TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/REFERENCE-FIBER-EVALUATION.txt
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000001 | ORIGINAL SEAM EVEN REFERENCE: COMPLETE FERMION FIBER EVALUATION
000002 | 3 November 2026. Original R032 source branches, no new lift or parameter.
000003 | 
000004 | Result
000005 | 
000006 | For the physical seam even Higgs class and the actual R032 bundle chart,
000007 | the reference coefficients simplify exactly to h_R,S=A gamma and
000008 | h_R,T=h_A. Their entire finite fermion fibers, coherent kernels, source
000009 | densities, sea factors and circle mode determinants are evaluated below.
000010 | All actual Yukawa, Majorana and background couplings remain. The known
000011 | sea factor is separated from the whole original reference prefactor;
000012 | no boson weight or whole Z_ref is set to1.
000013 | 
000014 | 1. The seam even premise and the actual reference lifts
000015 | 
000016 | [OMITTED: non-scientific conversation or execution metadata]
000017 | 
000018 | documents/archive--tfpt-45--source_extracts--05_metrology_source/sections/
000019 | 001-subsection-from-bridge-to-geometric-completion.tex:274-280 states that
000020 | the unique light original Higgs doublet is seam even: tau*Phi=+Phi.
000021 | 
000022 | The existing source proof
000023 | experiments/theory-contracts/tfpt-unified-source-derivation-20261029/
000024 | aps_source_review/ORIGINAL-HIGGS-BOUND-REFERENCE-PROOF.txt:48-62 explicitly
000025 | defines H_tau using the supplied scalar/bundle pullback. It intertwines L
000026 | and transports the given Yukawas and MR unchanged in THAT chart. Put
000027 | 
000028 |  H_even=(H+H_tau)/2, H_odd=(H-H_tau)/2.
000029 | 
000030 | Thus the physical even class has H_tau=H, H_even=H, H_odd=0 in this
000031 | actual equivariant field chart. A different internal transport would need
000032 | its original dictionary; evenness is not imposed on an unrelated raw
000033 | quaternion coordinate by notation. Also H_tau is the stated base/bundle
000034 | pullback, not the full conjugation of the Dirac operator by its spin lift.
000035 | 
000036 | The two actual R032 lifts give
000037 | 
000038 |  Dref,S=D0+Z L(H_odd),
000039 |  Dref,T=D0+Z[L(H_even)+DM].                             (1)
000040 | 
000041 | REFERENCE-SOURCE-COMPLETION.txt:31-45 in the existing
000042 | original-local-source-variable-background-20261102 contract transports
000043 | them through the same complete U and common original domain:
000044 | 
000045 |  h_R,S=A gamma-i gamma L(H_odd),
000046 |  h_R,T=A gamma-i gamma[L(H_even)+DM].
000047 | 
000048 | Therefore on the physical even class,
000049 | 
000050 |  h_R,S=A gamma,
000051 |  h_R,T=h_A=A gamma-i gamma[L(H)+DM].                    (2)
000052 | 
000053 | This follows without [A,Phi]=0 and is valid for the smooth mixed
000054 | backgrounds of the given fixed product chart. It chooses neither S nor T
000055 | as the physical global lift. Endpoint conditions, reflection and actual
000056 | reference domains remain the transported ones. It is the original R032
000057 | reference branch calculation, not an independently selected reference.
000058 | 
000059 | 2. Whole finite exterior fiber and its actual scalar
000060 | 
000061 | SEED-FIELD.txt:149-177 supplies, at a compatible finite original cutoff,
000062 | 
000063 |  Q_raw,j(t)=Gamma(exp(-t h_j)),
000064 |  Esea,j=Tr[1_(h_j<0)h_j],
000065 |  s_sea,j(t)=exp(t Esea,j),
000066 |  Q_heat,j(t)=s_sea,j(t)Q_raw,j(t),                      (3)
000067 | 
000068 | where j=A,R,S or R,T with its actual coefficient. The particle/hole
000069 | excitation form of Q_heat is exp(-t H_exc,j), H_exc,j=dGamma(|h_j|).
000070 | The original field map between signed old fields and excitations remains;
000071 | the grading gamma is not the spectral sign of h.
000072 | 
000073 | The entire ORIGINAL conditional caller can carry a further scalar or
000074 | remaining body factor. At this finite fixed boson fiber write exactly
000075 | 
000076 |  s_j=a_j s_sea,j,
000077 |  Q_j=a_j Q_heat,j=s_j Gamma(exp(-t h_j)).                (4)
000078 | 
000079 | Here a_j is the actual supplied remaining factor, not a new adjustable
000080 | normalization. The known sea factor in(3) does not calculate a_j. It can
000081 | include the given original scalar, reference, eta and other source factors
000082 | at that stage. Do not set a_R=1 unless that whole original caller says so.
000083 | If a remaining factor is operator valued, retain its given order outside
000084 | the fixed classical fiber operation. No determinant of an averaged or
000085 | noncommuting boson operator matrix is used below.
000086 | 
000087 | In the original charged coherent coordinates the complete fixed fiber is
000088 | 
000089 |  ((eta|Q_j(t)|xi))
000090 |        =a_j exp(t Esea,j) exp[bar eta exp(-t h_j)xi].    (5)
000091 | 
000092 | This includes every exterior minor and all the actual mixed fermion
000093 | components. It is not merely a determinant or a list of eigenvalues.
000094 | Original orientation and source ordering are those in SEED-FIELD:154-170.
000095 | 
000096 | The finite fermion trace is exactly
000097 | 
000098 |  Z_F,j=a_j exp(t Esea,j) det(I+exp(-t h_j)).             (6)
000099 | 
000100 | Proof: the exterior trace is the sum over all occupied subsets of the
000101 | products exp(-t lambda_i), hence product_i(1+exp(-t lambda_i)). This is
000102 | an identity on the existing complete coordinate exterior space; it adds
000103 | no second determinant/Pfaffian to a previously integrated physical source.
000104 | 
000105 | The determinant in(6) is the trace on the ENTIRE exterior/CAR coordinate
000106 | fiber of this given chart. It is not a count of96 independent physical
000107 | Standard Model fermions. If the original physical closure retains an
000108 | additional selector P on a different or coupled physical space, its
000109 | actual trace remains Tr_F(P Gamma(R)), with its source contacts and
000110 | remaining factors; it is not replaced by det(I+R). The positive reference
000111 | density derived on this one face from the actual whole Gref therefore
000112 | does not itself equal the physical joint Z_ref.
000113 | 
000114 | 3. Actual constant circle mode: S branch completely evaluated
000115 | 
000116 | Now specialize only to the already existing constant circle chart
000117 | A=p I_96, p=n+alpha, with its actual original alpha and t>0. No numerical
000118 | alpha, length or Yukawa is selected. Write P_plus/minus=(I+/-gamma)/2.
000119 | The actual full96 grading has ranks48/48. Thus
000120 | 
000121 |  h_R,S,p=p gamma,
000122 |  exp(-t h_R,S,p)=exp(-tp)P_plus+exp(tp)P_minus,
000123 |  Esea,R,S,p=-48|p|.                                    (7)
000124 | 
000125 | The raw exterior trace, with NO sea factor absorbed, is
000126 | 
000127 |  Z_F,R,S,p/s_R,S,p
000128 |   =(1+exp(-tp))^48(1+exp(tp))^48
000129 |   =[4cosh^2(tp/2)]^48.                                (8)
000130 | 
000131 | For the actual sea corrected heat source,
000132 | 
000133 |  s_sea,R,S,p=exp(-48t|p|),
000134 |  Z_heat,R,S,p=(1+exp(-t|p|))^96,
000135 |  Z_F,R,S,p=a_R,S,p(1+exp(-t|p|))^96.                   (9)
000136 | 
000137 | The pure fermion fiber normalization is therefore fixed exactly. The
000138 | factor a_R,S,p remains whatever its whole original source provides.
000139 | Equations(8) and(9) are different explicitly related conventions, not
000140 | two interchangeable choices of the same absolute reference scalar.
000141 | 
000142 | 4. Full mixed96 A and T fibers, including every Majorana entry
000143 | 
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)