TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-joint-source-20261023/RELATIVE-TRANSPORT.txt Original file SHA-256: cfe6ed1c5d80ceba8cb84ea73c162b9664cc134b9a8f08b2f5741be24e6606ed Inclusive original lines: 1-118 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 | MATH.SEAM.JOINT.RELATIVE.05 000002 | 23 October 2026 000003 | 000004 | Complete joint deck and clock comparison 000005 | 000006 | 1. Question and the supplied objects 000007 | 000008 | The predecessor COVERS.txt gives an actual integral clock intertwiner between 000009 | the unique Phi12 components of the native real8 compiler and the retained 000010 | Burau rank6 lattice model. Does that same map also intertwine the original 000011 | peripheral local system and the cover-induced Burau deck local system? This 000012 | is the first necessary comparison before 000013 | claiming a common parallel identification over the punctured base. 000014 | 000015 | Use exactly the supplied saturated cyclic bases B_E,B_C. Both clock actions 000016 | become the companion matrix A of Phi12(X)=X^4-X^2+1. In these same bases the 000017 | actual native meridian is J=A^9 and the actual oriented Burau meridian is 000018 | D=A^4. Infinity has native action I and Burau action D^-4=A^8. 000019 | 000020 | On the Burau side the local system is DEFINED by the existing marked cover 000021 | and its retained deck representation: 000022 | 000023 | U_B,per=(C3 x U_B)/((Delta3 p,u)~(p,D u)). 000024 | 000025 | Equivalently use the supplied C12 refinement and send its deck generator to 000026 | D. This is the standard induced local system from already supplied objects. 000027 | It is not an independently established geometric Gauss-Manin H1 family or 000028 | an identification of the integral Burau basis with geometric cycles. The 000029 | native side is the supplied all-J local system. The comparison below is 000030 | complete for precisely this pair of associated local systems. 000031 | 000032 | 2. Complete clock-intertwiner space 000033 | 000034 | Every rational clock intertwiner in the common bases commutes with A. Since 000035 | A is the companion matrix of the irreducible polynomial Phi12, 000036 | 000037 | Hom_clock = Q[A] = Q(zeta12), dimension4 over Q. 000038 | 000039 | The integral cyclic vector e0 satisfies(e0,Ae0,A^2e0,A^3e0)=I4. Thus an 000040 | integral commuting matrix H is determined by the integral vector H e0 and 000041 | is precisely a polynomial in A with integer coefficients. Consequently 000042 | 000043 | Hom_clock,integral = Z[A] = Z[zeta12], 000044 | Isom_clock,integral is a torsor under Z[zeta12]^units. 000045 | 000046 | The predecessor displayed one explicit representative of this torsor. It did 000047 | not derive a uniquely selected intertwiner. 000048 | 000049 | 3. Actual peripheral action and exact obstruction 000050 | 000051 | Parallel transport in the Hom local system acts by 000052 | 000053 | H -> D H J^-1 = A^4 H A^-9 = A^-5 H = A^7 H. (3.1) 000054 | 000055 | Every clock intertwiner is subject to the same relative character. Its 000056 | characteristic polynomial is again Phi12. In particular 000057 | 000058 | ker(H -> D H J^-1-H)=0. 000059 | 000060 | There is NO nonzero rational map that simultaneously preserves the actual 000061 | clock and both peripheral monodromies on this component. The complete 000062 | ordinary local-system comparison is therefore decided negatively; a constant 000063 | clock intertwiner must not be advertised as a parallel map of the original 000064 | local systems. No change of integral basis repairs this spectral obstruction. 000065 | 000066 | 4. Forced relative transport and the common cover 000067 | 000068 | The relative Hom system is nevertheless fully determined. Its finite 000069 | peripheral exponents and infinity exponent, in terms of A, are 000070 | 000071 | (7,7,7,7,8) modulo12. (4.1) 000072 | 000073 | The sphere relation holds because4*7+8=36=0 modulo12. Multiplication by A^7 000074 | has exact order12; moreover every nonzero H has an orbit of length12 because 000075 | Q[A] is a field and no A^(7k)-1 vanishes for1<=k<12. 000076 | 000077 | Let nu be the already supplied winding character taking each positive finite 000078 | meridian to1. The monodromy is A^(7 nu). Since7 is invertible modulo12, 000079 | 000080 | kernel(relative Hom monodromy)=kernel(nu modulo12). 000081 | 000082 | This is exactly the existing forced cover C12:s^12=x^4-1. Hence that cover 000083 | is the minimal connected cover trivializing the complete relative Hom system, 000084 | not merely a degree match between two unrelated constructions. The supplied 000085 | relative mismatch itself derives its degree12 requirement. 000086 | 000087 | On C12, parallel clock intertwiners exist after specifying their initial 000088 | value; deck transport of that value remains the explicit A^7 action. On the 000089 | base the complete object is the relative Hom local system (3.1), with this 000090 | carry retained. Introducing an extra line twist to erase the carry would 000091 | change the supplied comparison and is not done here. 000092 | 000093 | 5. Consequence for the joint construction 000094 | 000095 | The associated E8 lattice-VOA system of JOINT-CLOCK.txt carries the original 000096 | peripheral J and cyclic compiler action together without merging their base 000097 | roles. The present result specifies exactly how its common Phi12 clock 000098 | component compares with the retained Burau model: an integral clock 000099 | isomorphism torsor with forced faithful relative order12 transport. The 000100 | original forms and complementary spectra remain different. This gives the 000101 | full comparison object and its exact obstruction to a stronger identification. 000102 | 000103 | Executable certificate: 000104 | python3 -B experiments/theory-contracts/e-series-joint-source-20261023/relative_transport.py 000105 | 000106 | The certificate imports the original cover dictionary through its direct 000107 | existing execution path, retains both displayed bases and actual J/deck 000108 | matrices, solves the clock commutant, verifies its integral cyclic basis, 000109 | checks(3.1) on all four Hom generators, proves absence of invariant Hom, 000110 | and checks the exact relative order and infinity relation. 000111 | 000112 | Standard covering-space basis: 000113 | Allen Hatcher, Algebraic Topology, Section1.3, Propositions1.39 and1.40. 000114 | https://pi.math.cornell.edu/~hatcher/AT/AT.pdf 000115 | 000116 | Verdict: PASS, complete for the supplied clock and peripheral modules. The 000117 | flat-intertwiner dimension is exactly0, and the relative system trivializes 000118 | on exactly the existing C12 kernel cover.