TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C3-QUARTER-EQUIVARIANT-JOINT-LIFT.txt
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000001 | QUARTER-EQUIVARIANT COMMON K LIFT ON THE ACTUAL C3 SOURCE COVER
000002 | Independent analytic review, 2026-10-29. No enumeration or Atlas mutation.
000003 | 
000004 | 1. Exact original data retained
000005 | elliptic-moebius-source-bridge-20261026/CLOCK-Q-PACKET-COMPATIBILITY.txt:
000006 | 35-54 gives Q(z,w,eta)=(iz,1/w,-i eta/w^3), Q^4=1 on C;
000007 | 61-84 gives Q*L_a=L_(-a-1), hence L0<->L2 and L1 fixed;
000008 | 125-170 gives the actual flat coefficient lift and its old F basis:
000009 |   A_q=((0,0,-1),(0,1,0),(1,0,0)),
000010 |   K=((0,1,0),(1,0,0),(0,0,-1)),
000011 |   R_Q=K A_q K^-1=((1,0,0),(0,0,1),(0,-1,0)).
000012 | The existing line identifications, not just their divisor classes, are used.
000013 | A_q^2=diag(-1,1,-1), A_q^4=I, det A_q=1.
000014 | Positive t=0 flat-solution monodromy is the inverse, as in the original.
000015 | Atlas finding:negative-sign-packet-has-source-selected-quarter-clock-lift;
000016 | research-source:ba2e3cf072f3c6c8a083e8ba, SHA256
000017 | fe37f20bbb2d959dfe678ca1bf435a69067cfcb859a3d36769294bbe9bf61c32.
000018 | 
000019 | 2. Exact compatible assignment and its fourth line
000020 | Over X x C keep the GIVEN split E,D=det E and take in q order
000021 |   m_q=(0,-1,0,-1),
000022 |   W=Lambda-even E tensor [L0 + D^-1 L1 + L2 + D^-1].
000023 | Extend the actual coefficient Q lift by the identity on the fourth line.
000024 | Its family matrix in flat q frames is Q4=diag(A_q,1), with determinant one.
000025 | The two exchanged lines have EQUAL determinant power, namely zero.
000026 | The fixed L1 and the fourth trivial flat line each have power -1.
000027 | Since Q acts trivially on X, the D powers have their canonical pullback maps.
000028 | No map between unequal D powers is required.
000029 | 
000030 | In the original F order this is
000031 |   m_F=(-1,0,0,-1), Q4_F=diag(R_Q,1).
000032 | Using the SAME K, the native family phases are equivalently
000033 |   R_native,F=diag(omega,omega^2,omega^2,omega),
000034 |   R_native,q=diag(omega^2,omega,omega^2,omega).
000035 | Both M_q and R_native,q commute exactly with Q4.
000036 | K is the old printed comparison, but identifying these F slots with the
000037 | native E8 family-four frame is a specified joint assignment, not an
000038 | originally derived microscopic source map.
000039 | 
000040 | 3. Actual global K equivariance
000041 | Reuse the September22 natural lift
000042 |   iota_m(g)=[s(g,z),diag(z^(2m_j+1))], z^2=det g,
000043 |   Kgroup=(Spin10 x SU4)/diagonal Z4.
000044 | The actual flat sign packet has transitions l=diag(l0,l1,l2,1) in SU4.
000045 | The common transitions are k_ij=iota_m(g_ij)[1,l_ij].
000046 | The original Q line lift satisfies its transition compatibility with the
000047 | pulled-back flat packet. Since [Q4,diag(z^(2m_j+1))]=0 and X is unchanged,
000048 | that compatibility extends to the full k_ij cocycle, not just to W matrices.
000049 | The choice z->-z multiplies both local half factors by (-1,-I4), already
000050 | trivial in Kgroup. Q commutes with this kernel; no extra root choice appears.
000051 | The given determinant trivialization of L0 L1 L2 and det Q4=1 preserve SU4.
000052 | Four Q applications are the identity on every factor, including the
000053 | fourth line. Thus this is an actual C4-equivariant common principal K bundle.
000054 | Every associated E8 branch and its Lie brackets/forms inherit that lift.
000055 | 
000056 | The given split connection plus the canonical flat line connection is
000057 | Q-equivariant: E is pulled from X and the actual line lift is parallel.
000058 | The adapted native kappa is central in Kgroup, and sigma0 has the above
000059 | family phases and the color center on its Spin part. They commute with Q.
000060 | Hence the whole C4-equivariant E8 bundle carries the common global parallel
000061 | native-class C12 clock kappa sigma0 of the existing joint-lift source.
000062 | 
000063 | This is compatibility of the two specified actions. Q4 is NOT thereby equal
000064 | to kappa or to the unchanged original marked Jhat: on W, kappa is scalar
000065 | whereas Q4 has eigenvalues 1,1,i,-i. No such extra identification is claimed.
000066 | 
000067 | 4. Entire norm-minimum family without enumeration
000068 | For sum m_j=-2, sum(m_j+1/2)^2>=1. Equality requires each m_j in {0,-1}
000069 | and exactly two -1 entries. These are the six existing minima.
000070 | Q invariance in F order requires m2=m3 (one-based slots).
000071 | Therefore exactly TWO minima remain:
000072 |   (-1,0,0,-1), and (0,-1,-1,0).
000073 | They form ONE dual pair. In q order the pair is
000074 |   (0,-1,0,-1), and (-1,0,-1,0).
000075 | The total toral norm is two when the already fixed D5 contribution is one.
000076 | This norm statement is not a minimum of the original P1 action.
000077 | 
000078 | 5. Verdict and separation of the other deck action
000079 | The proposed C4 equivariant K/connection/clock compatibility is correct on
000080 | the actual given cover and with the stated frame assignment.
000081 | The independent c:w->omega w still cycles the three lines: it does not
000082 | preserve these fixed M/R marks. The earlier C3 transported-orbit result
000083 | remains unchanged; a C4 lift must not be called a fixed C3 descent.
000084 | No charged CAR identification, insertion-domain identity, state selection,
000085 | OS action or physical time is deduced. The full given Seed retains its
000086 | existing state and time. Original P1 selection of this joint assignment
000087 | is not proved by this compatibility.
000088 |