TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/COMMON-CURVATURE-RESPONSE-PROOF.txt
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000119 | 5. Whole existing Higgs source gives a distinct curvature response
000120 | R031/R045 give, conditional on their actual fixed-H reference branch,
000121 |  SH=int[|DHc|²-mu0²|Hc|²+lambda|Hc|^4+xiR|Hc|²],
000122 |  xi=1/6,lambda=8pi² B/(d A²)>0,
000123 |  mu0²=2(c2/d)chi²-4C/A,
000124 |  A=Tr96DY²,B=Tr96DY^4,C=Tr96(DY² DM²)>=0.            (8)
000125 | AllY/MR stay supplied. Inner change remainsL(H)-DY, no secondPhi.
000126 | No numerical physical vacuum or mass is fitted. On the specified flat
000127 | classical broken stationary branch mu0²>0,v²=mu0²/(2lambda), use the existing
000128 | weak gauge-orbit chart Hc=(0,v+rho/sqrt2). Its radial kinetic is canonical.
000129 | Then m_rad²=4lambda v²=2mu0² and g_Rrho=sqrt2 xi v.
000130 | For fixed background metric, the full QUADRATIC radial operator is
000131 |  K_R=-Delta+m_rad²+xiR; at orderR² use K0=-Delta+m_rad².
000132 | Gaussian integration gives the tree curvature form factor
000133 |  DeltaGamma_tree=-xi²v² int R K0^-1 R.               (9)
000134 | For spectral momenta much smaller than m_rad, this tends to
000135 |  -xi²/(4lambda) intR²=-d A²/(1152pi² B) intR².       (10)
000136 | Independent algebraic elimination without derivatives gives
000137 |  u_star=(mu0²-xiR)/(2lambda),
000138 |  V_min=-(mu0²-xiR)²/(4lambda), mu0²>xiR.             (11)
000139 | It reproduces(10), as well as xi v²R and the constant vacuum contribution.
000140 | The Einstein shift xi v²R must be retained in the same common action.
000141 | On the unbroken classical branch the linearRrho source is absent; loop
000142 | effects are not excluded. This tree coefficient does not apply across the
000143 | branch boundary or beyond its low-momentum/curvature expansion.
000144 | 
000145 | This supplies a concrete connection: the existing Yukawa traces fix the
000146 | leading local susceptibility through A²/B; Majorana C/A changes the radial
000147 | scale and hence the momentum range of that local limit. No newR² knob is
000148 | chosen. The pole of(9) is the existing fixed-background radial response.
000149 | When gravity is dynamical, mixing can change physical poles, but integrating
000150 | an existing field does not add a new independent scalar degree of freedom.
000151 | An extra copy of(9)/(10) beside the unchanged full dynamicH would double
000152 | count this response. This is not the independent TFPT scalaron derivation.
000153 | 
000154 | 6. Preserve the whole source, determinant, reference and higher vertices
000155 | At the same finite regulator and actual integration domain, the quadratic
000156 | radial source is proportional to
000157 |  Z_quad[R,J]=exp(-Sbase[R]) det(K_R)^(-1/2)
000158 |        exp[1/2 (J-gR) K_R^-1 (J-gR)].               (12)
000159 | The determinant contributes+1/2logdetK_R to Gamma=-logZ. It is not discarded
000160 | from Z[0]. The real full radial potential also has sqrt2lambda v rho³ and
000161 | lambda rho^4/4. AngularH,gauge,fermionic and original additional vertices
000162 | remain in the full R026/R027 functional and its mixed words. Equation(12)
000163 | is an exact regulated QUADRATIC subintegration under its Gaussian existence
000164 | conditions, not an approximation silently substituted for the whole source.
000165 | Actual H-dependent reference jets must be integrated/subtracted with their
000166 | own original prescription; a fixed reference is not freely selected here.
000167 | Loops, coupled metric integration and all higher source cumulants remain
000168 | different operations. CCM section5.5 explicitly discusses radiative running