TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/MOVING-NORMAL-KERNEL-PROOF.txt Original file SHA-256: e5fca7381726e3744f8288ad905a1cde762a9daa3de424ec3afb4f2f6adbfc1d Inclusive original lines: 1-159 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 | ORIGINAL MOVING NORMAL KERNEL: CONNECTION, SOURCE DERIVATIVE AND SCALE COVARIANCE 000002 | 31 October2026. R051. F1 continuation after R049/R050. 000003 | Execute R035 section8's derivative instruction, not another index/filter scan. 000004 | 000005 | 1. Original inputs and the first additionally required data 000006 | The normal B_Ea=sigma_normal nabla^(a)+Phi_a is defined on the original 000007 | normal S2,carrier007:95-139. The separate metrology001:30-126 places Brel 000008 | in the geometric package and fDgeo f+Brel. R035:171-198 already gives its 000009 | conditional actual normal kernel transport 000010 | W=f^-1, G=Sdagger W²S, T=W S G^-1/2, Tdagger T=I. (1) 000011 | It explicitly requires the induced connection and exterior derivatives 000012 | for moving modes. That instruction, not the pointwise isometry, is now 000013 | executed. The existing Wedge to96H coupling and all originalY stay given. 000014 | 000015 | Assume the actual normal realization B(x),or its given selfadjoint doubling, 000016 | has a smooth finite constant-rank kernel and a supplied unitary horizontal 000017 | connection nabla on its Hilbert bundle,including the actual normal measure 000018 | transport. Assume T is smooth on the common core,with stated domains and 000019 | the indicated derivative vectors in their form domains. These family and 000020 | connection data are NOT determined by the integer index. No normal metric, 000021 | potential,Phi,sigma profile,physical4D geometry or horizontal connection is 000022 | chosen here. If the original construction does not supply such family data, 000023 | the formulas below are conditional on them,not a completed origin map. 000024 | 000025 | The old NORMAL-CONNECTION-MARKS-COVER:1-76 already classifies the Kato 000026 | off-diagonal term for the specified scalar circle symbol. NORMAL-INDEX- 000027 | TRANSPORT:49-132 supplies the conditional exact product pushforward and 000028 | the independent horizontal flat-line freedom. Both are REUSED. We do not 000029 | relabel standard Berry/Kato geometry as a new general mathematical theorem. 000030 | The new application executes the derivative/source terms of this R035 T. 000031 | 000032 | 2. Exact connection and quadratic form of the SAME moving kernel 000033 | Set P=TTdagger,Q=I-P and 000034 | A_mu=Tdagger nabla_mu T, C_mu=Q nabla_mu T. 000035 | Differentiating Tdagger T=I with the supplied unitary connection gives 000036 | A_mu^dagger=-A_mu, nabla_mu T=T A_mu+C_mu. (2) 000037 | For the same original mode coefficient h, 000038 | nabla(Th)=T(d+A)h+C h. 000039 | Consequently the positive connection-kinetic form,IF this is the actual 000040 | original form being restricted,has the exact identity 000041 | ||nabla(Th)||²=||(d+A)h||²+hdagger U h, 000042 | U=g^mu,nu C_mu^dagger C_nu >=0. (3) 000043 | Here g is the actual positive Euclidean form metric,not a newly selected 000044 | spacetime dimension. The full block Gram(C_mu^dagger C_nu)_mu,nu is positive. 000045 | An individual mixed mu!=nu block need not be Hermitian or positive. 000046 | The curvature,with Fambient=[nabla_mu,nabla_nu],is 000047 | F_A,mu,nu=Tdagger Fambient,mu,nu T 000048 | +C_mu^dagger C_nu-C_nu^dagger C_mu. (4) 000049 | This sign follows by differentiating A: the tangential derivative products 000050 | contribute -[A_mu,A_nu],which is restored in F_A=dA+A wedge A. 000051 | Under a passive unitary normal-mode frame T->TV,h->Vdagger h, 000052 | A->Vdagger A V+Vdagger dV, C->CV, U->Vdagger U V, 000053 | F_A->Vdagger F_A V. (5) 000054 | The given Wedge/Yukawa vertex and sources transform with h. No frame choice 000055 | optimizes Y or identifies this connection with the physical weakSU2 field. 000056 | 000057 | 3. Direct derivative of the ORIGINAL normal operator, not a free C 000058 | If0 is isolated in the selfadjoint B with the actual transported domain, 000059 | let R_N=Q(B|_Q)^-1Q be its reduced inverse. Under differentiability sufficient 000060 | for nabla(BT)=(nabla B)T+B nabla T=0, one obtains 000061 | C_mu=-R_N(nabla_mu B)T, 000062 | C_mu^dagger C_nu=Tdagger(nabla_mu B) R_N²(nabla_nu B)T. (6) 000063 | Thus the coefficient in(3) is fixed by the same actual normal-family 000064 | variation and its WHOLE paired spectrum. In spectral notation this includes 000065 | sum_(lambda!=0) lambda^-2 000066 | Tdagger(nabla_mu B) P_lambda(nabla_nu B)T, 000067 | or its spectral integral. An index,rank or determinant winding loses these 000068 | weights and matrix elements. nabla B includes the original changes in the 000069 | normal connection,Phi and principal symbol. They are not independently fitted. 000070 | With a true normal spectral gap delta,||C_mu||<=||(nabla_mu B)T||/delta 000071 | when the displayed operator norm exists. No gap value is selected here. 000072 | Without an invertible reduced block,(6) is not silently applied; (2)-(4) 000073 | can still hold for a separately well-defined differentiable kernel map T. 000074 | 000075 | 4. Execute the original normal scale map without scalarizing G 000076 | Put R=G^1/2 in(1). The full induced connection is 000077 | A=R^-1 Sdagger W(nabla W)S R^-1 000078 | +R^-1 Sdagger W²(nabla S)R^-1+R nabla(R^-1). (7) 000079 | No derivative of S or the square root has been dropped. For G>0,the actual 000080 | matrix square-root derivative is the unique Sylvester solution 000081 | R deltaR+deltaR R=deltaG, 000082 | delta(R^-1)=-R^-1(deltaR)R^-1. (8) 000083 | It is not deltaR=(1/2)R^-1deltaG for a noncommuting matrix family. 000084 | 000085 | In the explicitly PARALLEL seed subclass nabla S=0,with W=e^(sigma/2) 000086 | and its horizontal derivatives scalar multiplications,write 000087 | h_mu=2W^-1nabla_mu W. 000088 | In a fixed normal trivialization this is partial_mu sigma; otherwise use 000089 | the actual horizontal derivative,including its normal coordinate transport. 000090 | Then M_mu=Sdagger e^sigma h_mu S=nabla_mu G and 000091 | M_mu,nu=Sdagger e^sigma h_mu h_nu S. Equations(7)-(8) reduce to 000092 | A=(1/2)[R^-1,dR], 000093 | C_mu=Q(nabla_mu W)S R^-1, 000094 | C_mu^dagger C_nu=(1/4)R^-1 000095 | (M_mu,nu-M_mu G^-1 M_nu)R^-1. (9) 000096 | The last expression is the exact ordered matrix normal covariance. It is 000097 | a whole positive block Gram; do not assert positivity of each mixed block. 000098 | In this scalar-derivative subclass [W,[Fambient,W]]=0. Since Fambient S=0 000099 | and Sdagger Fambient=0, sandwiching that double commutator gives 000100 | 2Sdagger W Fambient W S=0. Hence Tdagger Fambient T=0. Do NOT infer 000101 | Fambient T=0: a Hilbert-bundle curvature can contain vertical derivatives 000102 | and need not commute with W. If these scalar-derivative hypotheses fail, 000103 | retain the ambient term in(4) and use the general(7). Here M_mu,nu=M_nu,mu,so 000104 | F_A,mu,nu=(1/4)R^-1 000105 | (M_nu G^-1 M_mu-M_mu G^-1 M_nu)R^-1. (10) 000106 | No simultaneous diagonalization of G and its derivatives is imposed. 000107 | 000108 | If G=g(x)I on a neighborhood,then A=0 but U need not vanish. For rank1, 000109 | U=(1/4)g^mu,nu Cov_normal(h_mu,h_nu),with normalized normal weight 000110 | e^sigma|S|²/. For a positive base metric U=0 exactly when 000111 | all C_mu=0. A varying sigma by itself does NOT force nonzero curvature. 000112 | If sigma is constant ON the normal fiber,then T=S. In the parallel seed 000113 | both A and C vanish,exactly preserving R035's earlier uniform-scale result. 000114 | A nonparallel original S retains its original A/C even for uniform scaling. 000115 | 000116 | 5. Retain whole dynamics, reference and field measure 000117 | Equation(3) is an exact restriction of one specified quadratic form. It is 000118 | NOT an exact invariant-subspace theorem or an approximation to the whole 000119 | source. For the full field Psi=Th+q with Pq=0,both cross terms remain: 000120 | 2Re[+]. (11) 000121 | The P/Q blocks,all nonzero normal modes and higher vertices are kept in 000122 | R037's full Schur/source formula. Its effective operator is energy dependent; 000123 | the corresponding Q determinant and shifted source legs belong to Z[0]. 000124 | U already belongs to the restricted connection-Laplacian block. Adding it 000125 | again beside the same full kinetic form would double count it. A first-order 000126 | Dirac square needs its own Clifford contractions; (3) is not a universal 000127 | replacement for them. No new ambient bosonic kinetic term has been introduced. 000128 | The actual inner96 Higgs kinetic response from R031 is retained separately 000129 | until the original field/representation map identifies the same term. 000130 | 000131 | Under a whole original scale/coordinate transport,transform the ACTUAL 000132 | horizontal connection,metric,field measure,bound reference and source too. 000133 | Do not interpret U from a fixed-connection partial calculation as a new 000134 | uncancelled physical force after changing the rest of that source. R035's 000135 | flat coefficient Jacobian(det_C G)^-1 for two complex modes,or the actual 000136 | non-flat pushforward,remains in the full measure. Keep Eta,H0 and absoluteZ0. 000137 | 000138 | 6. Primary comparison, verdict and next discriminant 000139 | Wachsmuth/Teufel,arXiv0907.0351,pp6,21-22 distinguish Berry connection, 000140 | Born-Huang contribution and off-band coupling; their dynamical approximation 000141 | needs additional hypotheses. https://arxiv.org/pdf/0907.0351 . We use the 000142 | elementary exact form identities above,not their adiabatic theorem to claim 000143 | a TFPT spacetime or dynamically closed Higgs subspace. 000144 | 000145 | R051 closes this derivative response in the stated supplied normal family. 000146 | It does not choose the original B(x),horizontal rule or physical metric. 000147 | The previous flat-line counterexample remains: a common flat horizontal 000148 | phase can leave vertical B and local U/F unchanged while changing global 000149 | holonomy. These local formulas cannot select the original clock matching. 000150 | The next decisive original quantity is (nabla_mu B_Ea)T and its actual 000151 | relative/source insertion,or the independent whole mixed source that reads 000152 | it. Compare the SAME connection/Phi/symbol and domain with Brel; retain 000153 | the complete tensor Tdagger(nabla B)R_N²(nabla B)T and the original reference. 000154 | No arbitrary positive profile,weak gauge field,normal gap,counterterm or 000155 | scalaron parameter is chosen. No new index/Kato/ghost scan is needed. 000156 | F1 remains in progress. E8/clocks,memory,State/Time/RP,Yukawa/Flavor/Alpha 000157 | and the complete existing source remain part of the same requested theory. 000158 | 18 direct symbolic tangent and matrix controls pass; no new Lean or ledger 000159 | promotion and no complete physical B_Ea to Brel/TFPT closure is claimed.