TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C3-COVER-JOINT-CONNECTION-REVIEW.txt
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000001 | INDEPENDENT REVIEW: ACTUAL C3 COVER AND COMMON K CONNECTION
000002 | 2026-10-29. Analytic source review; no enumeration or Atlas mutation.
000003 | 
000004 | 1. Reused exact source
000005 | elliptic-moebius-source-bridge-20261026/SIGN-LINE-COVER-DICTIONARY.txt:
000006 | 11-31,62-87,139-168, especially c*L_a=L_(a-1) at 82-84.
000007 | Atlas finding:actual-v117-sign-lines-are-negative-weierstrass-triangle-two-torsion;
000008 | research-source:c1ff3dffc1d1a7f7e0ebc182, SHA256
000009 | 555f75a2d3c67347fa5f8e3c4f7f46e79d94cb4172176c84010224585dc842e7.
000010 | The original negative q packet is retained, not replaced by p or antipodal lines.
000011 | On C: eta^2=1-w^6, w^3=(z^2-1)/(z^2+1),
000012 | L_a=O(q_a-q_(a+1)) are distinct nonzero unitary flat 2-torsion lines,
000013 | L0 L1 L2=1, and pi*F_v117=L0 direct-sum L1 direct-sum L2.
000014 | Hence its holonomy ON C is the diagonal V4 character representation.
000015 | The irreducible base representation and this reducible pullback are compatible.
000016 | 
000017 | 2. Global joint lift, not just four local matrices
000018 | Let X carry the GIVEN split rank-five bundle E=E3 direct-sum E2 and
000019 | D=det E; pull E,D back to X x C and the L_a back from C.
000020 | Use m_A=(0,0,-1,-1) and
000021 |   W=Lambda-even E tensor [L0 + L1 + D^-1 L2 + D^-1].
000022 | The plus signs in this formula denote direct sums.
000023 | Choose flat sign transitions l_a in {+1,-1} with l0 l1 l2=1,
000024 | possible with the actual tensor-product trivialization of the q packet.
000025 | Then l=diag(l0,l1,l2,1) is an honest SU4 transition cocycle.
000026 | 
000027 | Reuse source-joint-spin-family-lift-20260922/PROOF.md:38-78:
000028 |   K=(Spin10 x SU4)/diagonal Z4,
000029 |   iota_m(g)=[s(g,z),diag(z^(2m_j+1))], z^2=det g.
000030 | The extra [1,l] commutes with iota_m(g), since both family parts are diagonal.
000031 | Thus k_ij=iota_m(g_ij)[1,l_ij] satisfies the K cocycle identity.
000032 | The z sign ambiguity cancels by the common (-1,-I4) kernel exactly as before.
000033 | No global square root of D or global trivialization of each L_a is needed.
000034 | Every E8 branch and its Lie products are consequently globally associated
000035 | to this SAME K bundle. Its (16,4) branch is literally the W displayed above.
000036 | 
000037 | 3. Parallel clocks on the whole associated E8 bundle
000038 | With the GIVEN split connection A_E, its induced connection on W is
000039 |   A_W=rho_Lambda-even(A_E) tensor I4 + I16 tensor (M a + A_L),
000040 |   M=diag(0,0,-1,-1), a=tr A_E.
000041 | A_L is the traceless diagonal flat line connection; flat local gauges have
000042 | A_L=0 but the nontrivial sign cocycle remains on overlaps.
000043 | Therefore [M,A_L]=0 globally as an endomorphism identity.
000044 | 
000045 | Reuse original-admissible-source-closure-20261025/
000046 | JOINT-LIFT-NATIVE-CLOCK-CONNECTION.txt:88-146:
000047 | kappa is central in K, and
000048 |   sigma0=Lambda-even(g_col) tensor diag(omega,omega^2,omega^2,omega),
000049 |   g_col=diag(omega I3,I2), omega^3=1.
000050 | The spin color center commutes with every split U3 x U2 transition/connection;
000051 | the family R commutes with the determinant factor AND the flat sign factor.
000052 | Hence kappa and sigma0 are global PARALLEL automorphisms on every E8 branch.
000053 | This removes the extra nonabelian-family obstruction ON this actual cover.
000054 | It is the adapted native-class clock statement of that source. The unchanged
000055 | originally marked clock still needs its actual connection/mark transport.
000056 | A full source Dirac commutator additionally needs its insertion and domain
000057 | conditions; a connection identity alone does not prove those conditions.
000058 | 
000059 | 4. Fixed-mark descent versus a transported orbit
000060 | The original c:w->omega w cycles all three distinct nontrivial V4 characters.
000061 | A lift intertwining this actual coefficient system must cycle its three
000062 | isotypic lines. Phases cannot change that forced character permutation.
000063 | If the determinant endomorphism M is required to remain the SAME parallel
000064 | mark, equivariance requires m0=m1=m2. Thus m_A fails this fixed-mark test.
000065 | Its three placements of the exceptional triplet entry form a three-element
000066 | orbit. This does not contradict its valid global construction on C.
000067 | Likewise a fixed diagonal R must have equal phase on the three cyclic lines.
000068 | The native two-plus-two R has eigenspaces of dimension two, so no placement
000069 | of the three lines inside its four slots can preserve that fixed phase mark.
000070 | Kappa remains invariant because it is central.
000071 | 
000072 | A jointly transported M,R orbit is allowed only with the actual supplied
000073 | bundle, transition, connection and marking maps. A permutation mixing unequal
000074 | D powers is not automatically a selfmap of one fixed bundle. The local
000075 | normalizer matrix alone is not a global fixed-bundle descent proof.
000076 | The triplet-equality condition is the exact necessary commutant condition
000077 | for these fixed marks. Full ordinary descent across branch points also needs
000078 | the chosen deck linearization and its isotropy action; no coarse-base descent
000079 | is inferred solely from this condition. Punctured/orbifold descent must be
000080 | specified with its actual category.
000081 | 
000082 | 5. Verdict
000083 | The proposed common K connection and its parallel adapted clocks ON C are
000084 | correct. They give a concrete constructive use of the already selected q
000085 | sign packet, despite the base irreducibility argument.
000086 | The original packet does not itself select its alignment with m_A/R, an
000087 | original P1 quantum field map, a physical state, or OS/physical time.
000088 | The full GIVEN operational Seed keeps its existing state and time throughout.