TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/CURRENT-RESPONSE-AND-DICTIONARY.md Original file SHA-256: db9163de1017810818e104f651e531247378a4c8295580c8936cb19e594b4b73 Inclusive original lines: 99-185 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. 000099 | 000100 | || -W-W(R5 tensor I4) ||_HS^2=48. 000101 | 000102 | Already infinitesimally, every traceless Q(H) acts trivially on C_a and 000103 | ALL their products/OPEs, but as H tensor I4 on F. Thus this literal 000104 | charged-current dictionary fails. This does not identify the original 000105 | native Cartan four-space with C4; that different realization is tested next. 000106 | 000107 | ## 4. Test the actual J-positive Cartan quartic realization 000108 | 000109 | Use original Construction-A roots r of raw norm squared four and the 000110 | physical metric dot/2. In orthonormal Cartan coordinates alpha=r/sqrt(2), 000111 | so alpha^2=2. The ORIGINAL positive polarization is 000112 | 000113 | u_j=(h_2j - i h_(2j+1))/sqrt(2), J u_j=i u_j. 000114 | 000115 | The plus-sign convention would be the conjugate J=-i space. Keep the 000116 | minus sign when comparing the actual native charts. 000117 | 000118 | The symmetric-tensor/Fock isometry sends the five actual code support 000119 | sums to homogeneous polynomials f_j(x), up to these NONZERO factors: 000120 | 000121 | sqrt(6)/24 [sum_i x_i^4], 000122 | sqrt(2)/4 [x0^2 x1^2+x2^2 x3^2], 000123 | sqrt(2)/4 [x0^2 x2^2+x1^2 x3^2], 000124 | sqrt(2)/4 [x0^2 x3^2+x1^2 x2^2], 000125 | x0 x1 x2 x3. 000126 | 000127 | Their Fock Gram is exactly I5. This is the canonical oscillator embedding 000128 | of the existing code in the GIVEN source, not a derived physical 000129 | preparation or a choice of five charged root currents. 000130 | 000131 | For b_j=(u_j,alpha)=(r_2j-i r_(2j+1))/2 the original lattice vertex law is 000132 | 000133 | Y(e^alpha,z) f(x) Omega 000134 | = exp(sum_(n>0) alpha(-n) z^n/n) f(x-b/z) e^alpha. 000135 | 000136 | A fixed nonzero root-field phase multiplies both sides without affecting 000137 | the argument. In the z^-1 coefficient, the PBW oscillator-partition (3) 000138 | component is EXACTLY 000139 | 000140 | (f(b)/3) alpha(-3) e^alpha. 000141 | 000142 | Only the completely shifted degree-four term and the z^3 creation term 000143 | contribute to this partition. Other partitions cannot cancel it. This is 000144 | a PBW coefficient, not a coefficient in a normalized projection basis. 000145 | 000146 | The five displayed quartics have no common nonzero complex zero. The last 000147 | forces a zero coordinate. The three pair quartics then force at most one 000148 | remaining nonzero coordinate, and the sum of fourth powers excludes that 000149 | last possibility. For every nonzero real root b is nonzero. Consequently 000150 | each root zero mode leaks at least one code vector into charge alpha. 000151 | Different root charges are orthogonal, so such root contributions cannot 000152 | cancel across a linear combination. Cartan zero modes annihilate the 000153 | entire neutral oscillator code. Thus the complex leakage map has rank240, 000154 | and its kernel is exactly the eight-dimensional Cartan algebra, acting 000155 | trivially on the code. `check_cartan_source_stabilizer.py` verifies the 000156 | actual 240 roots, all shift coefficients, Fock Gram and one-particle 000157 | number matrix sum_j =5 delta_ik exactly. 000158 | 000159 | This gives a GLOBAL result. Let H be the compact inner E8 subgroup 000160 | preserving the code as a subspace of grade four. It is a closed compact 000161 | stabilizer. The leakage theorem gives Lie(H)=t8 and H^0=T8. Every element 000162 | normalizes this identity component, so its Cartan action is Weyl. The 000163 | one-particle support of the code is exactly the original H+ four-plane; 000164 | preservation of the code therefore preserves H+. Its restriction is a 000165 | code-preserving U4 operator. The already proved complete native code 000166 | stabilizer theorem forces its logical action into phase*rho5(G31). 000167 | The existing marked R5 lies outside that projective image. Therefore: 000168 | 000169 | No compact inner E8 automorphism preserving this native Cartan 000170 | quartic code implements the marked R5 on it. 000171 | 000172 | This includes arbitrary sequences of inner current-zero-mode exponentials 000173 | that return to the code at their endpoint, even if intermediate states 000174 | leave it. It is not merely an infinitesimal obstruction. 000175 | 000176 | The original conformal generator does not repair this: all five code 000177 | states have exactly four mode-one oscillators and zero lattice charge, 000178 | so L0|Code=4I5. Consequently exp(-itL0) is scalar on the code, and its 000179 | conformal quarter exp(-i pi L0/2) is I5. Since [L0,Q(H)_0]=0, 000180 | adding conformal propagation to inner zero-mode pulses does not enlarge 000181 | their projective logical action. This follows from the existing conformal 000182 | grading, not from identifying L0 with physical P1 time. The current fields 000183 | themselves all have conformal weight one; their internal order-four 000184 | automorphism must likewise not be called conformal time evolution. 000185 | A cylinder Casimir shift L0-c/24 changes only the common phase.