TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-paper-source-reconciliation-20261102/shorttime_20261105M/INDEPENDENT-RANKFREE-AND-CONNECTED-REVIEW.txt
Original file SHA-256: 0cc8f37e4229f41762fc111439a780a7762410d8ad8f9333ded7c7326ba03881
Inclusive original lines: 157-234
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.

000157 | 
000158 | Rank free whole original counter mode continuation
000159 | The original constant product mode result extends to every supplied Yukawa rank at p!=0 without replacing a null source Pfaffian by its inverse. This removes the full rank limitation of formula(11) within exactly the same actual free product/counter mode/Euclidean mark class. It does not change the raw heat versus Euclidean source distinction or the remaining original weight/domain/history assumptions.
000160 | 
000161 | Correct complex family frame first
000162 | Let the actual original neutrino matrix have singular value decomposition Ynu=U D V^dagger, D=diag(y1,...,yr,0,...,0), yi>0. For the ORIGINAL block
000163 | Fnu=[[sqrt(u)Ynu,0],[MR^dagger,sqrt(u)Ynu^T]],
000164 | left/right multiplication by diag(U^dagger,V^T) and diag(V,bar U), respectively, produces
000165 | Fnu_new=[[sqrt(u)D,0],[M^dagger,sqrt(u)D]],
000166 | M=V^dagger MR bar V, M^dagger=V^T MR^dagger V.
000167 | M is still COMPLEX SYMMETRIC, because this is a unitary congruence of the actual symmetric MR. A unitary similarity V^dagger MR V would in general be the wrong Majorana transport. Singular values of Fnu and det(qI+Fnu^dagger Fnu) are unchanged by the displayed unitary left/right maps. No physical matrix value is selected by this coordinate change.
000168 | 
000169 | Put q=p^2>0, k=3-r, J=diag(y_i^(-2)) on the active r subspace, and write
000170 | M=[[A,B],[B^T,C]], B=M10, C=M00=C^T.
000171 | With G=diag(Ga,qI_k), Ga=u D_a^2+qI_r, the actual whole Schur identity is
000172 | Qnu=det(G)det(G+q M G^(-1)M^dagger).
000173 | For E=Ga^(-1), the second matrix S has exact blocks
000174 | Saa=Ga+BB^dagger+q A E A^dagger,
000175 | Sab=BC^dagger+q A E bar B,
000176 | Sbb=qI_k+CC^dagger+q B^T E bar B.
000177 | Taking its Schur determinant and using E=J/u+O(u^(-2)) gives
000178 | Qnu=a_nu u^(2r)+b_nu u^(2r-1)+O(u^(2r-2)),
000179 | a_nu=q^k [pdet(Ynu^daggerYnu)]^2 det_k(qI_k+C^dagger C),
000180 | b_nu/a_nu=2q TrJ+2 T_M,
000181 | T_M=Tr[J B(I_k+C^dagger C/q)^(-1)B^dagger]>=0.
000182 | Here pdet is the product of active eigenvalues, with empty product1.
000183 | 
000184 | The factor2 in the Majorana correction has a genuine complex symmetry justification. At order1/u the Saa determinant and one Schur term combine as
000185 | Tr JBB^dagger-Tr[(qI+CC^dagger)^(-1)C B^dagger J B C^dagger]
000186 | =Tr[J B(I+C^dagger C/q)^(-1)B^dagger]=T_M.
000187 | The other Schur term is q Tr[(qI+CC^dagger)^(-1)B^T J bar B]. Transposition, C^T=C and real diagonal J turn this into the SAME T_M. C need not be real, diagonal or Hermitian. A=M11 contributes only at the next lower order and has not been set to0.
000188 | 
000189 | Charged sectors and exact source integration
000190 | For f=u,d,e and rank r_f,
000191 | det_3(qI+uY_f^daggerY_f)
000192 | =q^(3-r_f) pdet(Y_f^daggerY_f) u^(r_f)
000193 |  *[1+q Tr((Y_f^daggerY_f)^+)/u+O(u^(-2))].
000194 | Combining its ORIGINAL powers6,6,2 with Qnu gives
000195 | K=6r_u+6r_d+2r_e+2r_nu,
000196 | Sigma_plus=Tr3[3(Yu^daggerYu)^+ +3(Yd^daggerYd)^+
000197 |                            +(Ye^daggerYe)^+ +(Ynu^daggerYnu)^+],
000198 | b_Q/a_Q=2[q Sigma_plus+T_M].
000199 | The whole genuine384 counter mode Pfaffian is Q^2. Its leading power is u^(2K), and its next coefficient relative to it is4[q Sigma_plus+T_M]u^(-1). The same actual four real Higgs integration therefore has Gaussian ratio1/[2(2K+1)] and the unchanged connected global Euclidean coupling mark gives the evaluated all rank result
000200 | lim_(r_IR down0) C_(u,S_Y,global,Euclidean)
000201 | =-8[p^2 Sigma_plus+T_M]/(2K+1).                     (12)
000202 | This has no newly chosen Yukawa rank premise: it uses the actual supplied ranks, active subspaces and original Majorana matrix. At K=0 every Yukawa vanishes and this Yukawa mark is identically0; the expression gives0 directly, without interpreting a negative degree polynomial term. At p=0 the genuine selfdual original mass pairing and its sourced minors must instead be used; formula(12) is not permission to divide by its potentially zero mass Pfaffian.
000203 | Full rank yields k=0,T_M=0,K=48 and precisely(11). If only neutrino rank is2 and the other three ranks are3, K=46 and the denominator is93. For its one dimensional null neutrino subspace,
000204 | T_M=[p^2/(p^2+|M00|^2)] ||D_a^(-1)M10||_F^2.
000205 | Thus Majorana active to null mixing CAN already enter the first subleading coefficient in this actual rank deficient branch. The previous absence of its first correction was a full rank statement, not a general exclusion of MR. All complex original phases are retained by the displayed congruence and Schur factors.
000206 | As before, the held fixed original remainder must possess the stated bounded, nonzero degree0 scaled limit on the actual ordinary constant Higgs coefficient domain. No such condition is imposed on the unknown whole physical H_B by fiat. Equation(12) is an actually integrated free product counter mode source coefficient. It is not the fixed time Y_ham(s), entire P restricted history source, or a sum over freely chosen new modes. No numerical fixture or additional test run was used in this independent Schur/Gaussian verification.
000207 | 
000208 | Robust finite regulator sign from the actual whole mode minor polynomial
000209 | There is a stronger exact consequence of the SAME original mode source that does not require the IR ray limit. At q=p^2>0 its radial polynomial Q(u)=sum_(j=0)^K a_j u^j has a_j>=0, a0>0. If any of the original Yukawa matrices is nonzero, K>0. This includes every actual rank and every complex symmetric Majorana matrix, not just invertible Yukawas.
000210 | 
000211 | The coefficient positivity is source specific, not a general assertion about complex Pfaffians. The original charged determinants det(qI+uY_f^daggerY_f) have nonnegative coefficients. For the exact whole neutrino factor use Cauchy–Binet on its ACTUAL6er matrix Fnu:
000212 | det(qI_6+Fnu^dagger Fnu)
000213 | =sum_(I,J subset{1,...,6}, |I|=|J|) q^(6-|I|) |det Fnu[I,J]|^2.
000214 | After the already proved original weak orbit transport,
000215 | Fnu=[[sqrt(u)Ynu,0],[MR^dagger,sqrt(u)Ynu^T]].
000216 | In a nonzero minor with selected top rows I_top and selected right columns J_right, every determinant term contains EXACTLY |I_top|+|J_right| factors sqrt(u). Top rows must use left columns; right columns must use bottom rows. The remaining factors come from the fixed MR^dagger block. Therefore
000217 | |det Fnu[I,J]|^2=u^(|I_top|+|J_right|)|det Fnu(1)[I,J]|^2.
000218 | Complex phases can cancel this fixed degree minor coefficient to0, but cannot create an interference term of a different u degree. This proves the claimed nonnegative coefficients with full Majorana mixing. The empty minor and q>0 give a0>0. A nonzero original Yukawa gives a positive u coefficient in a principal Cauchy–Binet contribution, hence a nonconstant Q. Multiplying the actual charged determinant powers6,6,2 preserves these facts. No new source is selected and no numerical coefficient test is needed.
000219 | 
000220 | For the whole actual384 counter mode Pfaffian P_pair=Q^2 and the SAME global Euclidean source g multiplying only the H linear fermion vertex, its conditional marked value is exactly
000221 | y_E(u)=-2u P_pair'(u)/P_pair(u)=-4u Q'(u)/Q(u).
000222 | There is no zero division here: P_pair(u)>0 for every u>=0 at q>0. Define the finite degree weights
000223 | pi_j(u)=a_j u^j/Q(u).
000224 | Then y_E(u)=-4 sum_j j pi_j(u), and the exact derivative is
000225 |  d y_E/d log u=-4 Var_pi(u)(j)<0 for u>0,
000226 | since a0>0 and at least one positive degree coefficient is nonzero. It extends continuously to y_E(0)=0 and is strictly decreasing on u>=0, bounded between -4K and0. These finite degree weights are an algebraic identity of the existing polynomial, not a newly chosen configuration measure.
000227 | 
000228 | Let the actual remaining H weight be any NONNEGATIVE source factor independent of this coupling mark, with an existing nonzero normalized integral, finite E(u), and nondegenerate u marginal. It may depend on Higgs direction or other retained variables. Because the supplied pair polynomial is radial, integrating those actual variables/angles gives the same positive marginal for u. The original marked expectation satisfies exactly
000229 | Cov(u,S_Y)=Cov_mu(u,y_E(u))
000230 |  =1/2 E_(mu tensor mu)[(u-u')(y_E(u)-y_E(u'))]<0.       (13)
000231 | Finite E(u) suffices for this expression because y_E is bounded. Strictness follows from nondegeneracy of u and strict monotonicity; a source concentrated at a single radius is the excluded degenerate equality case. If every Yukawa vanishes, y_E=0 and the covariance is0 instead.
000232 | 
000233 | Thus the ACTUAL original free product mode's global Euclidean Yukawa mark has a strictly negative connected norm response under EVERY actual positive nondegenerate finite moment H weight in this class. No IR insertion or degree0 ray limit is required for this SIGN statement. For example the already given positive local four component SH quartic caller, when paired with this genuinely matching original constant mode factor, supplies the required finite moments; its numerical value is not formula(11)/(12), whose ray assumptions are different. This is a qualitative source identity of the existing pair, not a freely chosen positive measure or a new physical model.
000234 | The positive remainder condition is essential. A whole physical selector can change the fermion polynomial; complex eta/reference signs cannot be replaced by a nonnegative factor. Extra H dependent fermionic vertices can change Q_Y and invalidate the Euler score used above. Nonconstant H mixes modes. The fixed time raw Hamiltonmark, whole OS histories and actual physical P source remain distinct. Equation(13) does not claim their already evaluated primitive C_UY(t,s), or transfer an unprojected coordinate mode sign to them without the actual original equalities.