TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-SEAM-STATE-FLAG-SELECTION.txt Original file SHA-256: bdbba2dae8386cd94e68663ca3c2291ba652df2de563b1a4e114ffa2af4ec1ee Inclusive original lines: 155-207 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. 000155 | 5. ACTUAL QUARTER FOLD AND STATE: THE CRUCIAL SCOPE CORRECTION 000156 | 000157 | FOLD-AUDIT.md derives the actual source fold, not eight replacement copies: 000158 | (F f)_j(x)=f((x+2pi j)/8)/sqrt8, 000159 | C_fold=F C_cover F*, 000160 | with the retained cyclic transition and full-circle phase e^(2pi i alpha). 000161 | The recorded original E8 dictionary is its EXACT matrix M; no new conjugacy 000162 | is chosen here. Define 000163 | mu_j=(j+alpha)/8, 000164 | nu_alpha=M(2mu), 000165 | d_alpha=M(mu-1/2). 000166 | 000167 | For alpha=1/4, 000168 | d=(-1/2,0,1/16,-1/2,0,-3/16,-1/4,-1/4). 000169 | For alpha=3/4, 000170 | d=(-1/2,0,3/16,-1/2,0,-1/16,-1/4,-1/4). 000171 | In both cases ||d||^2=85/128 and the recorded quotient conformal ground 000172 | weight is85/256. These are original values, not a fitted clock/temperature. 000173 | 000174 | On ALL roots the actual fold g^2 has phase exp(2pi i nu_alpha.alpha_root). 000175 | Its phase multiplicities for exponents0,1/2,1/4,3/4 are28,34,33,33; 000176 | each of1/8,3/8,5/8,7/8 occurs28 times. This is order8, and does NOT equal 000177 | the internal order4 character of section3. Replacing it by the internal 000178 | character would discard an actual source holonomy. 000179 | 000180 | The native g^2 stabilizer has order4 and logical image {id,(01)} for either 000181 | quarter case. Its fixed q are2,3,4,5, and it fixes16 individual flags. 000182 | Hence g^2-invariance alone gives no UNIQUE fixed flag. These are symmetry- 000183 | compatible candidates, not a derivation of a physical candidate menu. 000184 | 000185 | More decisively, retaining the FULL marked vector d has stabilizer1 in 000186 | native G31. It is sufficient to test all root pairings d.alpha_root: 000187 | roots span R8, so preservation of those pairings is exactly gd=d. 000188 | Consequently there is NO nontrivial native symmetry obstruction after 000189 | this source diagnostic is retained. The same is true if the internal 000190 | character is retained jointly. All60 flags pass the fixed-point condition. 000191 | 000192 | This is the stabilizer of the original shift/time diagnostic d, NOT a 000193 | reconstructed full CAR covariance or a 000194 | new E8 field-state intertwiner. In the ordinary E8 representation it is 000195 | also the stabilizer of K_alpha=L0+d_alpha.h_0. There h_0 Omega=0 and K_alpha 000196 | still acts as4I on the neutral quartic code: d is NOT . 000197 | Charged exterior responses can retain d. The relation nu_alpha-2d_alpha 000198 | =M1=2(e3+e6) in E8 gives the actual g^2 character exp(4pi i d_alpha.p). 000199 | d is not in E8 and cannot be substituted 000200 | for the neutral E8 vacuum vector. In the fixed quotient conformal reference, 000201 | the folded state has weight85/256, while the ordinary E8 vacuum has weight0 000202 | and its untwisted grades are integral. Identifying those states while 000203 | intertwining that SAME Virasoro action fails. A twisted/defect construction, 000204 | appropriate reference transport and the actual local field map could change 000205 | the realization; none is ruled out by this test. Absolute excitation-energy 000206 | shifts must not be confused with a fixed Virasoro-module intertwiner. 000207 |