TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-INNER-SCALE-PROOF.txt Original file SHA-256: 8d041b0da6aef4a5e90007fb4c155029133138097e895517b396cb4fee32ef58 Inclusive original lines: 114-219 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. 000114 | If sigma is later integrated, Q(sigma) stays in that joint integral and 000115 | can contribute to its scale response. It is not an ignorable constant. 000116 | 000117 | Let Z_tilde denote the transported integral in(7). At fixed sigma, 000118 | log Z_out[j]=logQ+sum(1-q)j.Hstar+logZ_tilde[qj]. 000119 | Thus mean(H_x)=(1-q_x)Hstar+q_x mean_tilde(h_x), 000120 | kappa_n,out(x1,...,xn)=prod_i q_xi kappa_n,tilde(x1,...,xn), n>=2. 000121 | The first moment shifts; all higher connected answers survive with their 000122 | actual external leg factors. There is no Gaussianization or removal of 000123 | the mixed/higher words from R026. Absolute Z_out[0]=Q Z_tilde[0], with the 000124 | transported whole action/measure and reference retained. These relations 000125 | are exact coordinate transport, not a choice of physical state or vacuum. 000126 | 000127 | [OMITTED: non-scientific conversation or execution metadata] 000128 | [OMITTED: non-scientific conversation or execution metadata] 000129 | [OMITTED: non-scientific conversation or execution metadata] 000130 | [OMITTED: non-scientific conversation or execution metadata] 000131 | [OMITTED: non-scientific conversation or execution metadata] 000132 | [OMITTED: non-scientific conversation or execution metadata] 000133 | [OMITTED: non-scientific conversation or execution metadata] 000134 | [OMITTED: non-scientific conversation or execution metadata] 000135 | [OMITTED: non-scientific conversation or execution metadata] 000136 | [OMITTED: non-scientific conversation or execution metadata] 000137 | [OMITTED: non-scientific conversation or execution metadata] 000138 | 000139 | Same-bundle ordinary trace Einstein cancellation and the R033 scale test 000140 | apply to the full in-family inner fluctuation(4) with its stated references. 000141 | Equation(7) preserves that SAME comparison after full field/source/background 000142 | transport. It does not assert unchanged Taylor coefficients at two different 000143 | origins, or forbid genuinely different original Brel/trace/domain producers. 000144 | 000145 | 7. The normal index FIELD is not a bare five-slot endomorphism 000146 | Take the GIVEN carrier V=(3,1)_(-1/3)+(1,2)_(1/2), and the GIVEN Higgs 000147 | H=(1,2)_(1/2). Claim Hom_G^R(H,End_C(V))=0, with conjugation on End. 000148 | Color center diag(omega I3,I2), omega³=1, omega!=1, acts trivially on H. 000149 | Equivariance therefore kills both off-diagonal color/weak blocks of W(H). 000150 | Weak center diag(I3,-I2) acts trivially on the remaining diagonal blocks, 000151 | but sends H to-H. Real-linearity then gives W(H)=W(-H)=-W(H), hence0. 000152 | This proof includes both H and its conjugate; it does not use selfadjointness. 000153 | End_C(V)=2(1,1)_0+(8,1)_0+(1,3)_0+(3,2)_(-5/6)+(bar3,2)_(5/6). 000154 | It contains no colorless weak doublet. The original weak2 gauge action is 000155 | not the normal2 spin action, and neither is the spin4 spacetime module. 000156 | 000157 | This excludes ONLY that bare-module shortcut. The original wedge/Yukawa 000158 | map acts on the already given full fermion module, which has weak singlet/ 000159 | doublet endpoints; R026 supplies its literal full96 L(H). It is preserved. 000160 | The normal B_Ea has a nonzero first-order symbol on a different bundle. 000161 | To treat it as a zero-order term of the same spacetime anchor requires its 000162 | actual reduction/representation, not dimension matching. If P is the actual 000163 | kernel projection of B or its selfadjoint doubling, BP=0 gives PBP=0. 000164 | The zero-mode FIELD can couple through the given wedge tensor; this coupling 000165 | is not obtained by restricting the annihilating index operator to its kernel. 000166 | An index alone does not specify the full projector, metric, insertion or 000167 | all paired modes. No claim against the source's supplied Higgs doublet follows. 000168 | 000169 | 8. Exact whole zero-mode metric transport, without fitting profiles 000170 | Suppose the ACTUAL normal realization is a given self-adjoint B with a finite 000171 | constant-dimensional kernel, and its stated scale operation really is fBf 000172 | with the transported domain. This assumption is not inferred from its index. 000173 | Let S:C^k -> kerB be the given isometric basis map. Then 000174 | ker(fBf)=f^-1 kerB, 000175 | G=S* f^-2 S>0, T=f^-1 S G^-1/2, 000176 | T*T=I, rangeT=ker(fBf). (8) 000177 | For an ACTUAL vertex X whose relevant mode vectors are in its domain, 000178 | T*(fXf)T=G^-1/2 S*XS G^-1/2, 000179 | T*X T=G^-1/2 S*f^-1Xf^-1 S G^-1/2. (9) 000180 | No scalar exponent is assumed: G need not be diagonal or scalar if f varies 000181 | across the normal fiber. These are full matrix identities. For q=f² constant 000182 | ON that fiber, G=q^-1I and T=S in the same original Hilbert metric. 000183 | The overlaps then scale by q for fXf, and by1 for X. Mode normalization 000184 | alone does not choose which vertex the original inserts. 000185 | 000186 | For the given k=2 COMPLEX mode coefficients, h_norm=G^1/2 H_raw, hence 000187 | d4H_raw=(det_C G)^-1 d4h_norm (10) 000188 | when that coefficient measure is flat. This is a DIFFERENT change from(5), 000189 | whose Jacobian is q^4. Do not confuse normalized mode coordinates, the affine 000190 | seed coordinate, and the conformally transformed physical Higgs field qH. 000191 | Use the pushforward for a non-flat original measure. For x-dependent modes 000192 | retain the induced connection T*nabla T and exterior derivative couplings; 000193 | the exact Schur/memory response is not discarded by retaining only zero modes. 000194 | No actual normal metric, profile or overlap insertion is chosen here. 000195 | Isolation of the zero eigenvalue is required for the usual smooth spectral 000196 | projection perturbation theorem, not for the pointwise algebraic kernel 000197 | identity(8). No extra isolation gate is imposed on that identity. 000198 | 000199 | 9. Decisive remaining ORIGINAL map 000200 | Retain the original wedge-field -> full96 L(H) arrow, all supplied Y/Flavor/ 000201 | Alpha, clocks, state/time and whole-source memory. The source-resolved next 000202 | comparison is B_Ea / actual bosonic-relative realization -> Brel inside 000203 | Ksigma, together with its original trace and zero-mode/field normalization. 000204 | Use the actual full normal-mode embedding, operator action and reference; 000205 | match its Higgs argument to the already given wedge consumer. Do not invent 000206 | a bare C5 mass, add Phi twice, choose a new Psi, or fit Y/scale coefficients. 000207 | If actual normal modes are eliminated, retain the exact exterior Schur/ 000208 | memory response from R022, not just PBP. The given two-sheet weight formula 000209 | is a normalization readout; its name does not specify that operator map. 000210 | 000211 | [OMITTED: non-scientific conversation or execution metadata] 000212 | [OMITTED: non-scientific conversation or execution metadata] 000213 | [OMITTED: non-scientific conversation or execution metadata] 000214 | [OMITTED: non-scientific conversation or execution metadata] 000215 | [OMITTED: non-scientific conversation or execution metadata] 000216 | [OMITTED: non-scientific conversation or execution metadata] 000217 | [OMITTED: non-scientific conversation or execution metadata] 000218 | [OMITTED: non-scientific conversation or execution metadata] 000219 | [OMITTED: non-scientific conversation or execution metadata]