TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-THREE-HYPOTHESES-AND-WHOLE-SOURCE.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000100 | 4. H3: THE ACTUAL FERMION STAGE AND ITS FIXED TRACE WEIGHT
000101 | Retain exactly the original Spin4/full96 constructor from E, with its
000102 | ordinary total Weyl embedding Eeps and R=C tensor KF. On a compatible
000103 | J-real selfadjoint Gamma-odd paired operator/domain,
000104 |  M=Eeps^T R^dagger D Eeps,
000105 |  M^dagger M=Eeps^dagger D² Eeps.
000106 | J commutes with D and exchanges the two total Weyl fibers. Hence the two
000107 | restrictions of D² are antiunitarily isospectral. Consequently, on an
000108 | invertible finite patch or with the SAME compatible spectral prescription,
000109 |  -log|Pf M|=-(1/4)Tr_Eeps log(M^dagger M)
000110 |            =-(1/8)Tr_full log D².
000111 | The phase of Pf M remains separate and MUST remain in the whole source.
000112 | This is not permission to replace a dynamical Pfaffian by its modulus.
000113 | Zero modes use the entire oriented source polynomial P[M,eta], as in E.
000114 | The factor1/8 belongs to this full-product trace convention, not to a new
000115 | physical species count. Eeps is NOT Padm. No arbitrary physical projector,
000116 | field-dependent domain or actual primitive determinant-line identity is
000117 | inferred from the finite fiber count.
000118 | 
000119 | The new exact matrix check uses the SAME symbolic full Y/MR producer and
000120 | all four real constant S components at zero momentum:
000121 |  D=gamma5 Phi+(3i/4)S_mu gamma^mu gamma5,
000122 |  D²=I4 tensor Phi²+(9/16)|S|² I384.
000123 | Both actual192 skew matrices obey the displayed squared-kernel identity;
000124 | the actual J fiber intertwiners are unitary and preserve P=D². No fitted
000125 | masses, new oscillator, alternate source or new physical regulator is used.
000126 | The analytic operator identity still requires the same paired domains.
000127 | 
000128 | The proper-time representation of this absolute logarithm has weight+1/8
000129 | on the ordinary full heat trace. As a UV DIAGNOSTIC in this declared chart,
000130 | its positive proper-time leading S² coefficient is
000131 |  (1/8)a2_S/epsilon=9dim(F)/(256pi² epsilon),
000132 | whereas the original isolated geometric -6chi²Res1 gives
000133 |  -6chi²a2_S=-27chi²dim(F)/(16pi²),
000134 | with a2_S=9dim(F)/(32pi²). These opposite local signs exclude their direct
000135 | identification in this positive-proper-time prescription alone. They are
000136 | not a selection of a primitive regulator, a proof of quantum instability,
000137 | or a no-go for the complete relative source with all original contributions.
000138 | The corresponding ordinary local a2 ratio of S² to R is -27/2, while the
000139 | declared Einstein-Cartan scalar is R-3S²/2. This is another isolated local
000140 | test; it does not identify the minimal parent as a proven bare component
000141 | of the primitive action or justify a new bare-plus-induced model.
000142 | 
000143 | Even the correctly normalized logarithm is NOT Tr f(D), Res/FP zeta, or
000144 | the original Tr f((Drel+Aseed)/chi)-Tr f(Dref/chi)+i*pi*DeltaEta/2.
000145 | Its numerator operator is different until the actual Aseed=Brel directsum
000146 | nabla_chi map has been evaluated. The whole regulated fermion polynomial,
000147 | with every external eta leg and its bosonic dependence, must be retained.
000148 | Its higher field jets follow from M and its inverse only on nonsingular
000149 | patches; the polynomial formulation, not an inverse, crosses Pfaffian zeros.
000150 | 
000151 | 5. TESTING THE STRONGEST KNOWN ANOMALY BRIDGE WITHOUT A NEW FIELD
000152 | Primary papers arXiv:1001.2036, equations4.16,4.19,5.22, and arXiv:1106.3263,
000153 | equations4.10-4.18, connect a spectrally regulated fermion determinant to
000154 | a spectral SCALE RESPONSE. They do not equate the whole determinant to
000155 | an arbitrary spectral trace. The 2011 paper separates Weyl invariant and
000156 | non-invariant parts explicitly and states that the split is not unique.
000157 | Its full source/operator/counting/regularization hypotheses must be matched;
000158 | an ordinary Dirac determinant cannot replace the actual paired192 Pfaffian.
000159 | 
000160 | On the actual nonzero source patch define the transported scale cocycle
000161 |  C(D,sigma)=log Zpsi(D)-log Zpsi(D_sigma).
000162 | It obeys C(D,0)=0 and the cocycle law under composed actual scale transports.
000163 | It fixes a difference/variation. A scale-invariant contribution is not fixed
000164 | by this difference. This does NOT prove a free physical parameter in TFPT:
000165 | the supplied whole omega/seed/reference may already fix that contribution.
000166 | Its actual evaluation on the common field chart remains necessary.
000167 | 
000168 | In the original constant scale anchor with unscaled reference and fixed
000169 | compatible domain, Ksigma=e^-sigma D when its original Brel vanishes,
000170 |  zeta_rel,sigma(s)=e^(2sigma s)zeta_D²(s)-zeta_Dref²(s).
000171 | The a2 residue scales with e^(2sigma); the regular value at0 is unchanged
000172 | by constant scaling. A zeta logarithmic determinant instead has a linear
000173 | sigma response at a regular0. A pole at0 carries its own finite-part terms;
000174 | it may not be suppressed. These typed scale responses cannot be exchanged
000175 | merely because they use the same Dirac spectrum. This reuses the existing
000176 | bound-reference scale anchor, without assuming arbitrary Brel vanishes.
000177 | 
000178 | Verdict H3: generic partial fermion integration is constructive and its
000179 | trace normalization is now fixed on the actual product. The stronger
000180 | historical/geometric/primitive WHOLE-source identity is not established.
000181 | The anomaly route fixes only its actual variation; a new dilaton integral,
000182 | chosen cutoff profile or a counterterm fitted to the desired coefficients
000183 | would change the authorized problem. No such replacement is introduced.
000184 |