TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/automatic_identification/CLOCK-CLASS-PROOF.txt Original file SHA-256: 875711d782a807196969e6bc11f39f054678216e6ebe786107904575560ea74f Inclusive original lines: 1-126 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 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | 000004 | 1. Actual data and the precise candidate 000005 | 000006 | [OMITTED: non-scientific conversation or execution metadata] 000007 | [OMITTED: non-scientific conversation or execution metadata] 000008 | [OMITTED: non-scientific conversation or execution metadata] 000009 | [OMITTED: non-scientific conversation or execution metadata] 000010 | [OMITTED: non-scientific conversation or execution metadata] 000011 | [OMITTED: non-scientific conversation or execution metadata] 000012 | [OMITTED: non-scientific conversation or execution metadata] 000013 | [OMITTED: non-scientific conversation or execution metadata] 000014 | 000015 | The known original native Jhat has full E8 weight-one multiplicities 000016 | (1,i,-1,-i): (60,64,60,64). 000017 | This is not inferred from an unspecified model: see 000018 | e-series-source-reconstruction-20261024/NATIVE-CLOCK-CONJUGACY.txt:54-76. 000019 | Its complete torus conjugacy class and full field conjugation to kappa are 000020 | ALREADY PROVED there. They must be reused, not posed as a fresh open problem. 000021 | 000022 | 2. Why a finite list covers the stated class 000023 | 000024 | The four-cycle has determinant-1, so its SU4 lift has eta^4=-1 and fourth 000025 | power-I4. All four possible eta give the same unordered four fundamental 000026 | eigenphases, namely (1,3,-3,-1)/8, and are SU4-conjugate. In the actual D3 000027 | Cartan with tetrahedral spinor weights(+++, +--, -+-, --+)/2 these eigenphases 000028 | are represented by 000029 | t3=(1/2,-1/4,0). 000030 | The quotient kernel identifies(-1_Spin,-I4), so g^4=1 in E8 requires 000031 | g5^4=-1_Spin. 000032 | 000033 | The centralizer of the fixed SM in Spin10 lies in its maximal torus: 000034 | no D5 root has zero restriction to the rank4 SM Cartan. Commutation with 000035 | the color/weak root groups then makes its torus coordinates equal modulo 000036 | integers within the first3 and last2 slots. The possible additional integer 000037 | shift parity is exactly the Spin central sign. Hence write 000038 | t5=(a,a,a,b,b)/4. 000039 | Because g5^4 is trivial on the vector10, a,b are integers. Its action on 000040 | spinors is the central sign(-1)^(3a+2b), hence a must be odd. The Spin10 000041 | period lattice is D5: a shift8 in a adds(2,2,2,0,0), while a shift4 in b 000042 | adds(0,0,0,1,1), both D5 periods. Thus a=1,3,5,7 and b=0,1,2,3 give16 000043 | COVERING representatives of the entire stated class. No assertion that 000044 | they are16 inequivalent conjugacy classes or physical sources is made. 000045 | 000046 | Their combined eight-dimensional torus vector has 000047 | 4v=(a,a,a,b,b,2,-1,0). 000048 | Its coordinate sum is even, so4v lies in E8. Its second multiple has mixed 000049 | integer/half-integer coordinates, so2v does not: the adjoint clock has 000050 | exact order4. All remaining SU4 assignments are conjugate and cannot 000051 | alter its full E8 character. 000052 | 000053 | 3. Decisive exact result on all240 original roots 000054 | 000055 | For every doubled original root r, the phase is i^((4v dot r)/2). 000056 | The exponent is integral. Add the8 fixed Cartan vectors. The complete 000057 | weight-one characters are 000058 | b even: (64,64,56,64), 000059 | b odd: (70,56,66,56). 000060 | Neither equals(60,64,60,64). check_clock_four_cycle_class.py verifies all 000061 | 16 covering representatives and all240 roots with exact integers, using 000062 | the unchanged original-root helper. Results are stored in 000063 | clock-four-cycle-class-test.json. No numerical tolerance or mass fitting. 000064 | 000065 | Since conjugate full-field operators have conjugate weight-one actions, 000066 | this character difference excludes equality with Jhat even after FULL 000067 | inner E8 field conjugation. It is a necessary-invariant failure; the proof 000068 | does not infer equivalence merely from matching characters. 000069 | 000070 | 4. Removing the SM restriction: a complete positive and negative classification 000071 | 000072 | The same source test covers ALL Spin10 torus possibilities, not only the 000073 | SM-commuting pair of parameters. Write t5=n/4 with integer n and odd sum. 000074 | The period lattice is D5. Representatives modulo4D5 are all n_i in0..3 000075 | with odd sum, and the additional central shift4e0. This gives1024 covering 000076 | representatives with g5^4=-1. Every compact Spin10 element is conjugate to 000077 | one of these; the family4-cycle conjugacy class is still fixed as above. 000078 | 000079 | On all1024 representatives the exact E8 character histogram is: 000080 | (60,64,60,64):320, 000081 | (64,64,56,64):320, 000082 | (66,56,70,56):120, 000083 | (70,56,66,56):224, 000084 | (82,56,54,56):40. 000085 | The320 native matches have exactly3 odd n_i and two even residues0,2. 000086 | The old COMPLETE native E8 conjugacy-class theorem then establishes full 000087 | inner equivalence for these320, rather than inferring it from dimension 000088 | alone. They are not320 new physical sources. 000089 | 000090 | There is a short decisive invariant proof. For the family4-cycle lift, 000091 | Tr4(g3)=Tr6(g3)=0, Tr15(g3)=-1. 000092 | Decomposing248 as45+15+10*6+16*4+conjugate gives 000093 | Tr248(g)=Tr45(g5)-1. 000094 | Let k be the number of odd n_i and d=#(n_i=0 mod4)-#(n_i=2 mod4). 000095 | Then Tr10(g5)=2d, Tr10(g5^2)=2(5-2k), so 000096 | Tr248(g)=2(d^2+k-3). 000097 | Since k is odd, a zero trace is possible only for k3,d0. This is exactly 000098 | the stated parity condition; the full character computation verifies it. 000099 | The16 SM-commuting representatives instead give trace8 or4, never zero. 000100 | 000101 | For each of the320 matches, the fixed D5 roots are the6 roots on the three 000102 | odd coordinate planes, an A2. The other two even planes have opposite 000103 | residues0,2 and supply no additional roots. The fixed Spin10 Lie algebra 000104 | is therefore A2 plus3 abelian directions. It contains no commuting 000105 | SU3 times SU2 inside THAT Spin10 factor. This does not prevent the full 000106 | E8 fixed algebra from containing a different mixed D5+A3, nor prevent 000107 | the corresponding jointly transported SM embedding there. The output 000108 | clock-general-spin10-class.json records this complete class. 000109 | 000110 | 5. Boundaries and the actual positive old result 000111 | 000112 | This is stronger than the separate fixed-Cartan odd-rank typing gate, but 000113 | has additional candidate assumptions: an actual family4-cycle lift AND 000114 | Spin10 action commuting with the fixed SM AND exact quarter order. 000115 | Drop one and the conclusion does not apply. It does not exclude all 000116 | familial clocks, different SM embeddings transported with all marks, 000117 | the actual C12/glue construction or TFPT. 000118 | 000119 | [OMITTED: non-scientific conversation or execution metadata] 000120 | [OMITTED: non-scientific conversation or execution metadata] 000121 | [OMITTED: non-scientific conversation or execution metadata] 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]