TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/MU-FINITE-ONEFORM-REAL-SPAN-SCOPE.txt
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000001 | REAL VERSUS COMPLEX ONE-FORM SPAN: PRECISE SCOPE SUPPLEMENT
000002 | 30 October 2026. The preceding registered report is unchanged.
000003 | 
000004 | The explicit Higgs witness in MU-CURRENT-DZ-FINITE-INSERTION-REVIEW.txt
000005 | has the quaternion q(H)=[[t4,bar(t3)],[-t3,bar(t4)]]. This is exact for
000006 | the actual chosen a_F=pi(0,q(H),0), b_F=pi(1,0,0).
000007 | 
000008 | For every original left-to-right one-form the structural factorization
000009 |   (a[D_F,b])_LR=q_a(b_R-q_b)M
000010 | is likewise exact. Any linear combination retains one COMMON 2x2 factor
000011 | Q multiplying the supplied Yukawa blocks, in all generations/sectors.
000012 | 
000013 | The additional assertion Q belongs to the quaternion REAL form is
000014 | preserved for REAL linear combinations of those quaternion expressions.
000015 | It is not preserved under arbitrary COMPLEX linear coefficients:
000016 |   H tensor_R C = M2(C).
000017 | For example i0 I2 is in this complex span but is not of the displayed
000018 | quaternion form, whose two diagonal entries must be conjugate.
000019 | Selfadjoint completion alone does not repair that inference: the block
000020 | [[0,QM],[M*Q*,0]] is selfadjoint also for nonquaternion Q.
000021 | 
000022 | Original verification/v254_inner_fluctuations.py:47-60 lists four real
000023 | quaternion generators. Its actual tests85-127 check oddness and support,
000024 | not a reality constraint on general expansion coefficients or the complete
000025 | real scalar dimension. Moreover in_span:135-139 uses numpy.linalg.lstsq
000026 | on a complex matrix, so that particular span test is a COMPLEX column-span
000027 | test. It must not be cited as enforcing a real-field condition.
000028 | 
000029 | The standard intended real algebra A_F=C+H+M3(C) can carry the real
000030 | one-form convention, but it must remain that explicit convention when a
000031 | full quaternion claim is used. No automatic real-field theorem is obtained
000032 | by allowing arbitrary complex coefficients in the same numerical span.
000033 | 
000034 | Therefore the durable unconditional result of the cited computation is:
000035 |   explicit quaternion weak witness; general common 2x2 factor Q.
000036 | The simultaneous double-conjugation relations of equation(8) in the prior
000037 | report apply to the quaternion-real Higgs contract, not to a freely
000038 | complexified one-form calculus. The common-family-factor condition itself
000039 | survives either convention. No original source, Atlas entry or prior hash
000040 | was changed by this supplement.