TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/RECURSIVE-WORD-SOURCE-PROOF.txt Original file SHA-256: 06d651e747e392a5a9c021c071fea3865687203aa58d75cee26fceb7035f5af5 Inclusive original lines: 112-241 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. 000112 | 4. Actual original source calculation 000113 | 000114 | Original source: joint-native-algebra-selection-20261016/ 000115 | check_source_return_memory.py: full_source, basis and its original Fock 000116 | metric. Same original witness beta=(0,0,0,0,0,0,-2,-2), original frame, 000117 | cocycle and R_neut branch as SOURCE-RETURN-MEMORY.txt. Its S maps the 000118 | complete neutral conformal-grade4 block (dimension726) into the charged 000119 | oscillator-grade3 block. L=Sdagger S uses the weighted actual adjoint. 000120 | Do not confuse oscillator grade with conformal grade in the charged sector. 000121 | 000122 | Use the original J-positive u_j polarization with Ju_j=i u_j and 000123 | J bar(u_j)=-i bar(u_j). On a neutral PBW monomial of arbitrary mode 000124 | indices J multiplies by i^(number of positive creators minus negative 000125 | creators). This is the native linear Cartan J, not the marked R5 or a 000126 | new physical time. It preserves the neutral grade4 block. 000127 | 000128 | The five original quartics have raw Fock Gram diag(96,8,8,8,1). 000129 | Apply exact Gaussian-rational Gram-Schmidt in this actual metric, starting 000130 | with those inputs and their actual L images. Continue under L,J,Jdagger 000131 | until every basis vector's generator images are reconstructed exactly. 000132 | No fitted vector or additional energy cutoff occurs. The result is: 000133 | 000134 | first layer span(P,LP) dimension10, 000135 | full reachable/observable dimension68, 000136 | visible code dimension5, 000137 | required exterior memory dimension63. 000138 | 000139 | The old rank10 lower bound was not a sufficient10D model. This calculation 000140 | derives the actual68D minimal linear realization for this larger alphabet. 000141 | Every one of the204 generator images is checked as an equality of ORIGINAL 000142 | Fock polynomials, not merely as a small numerical matrix residual. 000143 | G>0, Ldagger G=GL, Jdagger GJ=G, J^4=I, actual adjoint retention and 000144 | JP=P are checked exactly. Minimality then follows by section3. 000145 | 000146 | The exact native clock traces are tr(I,J,J^2,J^3)=(68,6,12,6). 000147 | Therefore its four eigenspace multiplicities at (1,i,-1,-i) are 000148 | (23,14,17,14). The five code directions lie in eigenvalue1, leaving 000149 | memory multiplicities (18,14,17,14). These are representation multiplicities 000150 | for this source/alphabet, not generations or spacetime dimensions. 000151 | 000152 | The ordered-word defect in the RAW five basis is 000153 | P L J L P - P L^2 P = 000154 | [[-76544/3, 10496 i/3, 0,0,0], 000155 | [-41984 i,-48128, 0,0,0], 000156 | [0,0,-10240,0,0], 000157 | [0,0,0,-10240,0], 000158 | [0,0,0,0,-6144]]. 000159 | Its rank is5. Minus this matrix is strictly positive in the actual code 000160 | metric. Its eigenvalues are 110464/3 +/-128 sqrt(150913)/3,10240,10240,6144, 000161 | all positive. Thus every nonzero input distinguishes these two ordered 000162 | responses via its own diagonal amplitude. It is not an event probability. 000163 | Algebraically the difference is Bdagger(J_Q-I)B: clock information is 000164 | stored outside the code and read back by the original source. 000165 | 000166 | At M=I+L the full elimination to5 equals elimination through the actual 000167 | first10 directions followed by elimination to5. Its inverse equals the 000168 | 5x5 block of the actual68 inverse. Invertibility holds because L>=0 in 000169 | the actual positive metric. This is an exact numerical-point application 000170 | of section2, not a substitute for that all-word analytic proof. 000171 | The previous source witness PL^2P-(PLP)^2 has (0,0)=203312/9 and is reused. 000172 | 000173 | Machine output: RECURSIVE-WORD-SOURCE.json, including all68 original Fock 000174 | basis polynomials, positive Gram, exact L/J/Jdagger matrices, ordered-word 000175 | defect, full nested Schur output and checks. Script: 000176 | check_recursive_word_source.py. Further character/defect controls: 000177 | RECURSIVE-WORD-SOURCE-STRUCTURE.json. 000178 | 000179 | 5. What this closes and what it does not 000180 | 000181 | F1.M4: stable exact nested observation with retained memory is established 000182 | on the declared original domain; the native specified source/alphabet has 000183 | an exact minimal68D all-word realization. A finite exhaustive closure 000184 | certificate answers infinitely many allowed word queries automatically. 000185 | 000186 | The complete E8 source and its all-grade OPE construction already exist. 000187 | This68D block is NOT their whole reachable space: new letters/charged end 000188 | sectors/grades may enlarge it. Whole-source composition uses section2 with 000189 | the actual complete alphabet and actual original readouts, rather than 000190 | declaring this finite demonstration the entire theory. 000191 | 000192 | Stable representation, stationary state and physical time are different. 000193 | The calculation supplies no originally selected intervention sequence, 000194 | interacting3+1D geometry or unique source selection. Existing flavor, alpha, 000195 | clock and other verified results retain their own precise hypotheses. 000196 | The next derivation must compose their actual source/readout maps within 000197 | one original full process and preserve the supplied state/time/reflection, 000198 | locality, domains, reference and whole action. No new Yukawa scan follows. 000199 | 000200 | 6. Method context and independent review 000201 | 000202 | Ordinary Feshbach/Schur elimination is established mathematics, not a new 000203 | TFPT principle. Primary context reviewed: Griesemer/Hasler, arXiv:0704.3244, 000204 | formula1 and theorem1; Bach/Ballesteros/Froehlich, arXiv:1308.0911, 000205 | sectionsIV/V. Arbitrary spectral recalibration need not preserve a reduction 000206 | semigroup. We keep the same variables and do not invent a physical RG time. 000207 | 000208 | [OMITTED: non-scientific conversation or execution metadata] 000209 | [OMITTED: non-scientific conversation or execution metadata] 000210 | [OMITTED: non-scientific conversation or execution metadata] 000211 | [OMITTED: non-scientific conversation or execution metadata] 000212 | [OMITTED: non-scientific conversation or execution metadata] 000213 | [OMITTED: non-scientific conversation or execution metadata] 000214 | [OMITTED: non-scientific conversation or execution metadata] 000215 | [OMITTED: non-scientific conversation or execution metadata] 000216 | 000217 | 7. The old memory defect and clock response are the same four-channel object 000218 | 000219 | Let E_w=(1/4)sum_(r=0)^3 w^(-r) J_Q^r for w=1,i,-1,-i on 000220 | the actual63D memory. These are orthogonal projections in its actual metric. 000221 | The four input forms F_w=Bdagger E_w B are positive in the original code 000222 | metric. Their ranks are (4,5,5,5), and their coordinate traces are 000223 | (274928/9,12288,113536/3,12288). F_i=F_-i exactly. 000224 | The original memory defect and the new clock response satisfy 000225 | Delta=sum_w F_w, 000226 | PLJLP-PL^2P=sum_w (w-1)F_w=-2(F_-1+F_i). 000227 | Both identities are checked exactly. Each nontrivial channel has rank5. 000228 | This connects the earlier positive memory result to the clock effect through 000229 | the SAME original source. No arbitrary clock coupling or normalization is 000230 | introduced. The four trace values are response strengths, not probabilities. 000231 | The entire ordered process still uses full matrices and all intermediate 000232 | returns; these first-return forms are not a replacement for longer words. 000233 | For the original unit vector f4=x0 x1 x2 x3 the response amplitudes are 000234 | =6144, =0. Repeated bare compression would instead 000235 | give =1024. These unnormalized operator response values 000236 | are exact; none is an asserted physical transition probability or time step. 000237 | 000238 | [OMITTED: non-scientific conversation or execution metadata] 000239 | [OMITTED: non-scientific conversation or execution metadata] 000240 | [OMITTED: non-scientific conversation or execution metadata] 000241 | [OMITTED: non-scientific conversation or execution metadata]