TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-SUPERCONNECTION-REDUCTION-PROOF.txt Original file SHA-256: 93e5aa95a231dec44ea173a680d6bcad7a5261dc1d7926f12957e42259b4afa6 Inclusive original lines: 123-200 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. 000123 | reduces K on that core. Even a perfect first-order projected vertex does 000124 | not imply that its square or heat trace is the projected whole square. 000125 | For K(t)=K0+tV with [P,K0]=0, the defect is exactly 000126 | t^2 P V Qp V P. (7) 000127 | If the original moving modes give [P,K0]!=0, retain all cross terms in(6); 000128 | do not invoke(7). The induced connection and external derivative terms of 000129 | R035 are among such original off-diagonal terms. 000130 | 000131 | An ACTUAL96 internal example, not a choice of the normal P: take the 000132 | existing finite chirality GammaF and P+=(I+GammaF)/2. Phi is odd, so 000133 | P+ Phi P+=0, yet P+ Phi^2 P+ is generally nonzero. 000134 | For its Higgs part the trace is A u/2; zero-order Majorana and its original 000135 | cross terms remain separately present in the whole Phi. This illustrates 000136 | the real consumer: projecting a coupling before squaring can erase the 000137 | mass response entirely. It does not identify P+ with a normal Higgs kernel. 000138 | For a positive finite-regulator heat trace, 000139 | d/dt_heat [trP e^(-t_heat K^2)-trP e^(-t_heat(PKP)^2)] at0 000140 | =-trP P K Qp K P. 000141 | Thus equality for all small heat times forces reduction in finite dimension. 000142 | This small-time derivative is not a continuum heat asymptotic assertion. 000143 | 000144 | 6. Exact full resolvent reader for the actual normal split 000145 | For z in the common resolvent sets and with the original domains, write 000146 | K-z=[[A(z),B],[C,E(z)]], R(z)=E(z)^-1, 000147 | S(z)=A(z)-B R(z) C. (8) 000148 | Finite regulated matrices make every operation below exact. For unbounded 000149 | operators use the actual valid closed block realization; no domain equality 000150 | or limiting trace class is inferred from formal blocks. 000151 | The complete inverse is 000152 | (K-z)^-1=[[S^-1,-S^-1 B R], 000153 | [-R C S^-1,R+R C S^-1 B R]]. (9) 000154 | The projected resolvent alone is S^-1. The full ordinary cyclic trace is 000155 | Tr(K-z)^-1=Tr_Q R-Tr_P[S^-1 partial_z S], 000156 | partial_z S=-I_P-B R^2 C. (10) 000157 | Consequently the nontrivial weight I+B R^2 C and the exterior trace must 000158 | both be retained. Tr S^-1 alone is NOT the whole trace. S(0) is also not 000159 | an energy-independent local Dirac replacement for this spectral family. 000160 | 000161 | For an actually declared weighted/graded trace apply that same linear 000162 | functional to ALL blocks in(9). With diagonal weight Wp,Wq this is 000163 | tr(Wp S^-1)+tr(Wq R)+tr(Wq R C S^-1 B R). 000164 | Off-diagonal weights require the corresponding cross blocks. The simple 000165 | log derivative(10) is not claimed for arbitrary weighted traces. No new 000166 | trace weights or independent a2/a4 renormalizations are chosen. 000167 | 000168 | 7. Carry the original heat/zeta pair through(9), not through S(0)^2 000169 | For any finite given self-adjoint invertible K and omega>0, 000170 | (K^2+omega^2)^-1=[(K-i omega)^-1-(K+i omega)^-1]/(2i omega). (11) 000171 | Use(9) at BOTH spectral parameters. This retains the projection-square 000172 | defect and all exterior excursions automatically. Functional calculus, or 000173 | the Mellin/Stieltjes formula for 0