TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/W8-SOURCE-TWOPOINT.txt Original file SHA-256: f8f03e184f8813ecaf7439f47903d3835896475e272bd0ea03f4e8d85c12f688 Inclusive original lines: 1-145 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 | W8 source two-point function in the existing quarter-folded current module 000002 | ========================================================================== 000003 | 000004 | Status 000005 | ------ 000006 | 000007 | PROVED exactly for the already constructed neutral primary W8 and the two 000008 | already supplied quarter-fold source charges. This calculation adds no state, 000009 | coupling, field direction, projection, or source model. Its exact executable 000010 | owner is check_w8_source_twopoint.py and its saved output is 000011 | w8-source-twopoint-results.json. 000012 | 000013 | Inputs retained 000014 | --------------- 000015 | 000016 | 1. quantum-octic-completion-results.json supplies the unique coefficients in 000017 | the declared 17-dimensional H8 plus contraction-monomial ansatz. The state 000018 | has all 1539 PBW monomials and norm 000019 | 000020 | ||W8||^2 = 65028096000/43. 000021 | 000022 | 2. native_folded_octic_origin.py supplies the original O/M/U transport and the 000023 | two quarter charges. In the canonical standard E8 root coordinates used by 000024 | W8 they are d=M delta: 000025 | 000026 | d_1/4 = (-1/2,0,1/16,-1/2,0,-3/16,-1/4,-1/4), 000027 | d_3/4 = (-1/2,0,3/16,-1/2,0,-1/16,-1/4,-1/4), 000028 | d_alpha^2 = 85/128. 000029 | 000030 | The checker independently verifies that O maps all 240 old physical roots 000031 | to the canonical standard E8 root set. 000032 | 000033 | Exact shifted-state expansion 000034 | ----------------------------- 000035 | 000036 | For a Heisenberg state polynomial P of total weight eight, 000037 | 000038 | z^8 Y_d(P,z)|d> = sum_(ell>=0) v_ell z^ell 000039 | 000040 | is obtained factor by factor from 000041 | 000042 | h_j(-n) -> (-1)^(n-1)d_j 000043 | + sum_(r>=n) binom(r-1,n-1) h_j(-r) z^r. (1) 000044 | 000045 | Equation (1) follows directly from 000046 | 000047 | Y(h_j(-n),z)=partial^(n-1)h_j(z)/(n-1)! 000048 | 000049 | and h_j(z)|d>=d_j z^-1|d> plus creation modes. Normal ordering is 000050 | preserved. The checker expands (1) sparsely through level eight and uses the 000051 | positive Heisenberg norm 000052 | 000053 | ||product_(r,j) h_j(-r)^m(r,j)|d>||^2 000054 | = product_(r,j) r^m(r,j) m(r,j)! . (2) 000055 | 000056 | No oscillator partition or contraction counterterm of W8 is removed. 000057 | 000058 | Smallest sign and source test 000059 | ----------------------------- 000060 | 000061 | Write w=_d and let F1=v_1. The existing primary relation gives 000062 | 000063 | d.F1 = 8w. (3) 000064 | 000065 | The direct expansion verifies in both quarters 000066 | 000067 | w = 25123488075/357824462848, 000068 | d.F1 = 25123488075/44728057856 = 8w, 000069 | ||F1||^2 000070 | = 93237295693374937125/15629680934138150912 > 0. (4) 000071 | 000072 | For alpha=1/4 the full vector is 000073 | 000074 | F1 = (-866874645/698875904, 000075 | 0, 000076 | -153470205/5591007232, 000077 | -866874645/698875904, 000078 | 0, 000079 | 3634816815/5591007232, 000080 | 1551720555/1397751808, 000081 | 1551720555/1397751808). 000082 | 000083 | The Cauchy consequence of (3) is also strict: 000084 | 000085 | ||F1||^2 >= 64w^2/d^2 000086 | = 126237930610933441125/265704575880348565504, 000087 | 000088 | ||F1||^2-64w^2/d^2 000089 | = 51311452775625/9345848836096 > 0. (5) 000090 | 000091 | At d=0, homogeneity instead gives v_0=v_1=...=v_7=0 and 000092 | v_8=W8. Thus a state, grading-generator, and W8-preserving identification of 000093 | the shifted module with the unshifted E8 vacuum is already excluded by (4). 000094 | The exact generator conversion is stated next. This is a diagnostic inside 000095 | the supplied source dictionary, not an exclusion of other P1 candidates. 000096 | 000097 | Radial generator and original grading 000098 | ------------------------------------- 000099 | 000100 | The Taylor level ell in this calculation is the spectrum of the quotient 000101 | relative radial generator 000102 | 000103 | N = Lhat_0-h_d, h_d=85/256. (6) 000104 | 000105 | The already proved cover/fold identity is 000106 | 000107 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_(n,0). 000108 | 000109 | For either quarter, the original twisted ground term is 000110 | (alpha-1/2)^2/2=1/32. Hence on the folded neutral sector 000111 | 000112 | N=(Lcal_0^alpha-1/32)/8=K_alpha, 000113 | H0=Lcal_0^alpha-1/32=8K_alpha=8N. (7) 000114 | 000115 | Here H0 is the unscaled relative excitation generator used in the complete 000116 | Hall/current decomposition. Its original current modes satisfy 000117 | 000118 | [K_alpha,J_-m]=(m/8)J_-m. 000119 | 000120 | The quotient creation mode h_j(-r) is carried by the original m=8r modes, so 000121 | it has N-level r and H0-grade 8r. Therefore the shifted response begins at 000122 | N-level one, equivalently original H0-grade eight; the unshifted W8 vacuum 000123 | state begins at N-level eight, equivalently H0-grade 64. 000124 | 000125 | When the radial ratio is written q_rad=exp(-tau), tau is the quotient radial 000126 | cylinder separation generated by N. Equivalently 000127 | 000128 | exp(-tau N)=exp(-(tau/8)H0) 000129 | 000130 | on this folded sector. The original real source evolution is separately 000131 | written U(t)=exp(-it K_alpha), and its Hall modes acquire exp(-itm/8). 000132 | These algebraic identities do not identify the collar length, quotient radial 000133 | parameter, or t with an independently calibrated physical duration. 000134 | 000135 | Why nine coefficients determine the entire separated function 000136 | --------------------------------------------------------------- 000137 | 000138 | Let q_rad=w_coordinate/z_coordinate with |q_rad|<1 and set 000139 | 000140 | F_d(q_rad)=z^8 w_coordinate^8 . 000141 | 000142 | The state expansion and radial adjoint give 000143 | 000144 | F_d(q_rad)=sum_(ell>=0) a_ell q_rad^ell, 000145 | a_ell=||v_ell||^2. (8)