TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/P1-COVARIANT-INSERTION.txt Original file SHA-256: ebb9e0cb15cdc017914d8567131c8d8b75855f20e05551b6f7b350989061e152 Inclusive original lines: 26-146 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. 000026 | 2. Differentiate the original collar before naming an internal coupling 000027 | 000028 | Use a fixed smooth collar coordinate and common appropriate domain/core. If 000029 | the normal Clifford symbol may vary, differentiating the original formula gives 000030 | 000031 | partial_a B = gamma_n^-1 partial_a D_+ 000032 | -gamma_n^-1 (partial_a gamma_n)(partial_n+B). 000033 | 000034 | A moving normal coordinate has an additional derivative-of-partial_n term. 000035 | Here partial_a may denote a smooth smeared background-field variation if such 000036 | a source family exists; no finite-dimensional or pointwise Phi parametrization 000037 | is asserted. A local distributional vertex requires that additional analysis. 000038 | Only for a fixed normal symbol and coordinate is the shorter identity 000039 | partial_a B = gamma_n^-1 partial_a D_+ 000040 | valid. This is an identity propagating the variation of D_+; it does not 000041 | produce a missing microscopic field variation. 000042 | 000043 | Likewise, on the smooth multiplication core the exact commutator is 000044 | 000045 | [D_+,b]=[gamma_n,b](partial_n+B) 000046 | +gamma_n(partial_n b+[B,b]). 000047 | 000048 | One cannot silently replace this by gamma_n[B,b] for arbitrary represented 000049 | algebra elements. Bounded spectral one-forms, compatibility with the normal 000050 | symbol, normal independence, real completion, adjunction and boundary-domain 000051 | preservation must all be checked on the ACTUAL represented algebra. The 000052 | ordinary real one-form prescription also requires its order-one hypothesis; 000053 | without it retain the generalized quadratic correction (arXiv:1304.7583). 000054 | None of these identities selects a missing algebra representation or Phi 000055 | multiplet. The earlier scalar-line one-form classification remains retained; 000056 | it is not rerun or promoted to a physical Higgs derivation here. 000057 | 000058 | There is also a necessary equivariance test before assigning sector blocks. 000059 | At a symmetry-preserving background, if a given linearized field action R(g) 000060 | obeys U_g B(Phi) U_g^-1=B(R(g)Phi), differentiation requires 000061 | U_g V_a U_g^-1=sum_b R(g)_(ba) V_b. 000062 | It therefore needs the original transformation law of Phi as well as the 000063 | actual fermion-sector representation. Geometric seam-evenness of Phi is not 000064 | that internal transformation law. The later S2 statement Phi in Gamma(E2) 000065 | uses the already named E2 branch; it is not an upstream derivation of R(g). 000066 | 000067 | 3. An ordinary off-diagonal derivative need not be an interaction 000068 | 000069 | Let U(t) be a differentiable unitary on the source Hilbert space. Transport the 000070 | ENTIRE datum: B_t=U(t)B_0U(t)^*, D_t=U(t)D_0U(t)^*, the algebra, state, J, 000071 | grading, reflection, boundary domains and time action. Every unitary-invariant 000072 | source condition is unchanged. In particular spectral invariants, chi, eta and 000073 | invariant defect values do not distinguish the representatives. No assertion 000074 | about arbitrary perturbations preserving a fixed APS domain is used: its 000075 | domain is transported too. 000076 | 000077 | With G=dot U U^* anti-Hermitian, dot B=[G,B] can be nonzero and off diagonal. 000078 | Nevertheless, with psi_t=U(t)psi_0, 000079 | =. 000080 | The simultaneous transport of state/time and field insertions preserves all 000081 | corresponding correlation functions. Thus calling such a derivative a new 000082 | Phi interaction, while holding its field labels fixed, is an identification 000083 | error. These are equivalent source descriptions, not two physically distinct 000084 | P1 worlds or a counterexample to a uniqueness theorem modulo equivalence. 000085 | This is a pointwise comparison of fully transported descriptions. It does not 000086 | prove implementation of an anomalous gauge transformation in a fixed sea, 000087 | trivial determinant-line holonomy, or equivalence after holding the old state 000088 | and physical fields fixed. 000089 | 000090 | After an embedding and fixed source/carrier charts are given, its frame 000091 | connection follows without a separate choice. Let Q(Phi) be a differentiable isometric embedding of a 000092 | fixed finite carrier coordinate space into a fixed source Hilbert chart, with 000093 | Q^*Q=I. Define 000094 | 000095 | a_a=Q^* partial_a Q, a_a^*=-a_a, 000096 | N_a=(I-QQ^*) partial_a Q, partial_a Q=Q a_a+N_a. 000097 | 000098 | For C_F=Q^* C_Sigma Q the exact covariant derivative is 000099 | 000100 | nabla_a C_F := partial_a C_F+[a_a,C_F] 000101 | =Q^*(partial_a C_Sigma)Q 000102 | +N_a^* C_Sigma Q+Q^* C_Sigma N_a. (1) 000103 | 000104 | Proof: substitute the displayed decomposition into the three derivative 000105 | terms of Q^* C_Sigma Q. The tangential terms are [C_F,a_a] and cancel. 000106 | This does NOT discard a physically changing embedding: its normal motion 000107 | N_a remains. For a moving source Hilbert chart its source connection must be 000108 | included as well; (1) uses the stated fixed ambient chart. 000109 | 000110 | Under a unitary carrier-frame change Q'=Q S, 000111 | C_F'=S^*C_F S, a_a'=S^*a_a S+S^*partial_a S, 000112 | so nabla_a C_F'=S^*(nabla_a C_F)S. The connection is induced by the specified 000113 | Q; it does not select Q from the primitive source. 000114 | It is the induced connection of a field frame, not a newly derived physical 000115 | gauge field or a scalar coupling. 000116 | 000117 | For the existing finite faithful quasi-free reader h(C)=log(I-C)-log C, 000118 | unitary equivariance of h gives 000119 | 000120 | nabla_a h(C_F)=D h(C_F)[nabla_a C_F]. (2) 000121 | 000122 | Proof: differentiation gives the Frechet term for partial_a C_F, and 000123 | D h(C)[[a,C]]=[a,h(C)] by differentiation of unitary conjugation. 000124 | Equations(1)-(2) retain the noncommuting divided differences proved in 000125 | P1-INSERTION-TANGENT.txt; no commuting-matrix shortcut is made. Scale, 000126 | chirality projection and their source variations remain separate data. 000127 | This quasi-free reader is not a reconstruction of an arbitrary interacting 000128 | state from its covariance alone. 000129 | 000130 | For an actual B_F represented in this same carrier chart the corresponding 000131 | operator insertion is nabla_a B_F=partial_a B_F+[a_a,B_F]. If physical fields 000132 | were identified there, its matrix elements, rather than a basis derivative, 000133 | would test a genuine field variation. In general do not set B_F=Q^*B_Sigma Q 000134 | or interchange compression with h; functional calculus need not commute with 000135 | compression. These extra equalities also need proof. 000136 | 000137 | An exact sign control uses C=diag(1/5,4/5), a=[[0,1],[-1,0]], and pure frame 000138 | motion dot C=[C,a]. Then dot C has off-diagonals -3/5, while dot h(C) has 000139 | off-diagonals log16. Both covariant derivatives vanish EXACTLY. Calling the 000140 | nonzero log16 entry a Yukawa coefficient would misread pure frame motion. 000141 | This is an algebraic sign control, not a physical P1 source model. The 000142 | existing logit_tangent reader reproduces it with numerical residual0. 000143 | The normal-motion terms in(1) were also checked symbolically for arbitrary 000144 | normal displacement in the canonical Q=[I;0] frame; they remain nonzero 000145 | when the ambient covariance has the corresponding cross entries. 000146 |