TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/NEUTRAL-QUADRATIC-AND-MIXED-RESPONSE.md Original file SHA-256: e6c6fa871adde32db55168103d19aed89b41844e9ef32eca90ed0e58b363b83d Inclusive original lines: 117-202 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. 000117 | ## 4. Do NOT combine separate exclusions: derive the mixed response 000118 | 000119 | Root-current zero-mode rotations and neutral quadratic zero modes need 000120 | not remain in either separate group. In particular conjugating R_neut by 000121 | a root operation produces lattice-charged weight-two fields. The previous 000122 | inner-E8 exclusion and the new neutral-quadratic exclusion do NOT exclude 000123 | the combined source. This is the next justified connection to inspect. 000124 | 000125 | Its first actual response already follows from the SAME lattice formula. 000126 | For an original norm-two lattice root beta (beta^2=2), 000127 | 000128 | E_beta,0 T_A = (v_beta.h(-1)) e^beta, 000129 | v_beta = -A beta + (beta^T A beta) beta/2, 000130 | beta.v_beta=0. 000131 | 000132 | Indeed translate (1/2)x^T A x by -beta/z in the lattice vertex formula. 000133 | The z^-1 coefficient has -A beta.x from the first shift and 000134 | (beta^T A beta) beta.x/2 from creation times the complete shift. No 000135 | field identification, fitted coupling or new response rule enters. 000136 | 000137 | For the original AR and ALL240 original doubled Chevalley roots r, 000138 | beta=r/2 (not the Construction-A r/sqrt2 convention), the exact distribution is 000139 | 000140 | | squared norm of v_beta | root directions | 000141 | |---:|---:| 000142 | | 0 | 46 | 000143 | | 14 | 96 | 000144 | | 30 | 32 | 000145 | | 32 | 60 | 000146 | | 64 | 6 | 000147 | 000148 | Thus194 actual first root responses are nonzero. These are complex 000149 | current-basis response directions, not194 elementary particles, independent 000150 | predictions or physically selected primitive instruments. Compact real 000151 | rotations pair the adjoint root responses. 000152 | 000153 | Each nonzero response also has an exact action test on the actual code. 000154 | Let b=U^T beta, c=U^T v_beta in the original positive polarization. The 000155 | energy-preserving zero mode of (v_beta.h(-1))e^beta has, on f(x)Omega, 000156 | the PBW partition-(3) component 000157 | 000158 | [f(b) v_beta(-3) - (D_c f)(b) beta(-3)/3] e^beta. 000159 | 000160 | Normal ordering gives the first term from v_beta(-3) times the completely 000161 | shifted quartic, and the second from the v_beta(1) derivative of f followed 000162 | by the completely shifted cubic and beta(-3) creation. Their signs differ 000163 | because the two shifted polynomial degrees have opposite parity. 000164 | The actual five quartics have no common nonzero projective zero; for every 000165 | real nonzero beta at least one f_j(b) is nonzero. Since v_beta is nonzero 000166 | and perpendicular to beta, these two vector contributions cannot cancel. 000167 | Every one of the194 individual nonzero directions therefore leaks from 000168 | code, and its compression is zero by its nonzero lattice charge. Existence 000169 | of leakage is invariant under common real orthogonal Cartan transport. 000170 | This is a FIRST-ORDER result, not a no-go for second responses or histories. 000171 | 000172 | ## 5. Actual continuation, scopes and review 000173 | 000174 | The combined source now has an explicit derivative with coefficients fixed 000175 | by original AR and beta, rather than a freely chosen matrix pulse. The next 000176 | decisive object is an ORIGINAL ordered charged-pair return response on the 000177 | actual code and erasure carrier, together with its source-selected history. 000178 | An internally nontrivial return matrix alone would establish mathematical 000179 | availability, not physical permission, polar selection, a time scale or 000180 | unconditional 3+2. An inserted projection, detector, parent extension to 000181 | the leaked field sectors or fitted coupling cannot supply the missing law. 000182 | 000183 | Preserve these exclusions exactly: neutral local weight-two zero modes 000184 | alone cannot give R5; individual first mixed responses leak. Combined 000185 | root/quadratic histories, individual K48 mode roundtrips, higher-weight 000186 | fields, other mode products and non-automorphism processes remain outside 000187 | the global exclusion. The prior36-direction response tomography remains 000188 | valid; it is not a claim that executing those fields yields the clock. 000189 | 000190 | Original sources: neutral_source_response.py (_native_neutral_branch, 000191 | build_neutral_source_response_data), process.py (_code_basis), the prior 000192 | actual Cartan/Fock embedding and complete code stabilizer in this contract. 000193 | Primary formula: Dong and Nagatomo, Section2.2, pp4-5, 000194 | https://arxiv.org/pdf/math/9808088#page=5 . 000195 | The existing clock-jordan-response-reconstruction-20261015 provides all36 000196 | neutral quadratic response directions; it did not establish this code action. 000197 | 000198 | Independent analytic review confirmed the full44-dimensional neutral 000199 | weight-two classification, zero-mode formula, leakage constants and bounds, 000200 | global endpoint-stabilizer argument, the source194-count and first mixed 000201 | PBW formula. The exact checker is replayed with Python -OO; checks do not 000202 | depend on assert statements. No physical field/coupling selection is certified.