TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/REFERENCE-FIBER-EVALUATION.txt
Original file SHA-256: b816066a532162b0b6dbe2f49203c484ab1f91662b290a759f3e9223963c4ee5
Inclusive original lines: 144-188
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000144 | For the same constant circle coefficient let
000145 | 
000146 |  Phi=L(H)+DM, Phi*=Phi, {gamma,Phi}=0.
000147 | 
000148 | In its actual grading decomposition Phi has blocks [[0,B],[B*,0]] with
000149 | B a48x48 matrix. This B is the ENTIRE original coupled block, including
000150 | Yukawas, conjugates, colors, families and Majorana mixing. Do not separate
000151 | its eigenvalues into independent Yukawa and MR contributions. Let sigma_a
000152 | be the singular values of that actual B, a=1,...,48, and set
000153 | 
000154 |  E_a=sqrt(p^2+sigma_a^2).
000155 | 
000156 | The full h_A,p has blocks [[pI,-iB],[iB*,-pI]]. Squaring gives p^2+Phi^2,
000157 | and its exact eigenvalues are +E_a,-E_a. Consequently
000158 | 
000159 |  det(I+exp(-t h_A,p))=product_(a=1)^48 4cosh^2(tE_a/2),
000160 |  Esea,A,p=-sum_a E_a,
000161 |  Z_heat,A,p=product_a(1+exp(-tE_a))^2.                  (10)
000162 | 
000163 | On the even T branch h_R,T,p=h_A,p, so the same exact formulas apply to
000164 | its reference fermion kernel and sea factor. In particular the null source
000165 | raw determinant ratio is1, and the finite sea factor ratio is also1.
000166 | The WHOLE original factor is still
000167 | 
000168 |  Z_F,A,p/Z_F,R,T,p=s_A,p/s_R,T,p=a_A,p/a_R,T,p.          (11)
000169 | 
000170 | It is not set to1 merely from h_A=h_R,T. Equal actual complete original
000171 | prefactors would make this particular fiber ratio1; that is an additional
000172 | equality of already given source factors to check, not a new freedom.
000173 | 
000174 | For S the exact finite fermion ratio is instead
000175 | 
000176 |  Z_F,A,p/Z_F,R,S,p
000177 |   =(s_A,p/s_R,S,p) product_a
000178 |        [cosh(tE_a/2)/cosh(tp/2)]^2
000179 |   =(a_A,p/a_R,S,p) product_a
000180 |        [(1+exp(-tE_a))/(1+exp(-t|p|))]^2.              (12)
000181 | 
000182 | Thus the fermionic reference contribution has actually been evaluated
000183 | for these original branches without selecting new inputs. The entire
000184 | remaining original prefactor is displayed, not fitted or discarded.
000185 | The per mode prefactor notation in(8)-(12) is for a supplied mode fiber.
000186 | When forming the entire circle product, a global original scalar/body
000187 | factor is retained ONCE. It is not copied into each Fourier mode unless
000188 | the original caller actually factorizes it in that way.