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Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/HIGGS_VARIATION_PROOF.txt
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000135 | 6. Derive the actual existing finite-triple Higgs dependence, leaving inputs visible
000136 | 
000137 | The finite path is read from the owning implementation, not from an inferred
000138 | Atlas annotation:
000139 |   v252_full_finite_triple.py:64-69 explicitly enumerates nu/e and u/d, with3 colors;
000140 |   :86-99 defines dirac_block(Yu,Yd,Ye,Ynu) by inserting those four matrices;
000141 |   :139-168 fixes rep(lambda,q,m) for C+H+M3 with the weak/color actions;
000142 |   v254_inner_fluctuations.py:80-87 uses random Yukawa blocks for its example.
000143 | No such random block or chosen physical Yukawa value is used in this annex.
000144 | 
000145 | For one sector with the GIVEN right-to-left seed
000146 |   M=diag(Y_up,Y_down),
000147 | let e_R be the represented lambda=1,q=0 element and a_q the represented
000148 | lambda=0, quaternion q element. The actual one-form has LR block
000149 |   (a_q[D_F,e_R])_LR=q M,
000150 | where
000151 |   q=[[phi1,phi2],[-conjugate(phi2),conjugate(phi1)]].
000152 | Hermitian completion supplies M* q* in the reverse block. Including the seed
000153 | replaces q by H=I+q, which has the same quaternion coordinate space. Thus
000154 |   M_q(H)=[[phi1 Yu, phi2 Yd],[-conjugate(phi2) Yu,conjugate(phi1) Yd]],
000155 | with an identical color copy already present in the input representation.
000156 | The lepton formula uses Ynu,Ye. The physical full real-triple completion and
000157 | Majorana requirements retain their separate first-order conditions.
000158 | 
000159 | This also describes EVERY particle LR one-form for this supplied SM action:
000160 | if b acts as q_b on the left and r_b=diag(lambda_b,conjugate(lambda_b)) on the
000161 | right, the commutator and left multiplication give
000162 |   (a[D_F,b])_LR=q_a(M r_b-q_b M)=q_a(r_b-q_b) M.
000163 | The family matrices commute with r_b because its entries are scalars on the
000164 | family index. Both r_b and q_b are quaternions, hence q_a(r_b-q_b) is one.
000165 | Arbitrary sums retain this structure, and b=e_R realizes every q M. No Yukawa
000166 | value was chosen to obtain this identity. The Majorana sector and real
000167 | completion remain separate as stated above.
000168 | 
000169 | The scalar interaction derivative is therefore proportional to the EXISTING
000170 | Y input. Inner fluctuation does not supply a missing numerical or structural
000171 | Yukawa seed. For D_F=0, all [D_F,b] and therefore all finite one-forms vanish.
000172 | For a fixed represented algebra, every block reducing both D_F and the algebra
000173 | also reduces every a[D_F,b]. None of this selects an absent 3+2 representation.
000174 | 
000175 | The independent block formula was checked against the actual original
000176 | dirac_block and rep code on the complete72-real-coordinate basis of the four
000177 | complex3x3 input matrices and all4 quaternion coordinates. All288 block
000178 | identities, comprising663552 Gaussian-integer entries, agree exactly. These
000179 | coordinate probes certify a linear identity; they are not chosen physical
000180 | couplings or an origin test. The zero-seed control is likewise not claimed as
000181 | a P1/P2-admissible alternative world. We do not rerun a target-mass inverse-KMS
000182 | construction as a Yukawa derivation.