TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-HEAT-PROOF.txt Original file SHA-256: 5904d71d3b9cc978ef5025b86a43931db227d7ee9e87d894384f844b10f1d805 Inclusive original lines: 111-216 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. 000111 | Clifford/internal factors. An arbitrary Higgs change cannot be declared 000112 | reference-independent. The primitive source itself explicitly says an 000113 | almost-commutative factorization is a further chosen realization, not an 000114 | extra primitive reconstruction output. That conditional distinction is 000115 | preserved in section4 below. 000116 | 000117 | For constant sigma and B=0, Dsigma=exp(-sigma)D implies 000118 | zeta_Dsigma²(s)=exp(2sigma s)zeta_D²(s). 000119 | Res_1 scales by exp(2sigma); at a regular zero the value stays unchanged. 000120 | At a pole zeta=r/s+f+..., FP becomes f+2sigma r. If the reference is fixed, 000121 | its FP is subtracted linearly, without this scale shift. If B is fixed, 000122 | Dsigma+B is NOT exp(-sigma)(D+B). Neither dimensional correctness of the 000123 | extra chi² factor nor physical double counting follows from notation. 000124 | Retain the actual scale/coordinate/reference contract before deciding it. 000125 | 000126 | 4. Full supplied96 Higgs family through the conditional product consumer 000127 | Use EXACT original v252 full_D and the R026/R028 one-form map, all supplied 000128 | complex Yu,Yd,Ye,Ynu and symmetric complex MR. Let DY=full_D(Y,0), 000129 | DM=full_D(0,MR), u=|H|². Define internal traces 000130 | A=Tr96 DY²=4Y2, B=Tr96 DY⁴=4Y4, C=Tr96(DY² DM²)=2T, 000131 | Y2=Tr3(3Yu*Yu+3Yd*Yd+Ye*Ye+Ynu*Ynu), 000132 | Y4=Tr3(3(Yu*Yu)²+3(Yd*Yd)²+(Ye*Ye)²+(Ynu*Ynu)²), 000133 | T=Tr3(Ynu*Ynu MR MR*)=||Ynu MR||_HS²>=0. (4) 000134 | Here * is adjoint. No family matrices are assumed commuting. A,B>0 for 000135 | nonzero Y, C>=0 because C=||DY DM||_HS². 000136 | 000137 | The whole original block proof, using the left rotation R028, gives 000138 | Tr96(Phi²-DM²)=A u, 000139 | Tr96(Phi⁴-DM⁴)=B u²+4C u, 000140 | Tr96(dPhi dPhi)=A |dH|². (5) 000141 | In radial coordinates Phi=rDY+DM. Every word with an odd count of DY has 000142 | zero trace by the original chirality/slot paths. DM DY DM=0, so the six 000143 | quartic mixed words reduce cyclically to four DY² DM² words, giving4C. 000144 | Polarizing the exact quadratic identity for arbitrary four real H 000145 | coordinates gives Tr(L_a L_b)=A delta_ab, proving the derivative identity 000146 | also for nonconstant H with the supplied fixed Y and constant MR. 000147 | For gauge-covariant derivatives use the actual gauge intertwiner and 000148 | covariantly constant invariant MR. Varying MR would add its own terms. 000149 | 000150 | The first extra spacetime realization is explicitly the STANDARD product 000151 | D=-i gamma^mu nabla_mu tensorI96+gamma5 tensorPhi(H), 000152 | with spin4, Euclidean Clifford algebra and a fixed local spin connection. 000153 | It is used as a conditional consumer of the given original finite map, 000154 | not claimed to be a derivation of the primitive-to-geometric spacetime map. 000155 | In a flat chart D²=-partial²+Phi²-i gamma^mu gamma5 partial_muPhi. 000156 | The same curvature convention yields 000157 | E=-R/4-Phi²+i gamma^mu gamma5 nabla_muPhi. 000158 | The derivative term has spin trace zero, and its square has spin trace 000159 | 4 delta_mu,nu times the internal derivative product, hence POSITIVE kinetic. 000160 | Keeping the H-dependent bulk terms and integrating total derivatives for 000161 | interior changes gives 000162 | a2_H=-A u/(4pi²), 000163 | a4_H=A |DH|²/(8pi²)+B u²/(8pi²)+C u/(2pi²) 000164 | +A R u/(48pi²). (6) 000165 | Spin4 has been traced separately from internal96. These spectral traces 000166 | do not introduce96 independent physical Grassmann fermions. An actual 000167 | uniform trace weight w multiplies A,B,C together. A source with a distinct 000168 | projection/trace must be evaluated in its own contract, not fitted here. 000169 | 000170 | For the SAME two-residue combination c chi²a2+d a4, d>0, A>0, 000171 | Z=d A/(8pi²), Hcan=sqrt(Z)H, 000172 | mu_can²=2(c/d)chi²-4C/A-R/6, 000173 | lambda_can=8pi² B/(d A²). (7) 000174 | These are the coefficients of V=-mu_can²|Hcan|²+lambda_can|Hcan|⁴, 000175 | NOT a calculated physical pole mass or full connected H covariance. 000176 | At R=0 the negative quadratic condition is c chi² A>2d C. Positive c is 000177 | necessary, not sufficient. For printed c=-6,d=1 the local coefficient is 000178 | positive for every nonzero given Y and every symmetric MR, since C>=0. 000179 | Keeping only a sign change to c=+6 is a candidate convention comparison, 000180 | not an original selection, and it leaves the independent factor6 problem 000181 | of section2 untouched. Its R sign and common Wick map must also agree. 000182 | 000183 | 5. Big picture and next actual origin arrow 000184 | Equation(6) connects the given Yukawas to kinetic normalization, quartic 000185 | strength and the positive neutrino/Majorana mass correction. They are 000186 | moments of ONE operator, not independently tunable Higgs coefficients. 000187 | One field normalization cannot turn TrPhi⁴ into (TrPhi²)² or remove T/Y2. 000188 | Do not reinterpret the printed scalar (Phi*Phi)² as a96er matrix trace. 000189 | The full96 finite source and scalar geometric Erel need an actual map. 000190 | 000191 | R030 rejects the literal SAME local heat producer for the printed linear 000192 | mass and curvature coefficients, with the explicit scope in sections1-3. 000193 | R031 evaluates the whole finite original class through a conditional4D 000194 | consumer with no new Y fitting. Neither closes F1 or the common original 000195 | field/state/time/reflection/RP/measure problem. The next step is the actual 000196 | primitive H insertion into Dgeo+Brel, its admissible compression and bound 000197 | reference, and the resulting principal symbol/endomorphism and scale map. 000198 | It must explain the ratio or identify the different original coefficient, 000199 | before using that weight for the full dynamical H/source integration. 000200 | No extra free counterterm, Gamma_H, or independent Higgs model is allowed. 000201 | 000202 | [OMITTED: non-scientific conversation or execution metadata] 000203 | [OMITTED: non-scientific conversation or execution metadata] 000204 | [OMITTED: non-scientific conversation or execution metadata] 000205 | [OMITTED: non-scientific conversation or execution metadata] 000206 | [OMITTED: non-scientific conversation or execution metadata] 000207 | [OMITTED: non-scientific conversation or execution metadata] 000208 | [OMITTED: non-scientific conversation or execution metadata] 000209 | [OMITTED: non-scientific conversation or execution metadata] 000210 | [OMITTED: non-scientific conversation or execution metadata] 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]