TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-NORMALIZATION-INVARIANT-SOURCE-MAP.txt
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000001 | HIST20261103N. Normalization-invariant matter coupling and nonlinear source map.
000002 | Anonymized continuation of the SAME geometric matter calculation.
000003 | 
000004 | 1. AN ACTUAL NORMALIZATION RESULT WITHOUT CHOOSING LAMBDA
000005 | Keep the exact premises and Lorentz/Euclidean boundary of
000006 | HISTORICAL-GEOMETRIC-MATTER-BACKREACTION.txt. For nonzero lambda*kappa*mu,
000007 |  f^2=lambda^2*kappa^2*mu^2/6,
000008 |  c=lambda*kappa^2/4,
000009 |  d=3*kappa^2/(16*mu^2),
000010 |  kappa_phys^2=kappa^2/mu^2.
000011 | Under the actual canonical coordinate map phi=f*a, p_phi=p_a/f, the
000012 | geometric derivative coupling is -(c/f)*dphi.J5. Exactly,
000013 |  (c/f)^2=3*kappa_phys^2/8,
000014 |  d=3*kappa_phys^2/16=(c/f)^2/2.
000015 | Its magnitude is fixed by the original gravitational coefficient and
000016 | is independent of |lambda| and the inserted calibration scale mu.
000017 | This closes the normalization of this geometric matter coupling within
000018 | the supplied Lorentz spin-lift source. It does not require lambda*kappa^2=2.
000019 | The sign follows the transported field orientation; do not reset it alone.
000020 | This does not prove that every coupling in the whole source is fixed.
000021 | 
000022 | 2. CORRECT AXIAL PHASE COORDINATE, NOT AN ADDED FIELD
000023 | Define theta=2*c*a. This is the phase whose half-angle rotation matches
000024 | the derivative coupling of the geometric Dirac operator, with the inverse
000025 | rotation for the inverse change of variables. Then
000026 |  phi=F_theta*theta,
000027 |  F_theta^2=f^2/(4*c^2)=2/(3*kappa_phys^2).
000028 | F_theta may carry the field-orientation sign; its square is unambiguous.
000029 | The SAME eliminated Lorentz action density becomes
000030 |  L_S,elim=(dtheta)^2/(3*kappa_phys^2)
000031 |           -(1/2)*dtheta.J5+(3*kappa_phys^2/16)*J5^2.
000032 | No free kinetic coefficient was inserted to obtain this expression.
000033 | The original phase-photon coefficient and current transform as
000034 |  g_theta=g0/(2*c),
000035 |  J_theta=F_theta^2*dtheta-J5/2+(g_theta/4)*C
000036 |         =S/kappa_phys^2+(g_theta/4)*C.
000037 | The conjugate momentum is
000038 |  p_theta=p_a/(2*c)=F_theta^2*dot theta-j0/2,
000039 | and Q_theta=int[p_theta-(g_theta/2)*A.B_can].
000040 | All source insertions and boundary phases must use the same map.
000041 | In particular theta has not been assigned a new period by notation.
000042 | A spinor half-angle, a bosonic field period and fermion parity are distinct
000043 | until the actual global field bundle and source establish their relation.
000044 | 
000045 | The photon term in the printed action remains -g_theta*theta*F*Ftilde/4.
000046 | With g0 independently held fixed, g_theta and g_phi=g0/f still contain
000047 | lambda. Hence the above matter result does NOT justify declaring the
000048 | whole printed photon/matter source parameter-free. Its actual anomaly
000049 | matching and field-period dictionary must be evaluated together.
000050 | 
000051 | 3. THE ORIGINAL TRACE/AXIAL MAP NEEDS THE ACTUAL MATTER SHIFT
000052 | The SAME auxiliary equation now reads, without lambda in these coordinates,
000053 |  S=(2/3)*dtheta-(kappa_phys^2/2)*J5.
000054 | Thus the bare axial one-form is not generally a scalar gradient:
000055 |  dS=-(kappa_phys^2/2)*dJ5.
000056 | Conservation div J5=0, even if it were true, would not imply dJ5=0.
000057 | The correct locally gradient combination is
000058 |  S_hat=S+(kappa_phys^2/2)*J5=(2/3)*dtheta.
000059 | On a contractible patch the earlier trace potential can therefore map to
000060 |  dphi_old=S_hat, or phi_old=2*theta/3+constant.
000061 | Using the EXACT Hodge identity already established for the original trace
000062 | ansatz, the unshifted contorsion is
000063 |  K_ax(S)=-*T_vec(dphi_old)-(kappa_phys^2/4)*epsilon*J5.
000064 | Dropping the last term is exactly the zero-current restriction. This is
000065 | a derived nonlinear field relation involving the actual matter current;
000066 | it is not a second comparison model or a chosen new interaction.
000067 | Global winding and original boundary data are transported, not discarded.
000068 | 
000069 | 4. WHOLE FINITE SOURCE MAP AND THE AUXILIARY INSERTIONS
000070 | At a fixed finite regulator, keep the Grassmann coordinates and every
000071 | other original field unchanged while shifting
000072 |  S_hat=S+(kappa_phys^2/2)*J5(psi).
000073 | The super-coordinate derivative is block triangular, with identity on
000074 | both diagonal blocks. Its Berezinian is therefore ONE. This alone does
000075 | not remove any field-dependent weight or normalize the source to one.
000076 | An original weight w(S,psi,...) becomes w(S_hat-alpha*J5,psi,...), and
000077 | its domain, contour and reference must transform with it.
000078 | 
000079 | For the auxiliary Gaussian actually specified in this local source,
000080 | write C_S=3*mu^2/(4*kappa^2), b=lambda*mu^2/2, u=3/4, and retain an
000081 | external torsion source J_S. Its complete exponent contains
000082 |  -C_S*S^2+B.S, B=b*da-u*J5+J_S.
000083 | The exact square identity is
000084 |  -C_S*[S-B/(2*C_S)]^2+B^2/(4*C_S).
000085 | If the original integration operation admits this same translation, its
000086 | factor is N_C*exp(i*B^2/(4*C_S)), with N_C defined by the ORIGINAL contour
000087 | and reference. Differentiating in J_S retains the actual mixed insertions
000088 | and contacts. Setting J_S=0 before comparing complete source functionals
000089 | would throw away a test of their identity. For a non-flat original
000090 | measure its transformed weight remains present; no Gaussian law is chosen.
000091 | Initial/final states, physical selectors and all other sources are also
000092 | pulled back by the same field map. A field reparametrization does not
000093 | select a new vacuum or a different physical time.
000094 | 
000095 | An older propagating-torsion beta term cannot be removed by this shift.
000096 | It produces derivative and mixed-current terms; a differential auxiliary
000097 | kernel would yield its actual inverse-kernel interaction rather than the
000098 | ultralocal contact above. Only the explicitly supplied algebraic P25 S
000099 | sector has been eliminated here.
000100 | 
000101 | 5. ANOMALY BOOKKEEPING IS A WHOLE-SOURCE IDENTITY
000102 | For an actual finite antisymmetric fermion kernel A and invertible field
000103 | map M, Pf(M^T*A*M)=det(M)*Pf(A), while D(M*chi)=det(M)^(-1)*Dchi.
000104 | The source covectors transform with M as well. This exact finite identity
000105 | requires no guessed Dirac/Majorana counting factor. Its infinite regulated
000106 | version must use the actual original pairing, regulator, reference and
000107 | boundary domain; a standalone local Jacobian is not the whole source.
000108 | In an ordinary Lorentz Dirac chart the half-angle map acts as U*D*U and
000109 | rotates the complete mass terms. This is not automatic for beta_new.
000110 | 
000111 | If the old local a*F*Ftilde term is already the result of a particular
000112 | fermion Jacobian, restoring the same unrotated fermions while adding that
000113 | Jacobian again double counts it. Conversely an independently supplied
000114 | bare topological term cannot be erased by calling it a Jacobian.
000115 | The actual source provenance decides which case holds.
000116 | That provenance is explicit in P25-L:196-205: the printed term is called
000117 | the Fujikawa Jacobian of exp(i*a*gamma5/2), not an independent bare term.
000118 | Restoring its same fermions must therefore transport that derivation,
000119 | not silently reinterpret this coefficient as a new independent coupling.
000120 | There is also a literal normalization conflict at P25-L:218: it writes
000121 | Delta L=e^2*E*theta*F*Ftilde/(16*pi^2), but its definition
000122 | a=e^2*E*theta/(4*pi) inserted into a*F*Ftilde/(8*pi) gives half that value.
000123 | This fixes the location of the first incorrect identification, not a
000124 | license to choose either coefficient as the physical anomaly. Its actual
000125 | regulated whole-source trace and all transported phase coordinates decide.
000126 | At a finite regulator Tr(gamma5) is not assumed zero on an arbitrary
000127 | chiral/APS domain; any exact cancellation must use that actual domain.
000128 | A local anomalous vertex alone is not the full physical photon amplitude:
000129 | all mass-phase and fermionic determinant contributions must be retained.
000130 | No axion mass or low-energy photon coupling is inferred from that vertex
000131 | alone. Constant-shift invariance of a genuinely invariant full fermion
000132 | factor remains the precise statement in the preceding source note.
000133 | 
000134 | 6. WHOLE TFPT RESULT AND REMAINING IDENTIFICATION
000135 | Recovered here: the normalization-invariant geometric matter coupling,
000136 | the corresponding correctly scaled phase coordinate, the nonlinear
000137 | matter shift in the original trace/axial map, and the complete finite
000138 | auxiliary-source bookkeeping. These are consequences of the same supplied
000139 | Lorentz action and geometric spin lift, not additional selected dynamics.
000140 | The original full Euclidean pairing/state/time/reference map must still
000141 | identify this chart with the primitive TFPT source. No complete S_adm,
000142 | physical H_tr,pos or common interacting spectrum has been claimed.
000143 | All existing compiler/E8/SM/flavor/alpha/clocks and full-source results
000144 | remain in their stated scopes. In historical RG papers the symbol
000145 | lambda(k)=Lambda(k)/k^2 denotes a cosmological coupling, not the axial
000146 | lambda in P25. Its reported fixed point cannot fix the latter by name.