TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-TORSION-WICK-PARENT.txt
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000001 | HIST20261104B. Original auxiliary torsion Wick map and exact current integration.
000002 | Anonymous source-reviewed scientific continuation. 4 November 2026.
000003 | No private identity, correspondence header, address, credential or private URL.
000004 | 
000005 | 1. THE SUPPLIED ORIGINAL PARENT, NOT A NEW ANSATZ
000006 | Use the historical P25-L manuscript and accompanying P25-S supplement.
000007 | P25-L SHA256 428c9f5707774afe442929fa122eafedbb80a3196fada1ec17fcc828f13659ad.
000008 | P25-S SHA256 f24a1e94ca72277e2ea31af812e879293d9cde2bce3270a2c4bfa4259a461c4d.
000009 | P25-L:127-155 fixes (+---), epsilon_lower0123=+1, torsionless Rhat,
000010 | K_mnr=epsilon_mnrs S^s/2, and the actual axial part
000011 |  L_aux,L=-A S_L^2+B partial_mu a S_L^mu-c0 S_L^mu J5L_mu,
000012 |  A=3mu^2/(4kappa^2), B=lambda mu^2/2, c0=3/4.
000013 | The minimal spin current coefficient c0 is derived from the same original
000014 | geometric spin lift, not fitted. J5 is the whole actual coupled current.
000015 | The original Maxwell, curvature, U, anomaly, boundary and reference terms
000016 | are not removed. The supplied 4D chart is not a fundamental spacetime proof. The simple
000017 | phase normalization below uses the source's fixed-coefficient sector with
000018 | nonzero real lambda and fixed mu,kappa. If c varies, dtheta=2c da+2a dc;
000019 | those extra terms must be carried instead of applying the constant formula.
000020 | 
000021 | 2. METRIC, HODGE AND CLIFFORD SIGNS MUST ALL CONTINUE TOGETHER
000022 | On its fixed flat chart take t=-i tau, gammaE4=gammaL0,
000023 | gammaEi=-i gammaLi, gamma5E=gammaE1gammaE2gammaE3gammaE4.
000024 | Then gamma5L=-gamma5E and g_L continued=-g_E.
000025 | The lowered contorsion K_mnr is a connection difference with an index lowered
000026 | by the metric. Consequently its geometric Euclidean continuation is
000027 |  K_E=-pullback(K_L),
000028 |  K_E4ij=+i K_L0ij, K_Eijk=-K_Lijk.
000029 | Simply pulling back a metric-independent covariant 3-form misses that sign.
000030 | Using epsilonE1234=+1 and the SAME dualization gives
000031 |  S_E4=S_L^0, S_Ei=-i S_L^i.
000032 | Direct contractions give S_L^2 -> S_E^2 and da_L.S_L -> i da_E.S_E.
000033 | No single component-independent rule S_E=-i S_L is correct here.
000034 | 
000035 | The actual Lorentz spin term is -3 S_L,mu gammaL^mu gamma5L/4.
000036 | The continued traditional Euclidean kinetic kernel has
000037 |  K_E=gammaE.partial+Phi-c0 S_Emu gammaE^mu gamma5E.
000038 | This agrees with the geometrically continued spin connection, including its
000039 | full lowered-contorsion sign. The supplied selfadjoint spectral convention
000040 | D_sa=-i gammaE.partial+gamma5E Phi+i c0 S_E gammaE gamma5E maps to it by
000041 |  i V D_sa V=K_E, V=exp(-i pi gamma5E/4).
000042 | This is a congruence of fermionic kernels, not unitary similarity of D_sa.
000043 | It keeps arbitrary original Phi matrices. For the full paired source,
000044 | transport the actual conjugation, Grassmann variables and regulated measure;
000045 | the constant rotation is not a theorem that its quantum Jacobian is one.
000046 | 
000047 | The primary discussion https://arxiv.org/pdf/1605.03231, section4 and
000048 | AppendixA, explains why Wick transport, charge-conjugation reduction and
000049 | fermion measure cannot be interchanged. It supplies no missing TFPT contour.
000050 | The local derivation here is explicit and does not import a new global time
000051 | foliation or curved-spacetime Wick theorem from that comparison.
000052 | 
000053 | 3. THE CORRECT EUCLIDEAN PARENT AND ITS EXACT REGULATED INTEGRAL
000054 | Since S_E=-integral L_L continued in this chart, its actual auxiliary part is
000055 |  L_aux,E=A S_E^2-S_E.(iB da+c0 J5E), A>0.
000056 | At each finite regulator cell, include an external source ell for S and set
000057 |  R=iB da+c0 J5E. With the displayed flat-cell Lebesgue normalization,
000058 |  integral_R4 d4S exp[-A S^2+S.(R+ell)]
000059 |     =(pi/A)^2 exp[(R+ell)^2/(4A)].
000060 | For a cell volume v the corresponding formula is
000061 |  (pi/(vA))^2 exp[v(R+ell)^2/(4A)].
000062 | A covariant measure carries its original cell/metric factors. None is set to1
000063 | by a new normalization. For a genuine finite matrix A the same statement
000064 | retains the appropriate determinant. Products over cells remain regulated.
000065 | 
000066 | The real Euclidean Gaussian is the convergent analytic continuation of the
000067 | original Fresnel Gaussian with its Feynman boundary value. The contour shift
000068 | is justified for this entire finite-regulator Gaussian and polynomial
000069 | Grassmann-even current sources. This proves no universal contour theorem for
000070 | an additional non-Gaussian primitive spectral body or moving APS projector.
000071 | If further factors depend on S, retain them inside the SAME integral.
000072 | Dropping them and calling the result the full S_adm would change the source.
000073 | 
000074 | All polynomial S insertions follow by ell derivatives. In particular
000075 |  <S_mu>_conditional=R_mu/(2A),
000076 |  <S_mu S_nu>_conditional=R_mu R_nu/(4A^2)+delta_munu/(2vA).
000077 | The second contact is lost by substituting only the classical saddle.
000078 | Every reference factor and every inserted original observable must use
000079 | this same pushforward, not an independently normalized second measure.
000080 | The actual quotient by Z_ref[0] remains one fixed original normalization.
000081 | 
000082 | 4. DERIVED EFFECTIVE ACTION WITH NO NEW COEFFICIENT
000083 | The exact local contribution after this original auxiliary integration is
000084 |  L_eff,E=B^2(da)^2/(4A)-i B c0 da.J5E/(2A)-c0^2 J5E^2/(4A).
000085 | Let c=Bc0/(2A), theta=2c a and kappa_phys=kappa/mu. Then
000086 |  Ftheta^2=2/(3kappa_phys^2), d=c0^2/(4A)=3kappa_phys^2/16,
000087 |  L_eff,E=Ftheta^2/2 [dtheta-i J5E/(2Ftheta^2)]^2.
000088 | The Lorentz effective contribution of the SAME original parent is
000089 |  L_eff,L=Ftheta^2/2 [dtheta-J5L/(2Ftheta^2)]^2.
000090 | These compact squares include both the derivative matter interaction and
000091 | the unique accompanying four-fermion contact; neither can be omitted.
000092 | No new decay constant, oscillator, counterterm or coupling is selected.
000093 | The old lambda cancels from Ftheta in this physical phase normalization.
000094 | Global phase periods and the original anomaly coefficient still require the
000095 | actual source map; their historical factor-two conflict is not repaired here.
000096 | 
000097 | 5. WHY THE FACTOR i IS NOW EXPLAINED, NOT DISCARDED
000098 | For the BOSONIC body J5E=0, the actual Euclidean saddle is
000099 |  S_E*=iB da/(2A)=(2i/3)dtheta.
000100 | The off-shell real geometric S integration variable is NOT this saddle.
000101 | Thus the real selfadjoint geometric spin operator and the antihermitian
000102 | axial derivative term on the imaginary saddle belong to different steps of
000103 | one parent integral. The extra i follows from that integral and Hodge map.
000104 | It need not be fitted away or treated as an independent imaginary Lorentz
000105 | coupling. After the original auxiliary integration the axial derivative
000106 | coupling can be moved into the actual mass phases by the full paired phase
000107 | map already proved in HISTORICAL-TOTAL-GRADE-PHASE-MAP.txt.
000108 | It cannot be so substituted into the bosonic trace of the independent S
000109 | operator, or into its bound reference, before doing that integration.
000110 | The full actual current equation is
000111 |  S_L=(2/3)dtheta-(kappa_phys^2/2)J5L.
000112 | Only S_L+(kappa_phys^2/2)J5L is the local gradient. Its curl is generally
000113 |  dS_L=-(kappa_phys^2/2)dJ5L.
000114 | 
000115 | The real geometric spin continuation is therefore compatible with the old
000116 | Lorentz auxiliary parent in this chart. This closes the specific local
000117 | factor-i ambiguity. It does not identify all of that parent with the current
000118 | primitive spectral action, whose independent S response and boundary body
000119 | must still be preserved. HISTORICAL-REAL-TORSION-OPERATOR.txt derives the
000120 | nonzero transverse a4 response of the same original geometric Dirac family.
000121 | That response is absent from the unintegrated algebraic historical parent.
000122 | A full identity must compute the actual reference/selector cancellation or
000123 | field restriction. On-shell gradient substitution is not that computation.
000124 | 
000125 | 6. ACTUAL TIME AND SOURCE CONSEQUENCES
000126 | The Lorentz phase momentum is p_theta=Ftheta^2 dot(theta)-J5L^0/2.
000127 | The real original current therefore enters the time Legendre transform.
000128 | Its temporal contact cancels in canonical variables as in the previous
000129 | geometric-matter note; this is not independent positivity of a fermion block.
000130 | The existing full Fock/see-reference transport and all original V_ji,
000131 | P_adm, domains and bosonic momentum connections remain required.
000132 | The finite Gaussian generates no independent propagating torsion-vector
000133 | pole. It produces the actual scalar/current terms of its supplied parent.
000134 | It supplies no exact masses of the complete interacting TFPT Hilbert space.
000135 | In particular no vacuum scalar mass is generated by this polynomial alone.
000136 | 
000137 | P25-L:196-205 attributes its printed photon term to the Fujikawa Jacobian.
000138 | Restoring the same unrotated fermions and adding that Jacobian again is not
000139 | this source. Keep the original regulated anomaly/eta line once, including
000140 | any constant-rotation phase and its bound reference. No cancellation from
000141 | an unregulated full trace gamma5 is inferred on an actual APS domain.
000142 | 
000143 | VERDICT: original local Wick/Hodge/spin map and exact regulated auxiliary
000144 | current integral established in the stated supplied chart. Source insertions
000145 | and prefactor are retained explicitly. The complete primitive S_adm,
000146 | H_tr,pos and common interacting spectrum have NOT been derived. Original
000147 | TFPT compiler, E8, flavor, alpha and coupled clocks retain their own scopes.