TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-incidence-algebra-20261024/CLOCK-COVER.txt Original file SHA-256: 78db186f0498400dfe76208fe8e6fb0bee27634f6d4a897d6ffc21abadc5a6d6 Inclusive original lines: 1-134 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.ESERIES.INCIDENCE.07.CLOCK-AUDIT 000002 | 24 October 2026 000003 | 000004 | Reconciliation with the independent report using the actual original files 000005 | 000006 | 1. Four representatives and one class: the precise restricted problem 000007 | 000008 | Keep the ORIGINAL even E8 lattice L, the ordered epsilon cocycle, the 000009 | integral basis B displayed in JOINT-CLOCK.txt, and the generator-normalized 000010 | field lift Jhat=S_J. The native standard J has 2 by 2 blocks 000011 | 000012 | [0 -1; 1 0]. 000013 | 000014 | The report displays inverse blocks [0 1; -1 0]. This changes the marked 000015 | orientation convention. It does not change the root census, J^2=-I or the 000016 | unrestricted class counts. Numerical masks may be compared only after 000017 | aligning the lattice basis, epsilon convention and clock orientation. 000018 | 000019 | [OMITTED: non-scientific conversation or execution metadata] 000020 | [OMITTED: non-scientific conversation or execution metadata] 000021 | [OMITTED: non-scientific conversation or execution metadata] 000022 | 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | 000025 | They are compared under character conjugations D_t preserving that same 000026 | S_J. This is not the unrestricted lift problem for J or for c. 000027 | 000028 | The exact affine equation for (1.1) in the ordered dual basis has byte columns 000029 | 000030 | (1,2,93,128,119,1,215,128), defect169, rank6. 000031 | 000032 | It has precisely four solutions, the masks31,62,151,182. Their fourth powers 000033 | are S_sigma D_b with b=63,30,183,150 respectively. The Jhat-preserving 000034 | character group has16 elements and its stabilizer on any solution has4. 000035 | There is one orbit of size4. No solution also has b=0. This is the complete 000036 | meaning of the original claim, verified on the eight lattice generators 000037 | and directly on all240 roots; oscillator descendants inherit the action. 000038 | 000039 | 2. Why the report's unrestricted numbers also hold 000040 | 000041 | On the SAME original E8 basis, the new audit obtains 000042 | 000043 | SNF(I-J)=diag(1,1,1,1,2,2,2,2). 000044 | 000045 | All256 sign lifts of J have order4: the character norm I+J+J^2+J^3 is zero, 000046 | and the original S_J has fourth powerI. Character conjugation is translation 000047 | by (I+J*)t over F2, of rank4, so it gives16 orbits of size16. 000048 | Keeping those256 SIGN-lift candidates unchanged and enlarging ONLY their 000049 | allowed gauge equivalence to fourth-root-valued characters, the Smith 000050 | invariants1 and2 show that every sign correction becomes a coboundary. 000051 | Then there is one class. If the candidate space ALSO includes every 000052 | fourth-root-valued lift, its quotient by fourth-root-valued gauge instead 000053 | has16 classes: the cokernel of I-J* modulo4 has size2^4. The one-class 000054 | statement is exactly about sign candidates under the larger gauge relation. 000055 | In this precise sense the counts agree with the independent report. 000056 | 000057 | For completeness, all256 unrestricted sign lifts of c have order12. 000058 | The original S_c has order12 and sum(k=0..11)c^k=0. The unrestricted sign 000059 | character gauge has rank6, giving four orbits of size64. Restricting to 000060 | (1.1) fixes Jhat and reduces both the candidate set and its allowed gauge, 000061 | giving the ONE orbit in Section1. These are distinct, explicit equivalence 000062 | problems, not inconsistent answers to one problem. 000063 | 000064 | The coefficient module is Hom(E8,{+1,-1}). Nothing here replaces the original 000065 | noncentral lattice-isometry lift extension by the older central mu4 000066 | extension of G31. The underlying lattice-VOA lift framework is described in 000067 | Dong and Nagatomo, Automorphism Groups and Twisted Modules for Lattice 000068 | Vertex Operator Algebras, https://arxiv.org/abs/math/9808088. 000069 | 000070 | 3. Actual matrix and ALL peripheral generators 000071 | 000072 | The original COVERS.txt and cover_dictionary.py supply saturated integral 000073 | bases BE,BC, retained byte-for-byte in the bundle. The common clock is 000074 | 000075 | A = [0 0 0 -1] 000076 | [1 0 0 0] 000077 | [0 1 0 1] 000078 | [0 0 1 0], 000079 | 000080 | with irreducible characteristic polynomial Phi12=x^4-x^2+1. In these actual 000081 | bases J BE=BE A^9 and deck BC=BC A^4. Rational clock Hom is Q[A]; its integral 000082 | lattice is Z[A]. On that Hom space, D H J^-1=A^7 H. 000083 | 000084 | The base is FIVE-punctured P1, with ordered generators g1,g2,g3,g4,gInfinity 000085 | and relation g1*g2*g3*g4*gInfinity=1. Its four finite punctures are1,i,-1,-i. 000086 | The supplied covers u_n^n=x^4-1 fix the winding homomorphism 000087 | 000088 | nu(g1)=nu(g2)=nu(g3)=nu(g4)=1, nu(gInfinity)=-4. 000089 | 000090 | The two ORIGINAL associated systems have ALL generator images 000091 | 000092 | g1 g2 g3 g4 gInfinity 000093 | native A9 A9 A9 A9 I 000094 | Burau deck A4 A4 A4 A4 A8 000095 | relative A7 A7 A7 A7 A8. 000096 | 000097 | Here Ak denotes A to power k. Infinity is the inverse product of the four 000098 | finite images in each row. The Burau system is DEFINED by its supplied 000099 | marked cover and deck representation. It is not a separately proved 000100 | Gauss-Manin homology family. That qualification is part of the theorem. 000101 | 000102 | 4. Global factorization and minimality, rather than one local period 000103 | 000104 | All generator images in the last row are powers of A, and the presentation 000105 | has no missing generators. Consequently, on EVERY word w in pi1, 000106 | 000107 | rho_rel(w)=A^(7 nu(w)). (4.1) 000108 | 000109 | This proves global factorization through nu modulo12. The order of A is12 000110 | and gcd(7,12)=1; hence the representation of C12 is faithful and 000111 | 000112 | ker(rho_rel)=ker(nu modulo12). (4.2) 000113 | 000114 | Connected covers correspond to subgroups of pi1. A cover trivializes the 000115 | relative local system iff its subgroup is contained in this kernel. The 000116 | index of the kernel is12, and it gives exactly the existing C12 cover 000117 | s^12=x^4-1. Every other connected trivializing cover has degree divisible 000118 | by12; the minimum degree is12. The same holds for the orbit of ANY nonzero 000119 | rational clock intertwiner, since Q[A] is a field and A^(7k)-1 is invertible 000120 | for1<=k<12. The parallel clock-Hom subspace on the base has dimension0. 000121 | 000122 | [OMITTED: non-scientific conversation or execution metadata] 000123 | [OMITTED: non-scientific conversation or execution metadata] 000124 | [OMITTED: non-scientific conversation or execution metadata] 000125 | [OMITTED: non-scientific conversation or execution metadata] 000126 | [OMITTED: non-scientific conversation or execution metadata] 000127 | [OMITTED: non-scientific conversation or execution metadata] 000128 | 000129 | Thus both previously inaccessible original conclusions pass the requested 000130 | test in their stated marked and induced-local-system category. Their 000131 | premises and actual matrices are present here and in the portable bundle. 000132 | The full field-associated-source construction and the distinction among 000133 | base rotation, deck action and fibre clock remain exactly as in the 000134 | unchanged original JOINT-CLOCK.txt.