TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ACTUAL-DIAGONAL-SPINE-ENDPOINT-TRANSPORT.txt Original file SHA-256: 8d9cd72f553b68854ce94bd101a847df6f2b9777225518361515b3c05732e0da Inclusive original lines: 1-160 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | FOUR OPEN D4 SPINE TRANSPORTS IN THE ACTUAL ORIGINAL FUCHS CONNECTION 000002 | Independent continuation, 2026-10-30. Originals and Atlas unchanged. 000003 | 000004 | RESULT 000005 | The four marked diagonal 0-to-infinity path classes now have exact full 000006 | transport matrices and source Hermitian endpoint metrics. Their target 000007 | quarter action differs from their source quarter action. A simultaneously 000008 | rotated open-word contribution has a NONZERO four-path sum, although the 000009 | sum with its insertion held fixed is zero. Thus an endomorphism trace 000010 | cancellation cannot be imposed on this typed open-endpoint consumer. 000011 | No original rule summing these four contributions is asserted here. 000012 | 000013 | 1. ORIGINAL SPINE SCOPE AND THE GEOMETRIC IDENTIFICATION 000014 | 000015 | Read source: archive45 em_flavor_source/sections/004-source.tex:164-244, 000016 | paper-source:7268c6c6db6ca8a62299c024, 000017 | SHA256 107d525221eb816043dbf4fb1a3c7e8244d3e24a5a5bcbfe1c2c34d8ee192f4a. 000018 | Lines180-197 name the unique D4-invariant geodesic spine, its canonical 000019 | Gamma_ij^min edges, and the Hol_G K / Hol_Gamma Yukawa consumers. The proof 000020 | at219-241 invokes that spine but supplies neither its base metric nor the 000021 | nine endpoint/word addresses. The Hermitian metric constructed at145-153 000022 | is the FLAT BUNDLE metric, not a Riemannian metric on the punctured sphere. 000023 | EM3:329-352 fixes the distinguished cycle basis, not those nine addresses. 000024 | The repeated earlier tfpt42/sections/018-source.tex:34-115 has the same 000025 | scope. Architecture document005:310-334 explicitly distinguishes flat 000026 | monodromy from parabolic/Hodge/metric word-length information. 000027 | This read does not rehabilitate the already corrected archived D/P 000028 | monodromy formula. All computations below use the actual v117 connection 000029 | and its already source-reviewed algebraic fundamental matrix G. 000030 | 000031 | There IS a concrete geometric realization without fitting any paths: 000032 | gamma_k(r)=r exp[i(pi/4+k pi/2)], r in[0,infinity], k=0,1,2,3. 000033 | Its complement has four once-punctured faces, one for each original mark. 000034 | The two original real reflections z->i bar z and z->-i bar z fix these 000035 | diagonals. In the canonical complete hyperbolic metric of the GIVEN 000036 | marked conformal sphere these reflections are isometries, so their fixed 000037 | arcs are geodesics. The same diagonals are round-sphere geodesics. 000038 | Using hyperbolic geometry here only identifies paths; the flat transport 000039 | does not depend on this auxiliary metric or their parametrization. 000040 | 000041 | The useful topological uniqueness reduction is conditional on an embedded 000042 | reduced spine with no leaves and four once-punctured complementary faces: 000043 | rotation z->iz permutes the four faces, so its fixed points0,infinity lie 000044 | in the spine. At each it permutes tangent germs in cycles of four; hence 000045 | both valences are at least four. The rank3 identity sum(deg(v)-2)=4 then 000046 | forces both valences exactly four and no other branching. A self-loop 000047 | would have a rotation orbit using all germs at that vertex and disconnect 000048 | the other fixed vertex. Thus the reduced graph is theta4. If the two real 000049 | reflections also preserve this graph, each fixes two of its edges. Their 000050 | restriction to an edge fixes both endpoints and is an involution of an 000051 | interval, hence is pointwise identity. The edges are the diagonal arcs. 000052 | Holomorphic D4 invariance alone must not silently be replaced by this 000053 | additional real-reflection invariance. The original displayed geodesic 000054 | declaration does not specify its metric. Accordingly the equality to its 000055 | unnamed spine is established under the natural complete-hyperbolic reading, 000056 | or under the stated real-reflection invariance, rather than asserted from 000057 | the word 'D4' alone. The four path transports themselves are unconditional 000058 | calculations in the already given original connection. 000059 | 000060 | 2. EXACT ENDPOINT LIMIT, WITH ALL KUMMER SIGNS 000061 | 000062 | Reuse ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt and the full marked G-frame report. 000063 | omega=exp(2pi i/3), a=2sqrt2/3, b=sqrt5/3, 000064 | w^3=(z^2-1)/(z^2+1), h_j^2=(w+omega^j)/(w+omega^(j+1)). 000065 | The selected lift over0 has w=-1, h0~rho z, h1(0)=tau=i omega^2, 000066 | rho=sqrt2 3^(-3/4) exp(-5pi i/12), h0 h1 h2=1. 000067 | Write N(w)=-(1+omega)w^3+(1-omega)w^2+(omega+2)w+omega. 000068 | The already proved G_j is g(omega^j w,h_(-j)), where 000069 | g=(h p1/(ab),h z p2/b,h z p3)^t and the p_i are the original G formulas. 000070 | 000071 | For gamma_0, w=e^(i phi) with 2pi/3infinity) z^(-D)G(z). 000080 | Its column j is exactly 000081 | h_(-j) [-2N(omega^(j+1))/(9ab),p2(omega^(j+1))/b, 000082 | p3(omega^(j+1))]^t. 000083 | The gauge-transformed connection z^(-D) A z^D-D/z is O(z^-3), so this 000084 | regularized frame is nonsingular at infinity, with no cutoff or pole reader. 000085 | Put T0=L Gbase^-1. Its determinant is -1. The finite matrices are recorded 000086 | in the accompanying JSON; no numerical ODE supplies their values. 000087 | 000088 | The source metrics are 000089 | H0=diag(10/27,5/(9sqrt3),1/(3sqrt3)), 000090 | Hinf=[[50/81,0,2sqrt10/81],[0,5/27,0], 000091 | [2sqrt10/81,0,17/81]]. 000092 | Exact identities: Gbase^* H0 Gbase=I, L^* Hinf L=I, 000093 | T0^* Hinf T0=H0. Hence T0 is the actual Hermitian parallel-transport 000094 | isometry Hom(E0,Einf), not a Euclidean unitary endomorphism in one frame. 000095 | 000096 | 3. ALL FOUR EDGES AND THE DECISIVE OPEN-WORD TEST 000097 | 000098 | The actual source quarter is U=diag(1,i,-i). The regularized target quarter 000099 | is V=i^(-D)U=diag(-1,1,-1). From G(Qy)=U G(y)KQ and the fixed base lift, 000100 | T_k=V^k T0 U^(-k), 000101 | not U^k T0 U^(-k). All four satisfy T_k^* Hinf T_k=H0. 000102 | This is the same existing signed Q lift on all paths and endpoints. 000103 | 000104 | FRAME TYPES MUST BE MATCHED: the displayed T_k use the raw Psi(0)=I 000105 | holomorphic source frame, while M0 is in its H0-normalized source frame. 000106 | Put R=H0^(1/2), T_hat,k=T_k R^-1, M_raw,k=R^-1 M_k R, 000107 | where M_k=U^k M0 U^(-k). Exact identities are 000108 | M_raw,0=Gbase Cs Gbase^-1, Cs=[[0,0,1],[-1,0,0],[0,-1,0]], 000109 | T_hat,k^* Hinf T_hat,k=I, 000110 | R T_(k+1)^-1 T_k R^-1=M_(k+1), indices modulo4. 000111 | The last identity recovers ALL FOUR original meridians from these paths. 000112 | An initial draft multiplied raw T0 by normalized M0. Parent review caught 000113 | that mismatch before publication; its earlier radical coefficient is 000114 | discarded. The following coefficients use matching frames throughout. 000115 | 000116 | Let P0=diag(1,0,0)=(I+U^2)/2 be the already marked endpoint line. Then 000117 | sum_(k=0)^3 T_hat,k M_k P0 =4 Pinf,+ T_hat,0 M0 P0, 000118 | Pinf,+=diag(0,1,0). 000119 | The exact first-ray row needed by this expression is 000120 | (T_hat,0)_(row1,:)=[-3sqrt30/10,3sqrt30(1-i)/20, 000121 | -3sqrt30(1+i)/20]. 000122 | In zero-based indices, 000123 | (T_hat,0 M0)_(1,0)=-3sqrt30/10 !=0, 000124 | and hence the whole covariant sum is 000125 | -6sqrt30/5 E10 !=0. 000126 | With the target row also orthonormalized by sqrt(Hinf_11), this is exactly 000127 | -2sqrt2 E10, 000128 | and its single first-path component is -1/sqrt2. Equivalently in matched 000129 | RAW frames, sum_k T_k M_raw,k P0=-4 E10 and the single component is -1. 000130 | In contrast sum_k T_hat,k M0 P0=0 when the insertion is frozen. Those are 000131 | different consumers. Both identities were checked exactly, including all 000132 | source/target phases and metrics, rather than inferred from dimensions. 000133 | Multiplication by the already evaluated kappa gives the same concrete 000134 | nonzero endpoint word; this fact does not insert kappa into the original 000135 | Zadm or Ftr action by itself. 000136 | 000137 | 4. WHAT THIS CLOSES, AND THE PRECISE NEXT ORIGINAL CALL 000138 | 000139 | The entire four-edge transport and the open endpoint covariance are now 000140 | explicit in the given original family source. They do not require a new 000141 | connection, metric fit, state, time, Higgs matrix or arbitrary path scan. 000142 | The four edges are not declared to be the nine Gamma_ij addresses. The 000143 | actual original next call is the packaging of these edges/closed words 000144 | and the given endpoint readers into Hol_G K or Gamma_ij. An original 000145 | sum/ordering/Sewing rule, if present, decides which of the two sums above 000146 | is called. The calculation proves why replacing this call in advance by 000147 | an endomorphism trace cancellation would erase relevant source data. 000148 | The full field/state/reflection/time transport remains the already given 000149 | source transport; this finite endpoint computation does not select a new 000150 | physical operation on it. The original real delta jet remains the closest 000151 | already explicit action consumer and must retain both C6 branches and 000152 | their adjoints, as proved in ORIGINAL-DELTA-JET-READS-THE-SAME-OPEN-WORD.txt. 000153 | 000154 | REPRODUCIBLE EXACT OUTPUT 000155 | Workspace outputs/multi-model-source-search-20261029/ 000156 | ACTUAL-DIAGONAL-SPINE-ENDPOINT-TRANSPORT-CHECK.json 000157 | SHA256 46787929dff0f7517949d225263fb4301bb4d3fe553b7bdfb4b937c6b6beae7d. 000158 | Fourteen exact Sympy identities passed on this actual new endpoint input. 000159 | A continuous numerical root replay only confirmed the analytically fixed 000160 | signs; it is not the proof of the endpoint transport or nonzero result.