TFPT scientific source excerpt Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/MICROSCOPIC-TO-OCTIC-COMPOSITION.txt Original file SHA-256: eb0d6d436abeab7ace1ff094740c4b0d4748d3599331bfb3454dcb21e328ef5a Inclusive original lines: 201-333 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. 000201 | phases. This is the finite restriction of the unitary F, within the single 000202 | microscopic one-particle space. Its second quantization introduces no new 000203 | Fock factor. The unitary fold and the fixed finite matrices in (9) preserve 000204 | the strong convergence (1), including adjoints and timed words. 000205 | 000206 | A nonzero point split does not by itself turn an unsmeared fermion field 000207 | into a bounded CAR operator. The first draft of this section incorrectly 000208 | used the elementary finite-word theorem at that step. The independent review 000209 | identified the gap. SMEARED-OCTIC-ENERGY-BOUND.txt supplies its repair. 000210 | 000211 | For the actual shifted current polynomial and its actual adjoint it proves 000212 | 000213 | ||F8_m P_E|| <= C(1+E+|m|)^11, 000214 | ||F8(rho) P_E|| <= C_rho(1+E)^11, (11) 000215 | 000216 | on the true positive-source-energy cutoff. Smooth rho makes the mode sum 000217 | absolute in Hilbert norm, and the adjoint estimate proves closability. 000218 | Every fixed output/internal/fermion mode cutoff is a finite original CAR 000219 | polynomial. Replacing its elementary Fourier modes by the actual QWZ fields 000220 | and first taking N->infinity therefore gives a strong transported limit; 000221 | removing those cutoffs afterwards yields the full smeared F8 and its 000222 | adjoint. These Fourier approximants have global support. 000223 | 000224 | The local route instead separately smooths every elementary fermion with 000225 | a compactly supported mollifier of scale sigma. The original subtractions 000226 | must first be combined into the finite Wick/divided-difference expression, 000227 | including the derivative counterterms. At fixed sigma,epsilon>0 the smooth 000228 | CAR kernel has an absolutely summable Fourier-word expansion. The proof 000229 | then gives, for the field and its adjoint, the ordered local limit 000230 | 000231 | first N->infinity at fixed sigma,epsilon, 000232 | then sigma->0 at fixed epsilon, 000233 | finally epsilon->0. (12) 000234 | 000235 | The finite Wick expression has at most sixteen elementary fermions and 000236 | total conformal weight eight. The corresponding mode bound has polynomial 000237 | degree at most fifteen, absorbed by smooth rho. Fixed-regulator support lies 000238 | in shrinking neighborhoods of the same eight original preimage arcs. 000239 | No simultaneous lattice/ultraviolet limit or bounded point field is asserted. 000240 | 000241 | 5. The decisive energy-64 Ward test 000242 | ----------------------------------- 000243 | 000244 | The fold retains the exact source-mode relation 000245 | 000246 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_(n,0), 000247 | L_-1^d=Lhat_-1=Lcal_-8^alpha/8. (13) 000248 | 000249 | The state 000250 | 000251 | |F8;d>=F8hat(a_+,-1) Omega_d 000252 | 000253 | has folded oscillator level eight. By (13) its original source excitation 000254 | energy is 8*8=64. The Ward vector (L_-1^d)^8 Omega_d has the same folded 000255 | level eight and the same original energy 64. Both vectors are charge zero: 000256 | F8 is a current polynomial and L_-1^d is built from the neutral shifted 000257 | stress tensor. Thus the test never leaves the neutral source-energy-64 000258 | sector of the original QWZ scaling representation. 000259 | 000260 | For any homogeneous degree-eight oscillator polynomial P, 000261 | 000262 | 000263 | =8! conjugate(P(b)). (14) 000264 | 000265 | For the normalized F8 and the QWZ-equivalent negative-sea alpha=1/4 chart, 000266 | 000267 | A_(1/4) 000268 | =3 sqrt(35)(5416993+18690336 i)/2^31 != 0. (15) 000269 | 000270 | In original source normalization, (13) gives the actual amplitude 000271 | 000272 | 000273 | =8^8 A_(1/4) 000274 | =sqrt(35)(16250979+56071008 i)/128. (16) 000275 | 000276 | At energy at most 64 the truncation Gamma(Pi_M) in V_N is eventually the 000277 | identity. The finite polarized mode intertwiner I_N therefore reproduces 000278 | the finite oscillator polynomial entering (14) exactly for all sufficiently 000279 | large N. Replacing its polarized edge modes by the actual local top-edge 000280 | QWZ modes changes each fixed mode by the proved vanishing residual. Since 000281 | (14) contains a fixed finite-degree word, the actual microscopic amplitude 000282 | converges in the ordered limit (12) to (15), or to (16) with the unscaled 000283 | source mode. This is an amplitude of the full QWZ filled-sea Fock space, 000284 | not a value obtained only after an energy compression. 000285 | 000286 | The exact arithmetic certificate 000287 | 000288 | native_folded_octic_origin.py 000289 | 000290 | returns PASS_EXACT_ORIGINAL_FOLDED_OCTIC_OVERLAP and independently checks 000291 | (9), (13)-(16), all oscillator partitions in the Ward descendant, and the 000292 | nonzero values in both quarter sectors. 000293 | 000294 | 6. Exact remaining primitive-source question 000295 | --------------------------------------------- 000296 | 000297 | The neutral microscopic chain is now complete after its boundary data and 000298 | the original fold/current dictionary have been fixed: 000299 | 000300 | P1/P2 boundary data 000301 | -> width-eight, mass-one, holonomy-r=1 QWZ collar 000302 | -> negative-energy projector P_N and filled sea Omega_N 000303 | -> top-edge chirality chi_+ and boundary orientation/reflection 000304 | -> the quotient localization and unitary fold (4) 000305 | -> the fixed original matrices M, O and U_+ 000306 | -> the neutral field (10), state, time and Ward amplitude (16). 000307 | 000308 | The primitive step not derived by this composition is the first source 000309 | identification: the original seed must contain the appropriate local CAR 000310 | image and induce the same one-particle time, seam and reflection, with the 000311 | required collar domains and relative action. A microscopic regulator's 000312 | specific width and mass need not themselves be fundamental constants. 000313 | If the full graded CAR image lies in the original reconstructed vacuum 000314 | representation, its given positive energy and the same gapped time law force 000315 | the filled-sea state; that state is not another independent assumption. 000316 | An even-algebra image alone does not force it. 000317 | 000318 | P1 scalar source and preceding explicitly conditional premise merge: 000319 | 000320 | documents/tfpt_1_architecture_e8/sections/004-source.tex:190-243. 000321 | 000322 | The registered paper-source:7b58a2b8767cacd98fa2feb4 covers the preceding 000323 | context at lines1-200; the continuation at201-243 was also read in the actual 000324 | source. The former research_contracts path in this note was incorrect. 000325 | 000326 | This is narrower than a missing generic source-field map. The QWZ-to-CAR 000327 | field/state/time map and the one-CAR-to-neutral-F8 map are explicit and 000328 | compose. The separate eight-identical-copy synthesis remains a different 000329 | route for its stated pair-kernel result. It is neither used nor selected by 000330 | the present neutral-octic construction. Charged E8 root/spinor fields and 000331 | microscopic exhaustion remain separate questions and are not premises for 000332 | the result proved here. 000333 |