TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/dynamic_reconstruction_20261027/A7-E7-CLOCK-DECOMPOSITION.txt Original file SHA-256: 62c251a47e1b0b67f034027fec66a5d0ffaab35abf61dcb2e709395c3bc82581 Inclusive original lines: 1-175 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 | A7 CLOCK DIRECTION AND THE NEUTRAL E7 ROOT CUT 000002 | ============================================== 000003 | 000004 | 27 October 2026 000005 | 000006 | Question 000007 | -------- 000008 | 000009 | What is the exact Lie theoretic direction of the original eightfold source 000010 | offset 000011 | 000012 | delta_j=(j+a)/8-1/2, j=0,...,7, 000013 | 000014 | and how does its distinguished zero-grade root p* sit in the complete E8 root 000015 | system with the source charge Q(alpha)=sum alpha_j? 000016 | 000017 | Verdict 000018 | ------- 000019 | 000020 | The nonuniform part of the source offset is exactly the A7 Weyl vector with 000021 | Coxeter denominator eight: 000022 | 000023 | rho_A7=(7/2,5/2,3/2,1/2,-1/2,-3/2,-5/2,-7/2), 000024 | 000025 | delta=-rho_A7/8+(2a-1)q/16, q=(1,...,1). 000026 | 000027 | Consequently the folded source generator has the exact decomposition 000028 | 000029 | H8=8L0-rho_A7.a0+(2a-1)Q/2. (1) 000030 | 000031 | This corrects the proposed identification of the staircase with a D8 or E8 000032 | Coxeter direction. The denominator eight and rho_A7 are intrinsic to the 000033 | ordered eight-coordinate ramp. This statement does not identify physical 000034 | Euclidean time and does not replace the independent D8 and E8 Coxeter clocks 000035 | elsewhere in the theory. 000036 | 000037 | The complete E8 root census by Q is 000038 | 000039 | Q -4 -2 0 +2 +4 000040 | count 1 56 126 56 1. 000041 | 000042 | The 126 neutral roots are exactly the E7 subsystem orthogonal to q. Under 000043 | its maximal-rank A7 subsystem they split as 000044 | 000045 | 126 = 56 + 70 000046 | = roots(A7) + weights(Lambda^4 C^8). (2) 000047 | 000048 | The distinguished root 000049 | 000050 | p*=(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2) 000051 | 000052 | is omega_4(A7), has norm squared two, rho_A7.p*=8 and Q(p*)=0. It is thus an 000053 | E7 root with zero charge under the complementary source U1. Calling this 000054 | "electromagnetically neutral" is valid only through the already stipulated 000055 | source charge dictionary; algebraically the exact statement is Q(p*)=0. 000056 | It is not an E7 singlet. 000057 | 000058 | 1. Source and Atlas boundary 000059 | ---------------------------- 000060 | 000061 | The original retained formula is in 000062 | 000063 | experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt, 000064 | lines 17-26. 000065 | 000066 | That proof already records p*, Q(p*)=0 and delta.p*=-1 at lines 57-69, and 000067 | the complete charge histogram at lines 96-101. It does not identify the 000068 | staircase with -rho_A7/8 or refine the 126 neutral roots as 56+70. 000069 | 000070 | The active Atlas revision 4d210a8bebd5c91c7ee7ad75 was searched for the A7 000071 | Weyl-vector identity, omega4 and the 126=56+70 decomposition. No reviewed 000072 | finding containing this exact joint statement was found. The Atlas does 000073 | contain the older E8 charge histogram through the registered joint-source 000074 | contract, so the five charge multiplicities are supporting context rather 000075 | than a new result by themselves. 000076 | 000077 | 2. Exact offset decomposition 000078 | ----------------------------- 000079 | 000080 | The j-th entry of -rho_A7/8 is 000081 | 000082 | -(7-2j)/16. 000083 | 000084 | Adding the uniform component (2a-1)/16 gives 000085 | 000086 | -(7-2j)/16+(2a-1)/16 000087 | =(j+a)/8-1/2 000088 | =delta_j. 000089 | 000090 | Since L0=N+p^2/2 and Q=q.a0, multiplication by eight gives (1) exactly. 000091 | The parameter a occurs only in the uniform U1 direction. Every Q-neutral 000092 | state is therefore independent of a under the nonuniform grading. 000093 | 000094 | For a level-one root state J_(alpha,-1)|0>, the basic commutator is 000095 | 000096 | [H8,J_(alpha,-1)] 000097 | =(8-rho_A7.alpha+(2a-1)Q(alpha)/2) J_(alpha,-1). (3) 000098 | 000099 | For alpha=p*, rho_A7.p*=8 and Q(p*)=0, so the coefficient in (3) vanishes. 000100 | This is the precise source of the old zero grade. It is an A7 principal 000101 | grade cancellation, independent of the Wilson-line parameter a. 000102 | 000103 | Among all 126 neutral E7 roots, p* is the unique root with zero coefficient 000104 | in (3). The opposite root has coefficient 16. The complete neutral base 000105 | grade multiplicities for grades 0 through 16 are 000106 | 000107 | 1,2,4,6,9,10,13,14,8,14,13,10,9,6,4,2,1. 000108 | 000109 | 3. Full root census and the E7 cut 000110 | ---------------------------------- 000111 | 000112 | Use the standard E8 realization with 112 integer D8 roots 000113 | 000114 | +/-e_i +/-e_j 000115 | 000116 | and 128 half roots (1/2)(+/-1,...,+/-1) with an even number of minus signs. 000117 | 000118 | For the integer roots, Q=0 consists exactly of e_i-e_j and its negatives: 000119 | 2*C(8,2)=56 roots. The remaining integer roots give 28 roots at each of 000120 | Q=+2 and Q=-2. 000121 | 000122 | For a half root with m minus signs, 000123 | 000124 | Q=4-m. 000125 | 000126 | The allowed even values m=0,2,4,6,8 yield multiplicities 000127 | 000128 | 1,28,70,28,1. 000129 | 000130 | Adding both root types gives the five-charge census above. At Q=0 the 70 000131 | half roots have four plus and four minus signs. Choosing the four plus 000132 | positions identifies them exactly with the 70 weights of Lambda^4 C^8. 000133 | Together with the 56 A7 roots they form the 126-root, rank-seven, 000134 | reflection-closed E7 subsystem q-perp inside E8. This is the standard 000135 | adjoint branching 000136 | 000137 | 133 of E7 -> 63 of A7 + 70, 000138 | 000139 | with the common seven Cartan directions added to the root counts in (2). 000140 | 000141 | 4. Consequence and limit 000142 | ------------------------ 000143 | 000144 | The earlier source result that p* is a nontrivial D8 spinor-sector vector 000145 | and the present statement that it is Q-neutral are compatible. D8 000146 | superselection class and the complementary E7 x U1 charge are different 000147 | gradings. The new exact bridge is: 000148 | 000149 | ordered eightfold source clock 000150 | -> A7 rho grading plus uniform U1 000151 | -> Q=0 E7 root cut 000152 | -> p*=omega4(A7) as its unique zero-grade root. 000153 | 000154 | This explains the zero grade within the original eight-clock geometry. It 000155 | does not prove that H8 is physical OS time, that the p* current is an 000156 | admissible local observable, or that any E7 gauge interpretation is selected 000157 | by P1. Those source identifications remain exactly as scoped in the prior 000158 | local-time and admissibility reports. 000159 | 000160 | 5. Reproduction 000161 | --------------- 000162 | 000163 | Run 000164 | 000165 | python3 check_a7_e7_clock.py 000166 | 000167 | The checker constructs all 240 roots with exact rational arithmetic, verifies 000168 | the offset identity, the five charge sectors, their integer/half-root 000169 | refinement, the A7 plus Lambda4 neutral decomposition, rank seven and 000170 | reflection closure, p*=omega4, and the full commutator-grade distribution. 000171 | Its JSON output is a7-e7-clock-results.json. The checker contains explicit 000172 | failure exceptions and therefore runs identically with python3 and python3 000173 | -OO. 000174 | 000175 | No repository or Atlas file was changed.