TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-ORIGINAL-FIELD-MAP-AND-CURRENT.txt
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000057 | 4. DERIVED COMMON NOETHER CURRENT OF THE ACTUAL P25-L ACTION
000058 | Use the actually supplied P25-L:139-144 action, with fixed coefficients.
000059 | The displayed ordinary divergences here are on a constant-metric patch;
000060 | on a curved background use the LC covariant divergence, equivalently
000061 | d_m(sqrt(abs(g))*J^m)/sqrt(abs(g)), and the actual boundary density.
000062 | It already supplies the electromagnetic Chern-Simons current in:199-202,
000063 |  C^m=epsilon^{mnrs} A_n F_rs, d_m C^m=F_mn Ftilde^mn.
000064 | For a constant infinitesimal shift delta a=epsilon, the action variation
000065 | is the local boundary derivative -g0*epsilon*d_m C^m/4.
000066 | Its derivative momentum before eliminating S is
000067 |  partial L/partial(d_m a)=lambda*mu^2*S^m/2.
000068 | Noether's theorem therefore gives the ACTUAL mixed current
000069 |  J_shift^m=lambda*mu^2*S^m/2+g0*C^m/4.
000070 | Using the SAME auxiliary S equation yields exactly
000071 |  J_shift^m=f_eff^2*d^m a+g0*C^m/4,
000072 |  f_eff^2=lambda^2*kappa^2*mu^2/6.
000073 | The axion equation is d_m(f_eff^2*d^m a)=-g0*F*Ftilde/4,
000074 | so d_m J_shift^m=0 follows on shell, with the photon helicity term
000075 | retained. This is a current of the original nonlinear supplied action,
000076 | not a prescribed external-oscillator response.
000077 | 
000078 | 5. EXPLICIT ELECTROMAGNETIC ORIENTATION AND CANONICAL GENERATOR
000079 | P25-L:128 fixes tensor epsilon_lower0123=+1, signature(+---),
000080 | Ftilde^{mn}=epsilon^{mnrs}F_rs/2; hence epsilon_upper0123=-1.
000081 | With A_m=(A0,-A_vec), E=-dot A_vec-grad A0 and B_can=curl A_vec,
000082 | F_0i=E_i,F_ij=-epsilon_ijk B_can^k, direct tensor contraction gives
000083 |  F*Ftilde=+4 E.B_can, C^0=-2 A_vec.B_can.
000084 | Thus in this explicit chart Pi_A=-E+g0*a*B_can and the positive energy is
000085 |  H_aEM=p_a^2/(2f_eff^2)+f_eff^2|grad a|^2/2
000086 |         +|Pi_A-g0*a*B_can|^2/2+|B_can|^2/2.
000087 | On a flat patch with vanishing surface terms, the canonical shift charge is
000088 |  Q_shift=integral d3x [p_a-g0*A_vec.B_can/2].
000089 | Functional differentiation of integral A.curl A gives2B_can.
000090 | Consequently {a,Q_shift}=1 and {Pi_A,Q_shift}=+g0*B_can.
000091 | The combination Pi_A-g0*a*B_can, and hence E and H_aEM, is invariant.
000092 | This is a concrete common canonical transformation of both original fields.
000093 | 
000094 | The earlier historical energy/response notes used the oriented chart
000095 | F*Ftilde=-4 E.B_note. It is consistent with the same original epsilon when
000096 | B_note=-B_can. Substitute that relation EVERYWHERE: their Pi_A+g0*a*B_note
000097 | is Pi_A-g0*a*B_can and their G=g_phys*B_note is -g_phys*B_can.
000098 | All exact Omega_+- branches depend on G^2 or |G| and are unchanged;
000099 | the oriented off-diagonal time response must carry the same signed G.
000100 | No convention is silently replaced by the supplement's epsilon_upper=+1.
000101 | 
000102 | 6. SCOPE AT THE WHOLE TFPT SOURCE
000103 | This derives an actual original field dualization and a common classical
000104 | shift generator/current. Q_shift is a shift-symmetry charge, not the
000105 | physical time Hamiltonian H_tr,pos. Boundaries, nontrivial flux sectors,
000106 | large gauge transformations and the actual quantum measure determine
000107 | whether a continuous global shift survives; a local boundary identity
000108 | does not select an OS vacuum or a full interacting common spectrum.
000109 | P25-L:196 does specify a rotation exp(i*a*gamma5/2) on its displayed psi.
000110 | Its embedding into the actually integrated full96 fields, normalization,
000111 | periods, real pairing and the whole fermionic source still require the
000112 | original map. P25-L:218's rescaled theta-to-a matching is not repaired by
000113 | an arbitrary extra factor or by silently keeping the old boundary phase.
000114 | All previous compiler/E8/clocks/flavor/alpha and original-source results
000115 | remain in their established scopes. No new Holst action or extra field
000116 | has been chosen to supply the missing common physical source.