TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ROOT-WHOLE-MU01-SUPPORT-AND-PERIOD-REVIEW.txt
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000001 | ROOT REVIEW: THE COMPLETE SAME-INPUT mu01 HARDY READER
000002 | 30 October2026. Original data and earlier source conventions retained.
000003 | 
000004 | 1. All nine entries, independently from the old support/sign rules
000005 | 
000006 | The already reviewed input is E_s on q_s and JE_r on p_(r+1).
000007 | q_s meets only p_(s+1),p_(s+2). Thus r=s or r=s+1 are the only supported
000008 | entries. For the SAME h0q,h1p coefficient, eta-sheet signs are
000009 |  q=(-1,-1,+1), p=(+1,-1,-1).
000010 | They follow directly from the simple zero/pole of h0q at q0/q1 and h1p
000011 | at p1/p2: w-w_center has eta-order2, so its square root has odd parity.
000012 | The other marked centers are even. z is odd. Products+1 therefore cancel
000013 | between the actual two shared arcs; products-1 add, as in the oldM02 proof.
000014 | The supported diagonal products are all+1. For r=s+1, products for
000015 | s=0,1,2 are respectively+1,-1,-1. Hence the ONLY nonzero entries are
000016 | (r,s)=(2,1),(0,2). The three disjoint pairs and all four cancellations
000017 | give seven exactzeros. This is the whole SAMEinput matrix, not a choice
000018 | of two desired entries.
000019 | 
000020 | 2. The actual calibrated sign is forced by the already givenG andR
000021 | 
000022 | At the original lift nearz1, (h0,h1,h2)=(-omega,-omega,+omega).
000023 | G columns use h_(-s), so q0,q1,q2 values of scalarh0 are
000024 | (-omega,+omega,-omega). The givenR takes q_r to p_(r+1) and transports
000025 | h0q to h1p. Hence p0,p1,p2 values are(-omega,-omega,+omega).
000026 | Atw0,z1, alpha01=-h0q h1p/omega dw. For q2/p1 this is-omega dw;
000027 | for q1/p0 it is+omega dw. These determine the old and new continuous
000028 | zeta signs, with no separately chosen square-root reset.
000029 | 
000030 | 3. Independent continuous periods and exact elementary primitive
000031 | 
000032 | Along w=it use zeta=-omega sqrt(1-w)sqrt(1+omega w).
000033 | Along w=exp(-ipi/6)t use the SAME roots continued from1, with+omega.
000034 | On both paths w^3=-it^3. Their actual CCWz arcs have
000035 | z=+/-exp(-i arctan(t^3)), running tinfinity to0. The old NS/normal/graph
000036 | factor is therefore identical, (4Xstar/pi) for both integrals.
000037 | Root independently integrated BOTH actual continuous-root periods at70
000038 | digits. It reproducesI02 and
000039 |  I21=-0.4213752856971043792014444689166786680971744633140320
000040 |       +1.572593975275444788048708055806669722450561285370358i.
000041 | The ratio is real negative, -3.3475976640251801553608318488192678119... .
000042 | This numeric agreement is a replay; the proof uses the following primitive.
000043 | 
000044 | With u=zeta/(w-1), w=(u²-omega²)/(u²+1), a²=omega/(2omega+1),
000045 |  C=2*3^(-3/4)exp(-5ipi/12), Ca=-2omega/3, one has
000046 |  F=C[atan(u/a)-sqrt2 atan(u/(sqrt2a))].
000047 | Root independently simplifies the difference between F' and
000048 | zeta(w)w'/(1-w³) to EXACTzero as a rationalfunction. For the newpath,
000049 |  Im[(u/a)²]=sqrt3(t²-1)/(2(t²-sqrt3t+1)),
000050 |  (u/a)² att1=-3-sqrt3.
000051 | Actualu/a crosses the negative imaginaryaxis right-to-left below-i;
000052 | so does u/(sqrt2a), at -i sqrt((3+sqrt3)/2). Both continuedatans gainpi.
000053 | Endpoint oddness of the principal primitive gives-I02 and the increments
000054 | addpi(1-sqrt2)C. Thus EXACTLY
000055 |  I21=-I02+pi(1-sqrt2)C.
000056 | SinceI02=CL withL>0 and pi(sqrt2-1)>0, I21 is nonzero with the opposite
000057 | common phase. Its sign is calibrated, not fitted from the quadrature.
000058 | 
000059 | 4. Whole operator and already established KO class
000060 | 
000061 | Thus the whole family operator is
000062 |  M=(4Xstar/pi)(I02E02+I21E21), with the existing finiteP_gamma,+.
000063 | It has rank2,M²=(4Xstar/pi)²I02I21E01,M³=0. Root independently verifies
000064 | charpoly((M+M^T)(M+M^T)* )=lambda(lambda-r²)² for the common-phase real
000065 | coefficient form. This reuses the already proven GENERAL identical-family
000066 | KO theorem. In that declared96comparison, neutralB has rank32 and fullodd
000067 | KO completion rank64. Under a complex familyframe change carryJ_F and
000068 | all Cliffordlegs too; do not freezeJ_F and re-symmetrize anotherframe.
000069 | The complete sameinput answer alone is not the entire physicalY or the
000070 | selection of a wholeC6field insertion. No mass/CKM fit or new source is made.
000071 | Other originalmu inputs must be carried with their actual formpullbacks,
000072 | NS defect,J andendpointframes before their finiteoperators are inferred.
000073 | 
000074 | Independent outputs: ROOT-TWO-ACTUAL-HARDY-PERIODS-CHECK.json and
000075 | ROOT-WHOLE-MU01-ALGEBRA-CHECK.json. Original first-entry proof preserved.