105 | 000106 | There is also a direct proof excluding ANY invariant linear contraction 000107 | of this tensor. Torus invariance of a degree16 polynomial forces it to 000108 | be a multiple of z_+^8 z_-^8. The raising generator acts as 000109 | 000110 | z_+ partial_{z_-}(z_+^8 z_-^8)=8 z_+^9 z_-^7 !=0. 000111 | 000112 | Hence the only SU(2)-invariant vector is0. The same holds for invariant 000113 | linear functionals, since the representation is unitary. In particular 000114 | contracting two symmetric spin slots with epsilon^{alpha beta} gives0; 000115 | pairing sixteen spinor indices with eight epsilon tensors cannot repair 000116 | the pure internal epsilon determinant. 000117 | 000118 | The adjoint vertex carries the dual spin8 representation. Adding a 000119 | vertex to its adjoint does not create a singlet: a direct sum of two 000120 | spin8 representations still has no singlet. In the all-plus example its 000121 | two coefficients rotate with opposite nonzero weights. A nonzero fixed 000122 | phase theta in Omega+Omega^* changes which real component is selected; 000123 | it does not make that component invariant under arbitrary rotations. 000124 | Scalar density terms in the same Hamiltonian cannot cancel the missing 000125 | spin because they lie in the rotation-trivial representation. 000126 | 000127 | 3A. THE FACTORIZED REST HAMILTONIAN CANNOT HIDE THIS VARIATION 000128 | 000129 | For the precise embedding j(v)=chi tensor v choose the spatial rotation 000130 | axis along chi's Bloch vector. Then chi is an eigenvector of spin along 000131 | this axis with eigenvalue s=+1/2 (or -1/2 for the opposite convention). 000132 | The sixteen selected modes are invariant under THIS U(1) rotation, and 000133 | there is no mixing with the orthogonal spin polarization for this axis. 000134 | In the actual Fock split its spin generator is therefore 000135 | 000136 | J_n=s N_m tensor I + I tensor J_R 000137 | 000138 | up to an irrelevant scalar implementer phase. This uses the local 000139 | spin-only action specified above; orbital/directional completions remain 000140 | outside the premise. The rest may include the opposite polarization. 000141 | 000142 | Since [N_m,Omega]=-16Omega and [N_m,Omega^*]=16Omega^*, 000143 | 000144 | [J_n,Omega]=-16s Omega, [J_n,Omega^*]=16s Omega^*. 000145 | 000146 | P_0 and P_F commute with N_m. In the literal factorized target 000147 | H=I tensor H_R+2(I-P_phi) tensor I_R, the commutator with H_R has mirror 000148 | number charge0. The Omega and Omega^* commutators instead have mirror 000149 | number charges -16 and +16 and nonzero coefficients. Hence no rotation 000150 | variation of this H_R can cancel them. For example the mirror matrix 000151 | element <0|[J_n,H]|F> is nonzero and proportional to 16s e^(-i theta)I_R, 000152 | whereas <0|I tensor [J_R,H_R]|F>=0 exactly. 000153 | 000154 | This is a direct test of the coefficient I_R. It is not a supposition that 000155 | H_R itself is rotationally trivial. Choosing this axis also avoids a 000156 | potentially misleading degree-only argument for general spin-mixing 000157 | generators: the diagonal P_0/P_F vertices must not be forgotten there. 000158 | 000159 | A general covariant generating functional may still have spin8 field 000160 | insertions paired with dual external sources. Such a source response is 000161 | not a scalar Hamiltonian coefficient times Omega. Likewise a spacetime 000162 | 16-point function may carry angular dependence in its external legs. 000163 | The present test does not assert their vanishing. 000164 | 000165 | 3B. ONE ACTUAL SOURCE WORD DECIDES THE LITERAL SOURCE IDENTIFICATION 000166 | 000167 | Assume the SAME original equal-time CAR legs have a common s-wave spatial 000168 | smear about a fixed center, normalized with the actual CAR inner product, 000169 | and the common spinor chi above. The s-wave smear supplies no orbital 000170 | weight. Retain the native annihilation convention 000171 | 000172 | Omega_16=|0>0, and its pure-ground limit is 000196 | exp(i phi)/2. The sign/phase here are those of the native annihilation 000197 | Omega=|0>