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: 1-150 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 | HIST20261104A. Actual real geometric torsion, reference and full axial heat response. 000002 | Source-reviewed continuation of the original TFPT field/state/time question. 000003 | Anonymous scientific record. No private identity, email, address or URL. 000004 | 000005 | 1. ORIGINAL OPERATOR AND REALITY 000006 | Use the actual supplied ordinary Spin4/full96 constructor 000007 | experiments/theory-contracts/craig-original-source-bridge-20261103/actual_product_pairing.py 000008 | SHA256 6acaa9bc2c7a157a30730d7d6de75e944266afbf69b18a31c6fba654b8bd1c04. 000009 | Its four arbitrary complex Yukawa matrices and symmetric complex Majorana 000010 | matrix are retained symbolically. There is no fitted mass spectrum. 000011 | The tangent-first gamma matrices are Hermitian, gamma5=gamma1gamma2gamma3gamma4, 000012 | epsilon1234=+1, D=-i gamma^mu nabla_mu+gamma5 Phi, Gamma=gamma5 tensor gamma_F. 000013 | This is an already supplied four-dimensional geometric branch, not an 000014 | origin theorem for its spacetime, finite algebra or admissible field image. 000015 | 000016 | For the actual real contorsion K_muab=epsilon_muabnu S_nu/2, direct contraction 000017 | of the minimal spin connection gives 000018 | delta D=-(i/4)gamma^mu K_muab gamma^[a gamma^b] 000019 | =(3i/4) S_mu gamma^mu gamma5. 000020 | Antisymmetrization here has weight one half. This operator is Hermitian, 000021 | J-real and Gamma-odd with the original full J. A normal-first orientation 000022 | reverses gamma5 and the corresponding epsilon signs together. 000023 | 000024 | On a real geometric gradient S=(2/3)dtheta, write 000025 | W[theta]=(i/2)gamma^mu gamma5 partial_mu theta. 000026 | The previously derived unitary inner phase direction has, on total fiber 000027 | epsilon, the value (1/2)gamma^mu gamma5 partial_mu theta WITHOUT i. 000028 | Thus W restricted to that fiber is i times that phase direction. 000029 | The earlier full paired phase/companion dictionary remains valid. It is 000030 | not an identity between the two REAL off-shell geometric field paths. 000031 | 000032 | 2. ACTUAL BOUND REFERENCE, INCLUDING BOTH COORDINATE PARITIES 000033 | Use the SAME original linear deck and the two retained R032 lift charts. 000034 | The actual Sspin preserves tangential gamma, flips normal gamma and gamma5. 000035 | The T lift additionally carries gamma_F. The linear deck does not conjugate i. 000036 | It is not the physical antiunitary OS reflection and not the real structure J. 000037 | For theta_R=theta composed with normal reflection, direct calculation gives 000038 | tau W[theta] tau^-1=-W[theta_R] for BOTH existing lifts. 000039 | Let theta_e=(theta+theta_R)/2, theta_o=(theta-theta_R)/2. Then exactly 000040 | delta D_ref=W[theta_o], delta D_rel=W[theta_e]. 000041 | No intrinsic physical pseudoscalar parity is selected by this decomposition. 000042 | 000043 | With the original full Phi=L(H)+D_M, p=(H+H_tau)/2, m=(H-H_tau)/2, 000044 | D_ref,S=D0+gamma5 L(m)+W[theta_o], 000045 | D_rel,S=gamma5[L(p)+D_M]+W[theta_e], 000046 | D_ref,T=D0+gamma5[L(p)+D_M]+W[theta_o], 000047 | D_rel,T=gamma5 L(m)+W[theta_e]. 000048 | These are operators on their actual transported original domains. If APS or 000049 | Cauchy projectors move, retain their domain variation; compact interior 000050 | support alone does not fix the entire Calderon projector. 000051 | 000052 | For coordinate-even theta and seam-even H the pointwise multiplication 000053 | operators satisfy 000054 | {gamma5 Phi,W}=0, W^2=|dtheta|^2/4, 000055 | D_rel,S^2=Phi^2+|dtheta|^2/4, D_rel,T^2=|dtheta|^2/4. 000056 | These identities are NOT the square of the full first-order Dirac operator. 000057 | The previous unitary inner phase had the opposite coordinate-parity routing. 000058 | Do not substitute its reference jet for real geometric torsion. 000059 | 000060 | 3. EXACT FLAT HEAT JET OF THE WHOLE GEOMETRIC OPERATOR 000061 | On the already supplied flat product, hold the entire Phi constant and 000062 | vary theta smoothly. At second order in amplitude and ALL external momentum, 000063 | Tr exp(-t D_theta^2)|theta^2 000064 | =integral d4p/(2pi)^4 K_theta(p,t)|theta_hat(p)|^2, 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