TFPT scientific source excerpt Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-REAL-TORSION-OPERATOR.txt Original file SHA-256: 6a0b981eb3e314693c5d849fb5e976d4592d8828e2295f8bf83316efe513ea5e Inclusive original lines: 65-116 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. 000065 | K_theta=p^2/(8pi^2 t) G(t p^2) Tr_F exp(-t Phi^2), 000066 | F(x)=integral_0^1 exp[-x a(1-a)]da, 000067 | G(x)=F(x)-x F'(x)=[(x+6)F(x)-2]/4 000068 | =integral_0^1 [1+x a(1-a)]exp[-x a(1-a)]da. 000069 | This is the quadratic coefficient; the Hessian is twice it. 000070 | 000071 | For W(p)=-slash(p)gamma5/2 the two linear D^2 vertices have spin trace 000072 | p^4+4[k^2 p^2-(k.p)^2]. After the exact four-dimensional Gaussian momentum 000073 | integration it is p^4+6p^2/t. The quadratic W^2 contact contributes 000074 | -p^2/(16pi^2 t). Keeping both yields the displayed coefficient. 000075 | G=1-x^2/60+x^3/420-x^4/5040+..., positive for x>=0. 000076 | In particular the flat gradient a4 has NO fourth-derivative theta term: 000077 | a2_theta=dim(F)/(8pi^2) integral |dtheta|^2, 000078 | a4_theta=-Tr_F Phi^2/(8pi^2) integral |dtheta|^2. 000079 | 000080 | All original finite blocks enter through 000081 | Tr_F Phi^2=12Tr(Yu^dagger Yu+Yd^dagger Yd) 000082 | +4Tr(Ye^dagger Ye+Ynu^dagger Ynu)+2Tr(MR^dagger MR). 000083 | The 96 slots are not 96 independent physical particles. These are ordinary 000084 | unprojected spin4/full96 traces, not a new physical projection or Pfaffian count. 000085 | The formula uses no selected heat atom as physics and no new spectral profile. 000086 | If applying an actually supplied Laplace profile, integrate this coefficient 000087 | against its actual measure; f31(D/chi) uses t=s/chi^2, not f31(D^2/chi^2). 000088 | 000089 | 4. KEEP THE ORIGINAL INDEPENDENT TORSION BEFORE ITS FIELD EQUATION 000090 | The full geometric variable is b_mu=3S_mu/4, with W=i slash(b)gamma5. 000091 | Longitudinal and transverse refer to external momentum p, not particle types. 000092 | Rotational covariance allows p along axis1 and b=(b_L,b_T,0,0). 000093 | For k=q-a p the exact vertex trace average is 000094 | b_L^2(4p^2+24/t)+b_T^2[4p^2(1-4a(1-a))+24/t]. 000095 | The W^2 contact trace is 4(b_L^2+b_T^2). 000096 | The parameter identity 1=(1+x/2)F+2x Fprime follows by integrating 000097 | the derivative of (a-1/2)exp[-x a(1-a)] over a in[0,1]. 000098 | The FULL independent axial-vector quadratic coefficient is consequently 000099 | K_b=Tr_F exp(-t Phi^2)/(2pi^2 t) 000100 | [G(x)|b_parallel|^2+F(x)|b_transverse|^2], x=t p^2. 000101 | This is the same actual Dirac family, not a new vector-field model. 000102 | 000103 | Since F(x)=1-x/6+x^2/60+..., its local coefficients include 000104 | a2_S=9 dim(F)/(32pi^2) integral |S|^2, 000105 | a4_S=-3 dim(F)/(64pi^2) integral |dS|_form^2 000106 | -9 Tr_F Phi^2/(32pi^2) integral |S|^2, 000107 | where |dS|_form^2=sum_mu