TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-WHOLE-SOURCE-ASSEMBLY-AND-PAIRING.txt
Original file SHA-256: d977c34195dbe79f9b6fbd7598e47f9512e0b9fe39a26ae623786ee867a0515d
Inclusive original lines: 55-177
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.

000055 | identity before using four separate critical equations as the whole dynamics.
000056 | 
000057 | 3. THE ACTUAL FULL192 TORSION CURRENT, INCLUDING THE HALF FACTOR
000058 | Use exactly actual_product_pairing.py, SHA256
000059 | 6acaa9bc2c7a157a30730d7d6de75e944266afbf69b18a31c6fba654b8bd1c04.
000060 | Its actual matrices are
000061 |  gamma_i=sigma1 tensor sigma_i, gamma4=sigma2 tensor I2,
000062 |  gamma5=gamma1 gamma2 gamma3 gamma4, C=gamma1 gamma3,
000063 |  R=C tensor KF, J=R conjugation, Gamma=gamma5 tensor gammaF.
000064 | The actual full96 KF is real symmetric unitary with KF²=1 and
000065 | KF gammaF=-gammaF KF. Thus J²=-1 and J Gamma=-Gamma J.
000066 | Phi is the entire original Hermitian odd Yukawa/Majorana operator, with
000067 | KF barPhi=Phi KF. Eeps is the real coordinate embedding of Gamma=eps,
000068 | rank192. It is NOT Padm and counts no physical species by itself.
000069 | 
000070 | The actual product action and its skew form are
000071 |  SF=(1/2)psi^T M_eps(S)psi,
000072 |  M_eps(S)=Eeps^T R^dagger(D0+W[S])Eeps,
000073 |  D0=-i gamma.partial+gamma5 Phi, W[S]=(3i/4)S_mu gamma^mu gamma5.
000074 | Transpose includes the original spacetime integration by parts/domain.
000075 | Define the ACTUAL torsion-current matrix
000076 |  L_mu,eps=Eeps^T R^dagger(gamma^mu gamma5 tensor I96)Eeps.
000077 | Then, for both original total chiralities,
000078 |  L_mu,eps^T=-L_mu,eps, L_mu,eps^dagger L_mu,eps=I192.
000079 | Every species slot has a nonzero current coefficient. In particular there
000080 | is no zero obtained by isolating the symmetric internal KO6 mass pairing.
000081 | The precise current induced by this actual action is
000082 |  J5E_mu=-(i/2)psi^T L_mu,eps psi,
000083 | and therefore, with no coefficient fit,
000084 |  (1/2)psi^T deltaM_eps(S)psi=-(3/4)S_mu J5E_mu.
000085 | In the actual particle/antiparticle split each side has96 Weyl slots. The
000086 | current has only cross blocks; its two same-side blocks vanish. The factor
000087 | 1/2 is compensated by the two equal Grassmann cross terms. The Majorana
000088 | mass has nonzero same-side blocks, each with norm²=2Tr(MR^dagger MR),
000089 | on BOTH total chiral charts. Keep their anomalous contractions; an arbitrary
000090 | Dirac determinant over all96 internal labels duplicates the original fields.
000091 | This closes the coefficient and current-bilinear dictionary on the existing
000092 | full product chart. It does not by definition identify the historical
000093 | Lorentz integration variables, their quantum measure or the primitive
000094 | admissible source. Those require the actual whole field/time map.
000095 | 
000096 | Keep all Yukawa and Majorana terms. For q=Eeps^T(gamma5 tensor I)Eeps,
000097 |  q^T Mmass+Mmass q=2 Eeps^T R^dagger(I4 tensor Phi)Eeps.
000098 | This is the full axial mass insertion. There is no generally conserved
000099 | massless axial current in the massive original operator; the regulated
000100 | anomaly and boundary terms remain. TORSION20261104D's transverse identity
000101 | needs co-closed TEST SOURCES, not a divergence-free J5. The mass Ward term
000102 | therefore does not invalidate that earlier identity.
000103 | 
000104 | Primary comparison: Chamseddine/Connes/Marcolli, Gravity and the Standard Model
000105 | with Neutrino Mixing, section4 and Proposition4.1, supports the skew total
000106 | KO2 pairing and its half-weight Pfaffian. D'Andrea/Kurkov/Lizzi,
000107 | arXiv:1605.03231, sections4.1/4.2 and AppendixA, distinguishes Euclidean
000108 | pairing, Lorentz charge identification and the axial measure phase. These
000109 | standard mechanisms do not select TFPT's full boson integral or OS state.
000110 | 
000111 | 4. WHOLE BEREZIN SOURCE AND REFERENCE TRANSPORT, INCLUDING ZEROS
000112 | At the same finite original regulator use the full exterior source polynomial
000113 |  P[M,eta]=integral dpsi exp[-psi^T M psi/2+eta^T psi]
000114 | with the actual orientation. It is polynomial in M and eta, not an inverse
000115 | matrix formula. At zero source it is the oriented Pfaffian. At a zero
000116 | Pfaffian the higher-source coefficients, given by complementary Pfaffians,
000117 | must remain; do not divide them by the vanishing zero-source coefficient.
000118 | 
000119 | For an actual invertible coordinate change psi=V(b)chi with bosons b retained,
000120 |  M'=V^T M V, eta'=V^T eta,
000121 |  Dpsi=det(V)^(-1) Dchi,
000122 |  P[M',eta']=det(V) P[M,eta].
000123 | The determinant cancels the Berezinian. The full boson/fermion Jacobian is
000124 | block triangular even when V depends on b; no extra arbitrary density arises.
000125 | The spectral DeltaEta and the dynamical fermion Pfaffian need not be the
000126 | same line or regularized operator. There is no general license to replace
000127 | the latter by its absolute value; first compare their actual phase maps.
000128 | This is an exact coordinate identity on a finite regulator, NOT proof that
000129 | the limiting regulated anomalous Jacobian equals1. Carry its actual line,
000130 | eta/reference phase, all external legs, domains and existing state/time.
000131 | At varying bosonic b also retain
000132 |  partial_b M'=V^T(partial_b M)V+(partial_b V)^T M V+V^T M(partial_b V).
000133 | The two connection insertions cannot be discarded in the current response.
000134 | 
000135 | The whole common field source is the original boson/ghost/selector integral
000136 | of this full polynomial and its original weight, divided by the original
000137 | Zref[0]. Writing that integral is not its evaluation. QFT002:259-270 and
000138 | QFT008:143-152 define Zref as externally source independent. This does not
000139 | make Dref inside Sprim independent of the integrated fields, or make the
000140 | normalizer independent of varied background parameters. Keep both roles.
000141 | 
000142 | 5. EXPLICIT MIXED ORIGINAL VERTEX PREVENTS SECTOR ERASURE
000143 | On the existing ordinary flat bulk product, constant radial Higgs r and
000144 | fixed MR, the entire original96 trace is
000145 |  TrF Phi_r²=r² A_Y+2Tr(MR^dagger MR),
000146 |  A_Y=12Tr(Yu^dagger Yu+Yd^dagger Yd)
000147 |                          +4Tr(Ye^dagger Ye+Ynu^dagger Ynu).
000148 | All original entries remain symbolic. The prior exact geometric a4 term is
000149 |  -9/(32pi²) S² TrF Phi_r².
000150 | For a deck-even tangential S direction both actual bound references have
000151 | delta Dref=0, so the following ordinary bulk mixed coefficient survives:
000152 |  partial_S_tangent² partial_r² a4=-9 A_Y/(8pi²) per bulk volume.
000153 | It is nonzero for nonzero Yukawa matrices. It is a common original response,
000154 | not a four-sector fit or a heat coefficient promoted to a particle pole.
000155 | Arbitrary S-dependent Brel, projected/nonlocal traces and moving domains
000156 | are not evaluated by this ordinary product statement; their cancellation
000157 | cannot be assumed or globally excluded.
000158 | 
000159 | The correct common-source calculus keeps mixed variations. For the actual
000160 | finite regulated integral I(xi)=integral exp[-A(b,xi)]dmu(b), with all actual
000161 | Jacobian/reference/selector dependence included and no lost boundary terms,
000162 |  partial_a partial_b[-log I]=<A_ab>-<A_a A_b>_connected.
000163 | For Zrel=I/Iref subtract the analogous Iref expression; add the explicitly
000164 | present determinant, eta, transport and vacuum derivatives in M. One cannot
000165 | set every mixed derivative to zero from commuting filters or a determinant
000166 | block split. With Grassmann variables use the corresponding full insertion
000167 | identities before division; an inverse Pfaffian is unnecessary.
000168 | 
000169 | The historical auxiliary integration also fixes the intersector contact.
000170 | With the ACTUAL current just derived, J5=sum_f J5,f, it yields
000171 |  -3 kappa_phys²/16 (sum_f J5,f)².
000172 | For each unordered distinct pair the cross coefficient is
000173 |  -3 kappa_phys²/8 J5,f.J5,g.
000174 | There are no new independent intersector couplings. This is a concrete
000175 | common interaction of the historical parent, not a proof of primitive
000176 | action equality or of positivity of a truncated quantum Hamiltonian.
000177 |