TFPT scientific source excerpt Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/SMEARED-OCTIC-ENERGY-BOUND.txt Original file SHA-256: 9ba6c518b6f254ff1c4152fb27d8525154cbd92b47377c5c59ba597b8823f563 Inclusive original lines: 201-400 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 | convergent sums: 000202 | 000203 | =. (15) 000204 | 000205 | Consequently F8(rho) is closable and its Hilbert-space adjoint extends the 000206 | displayed smeared adjoint on D_fin. This is the required field-plus-adjoint 000207 | statement. It is a statement about smooth smeared fields, not point values. 000208 | 000209 | 5. Finite current cutoffs are finite original CAR words 000210 | ------------------------------------------------------- 000211 | 000212 | In the filled-sea representation each eigenchannel current mode has the 000213 | standard fermionic form 000214 | 000215 | j^a_n=sum_r :chi^a_r* chi^a_(r+n):_sea. (16) 000216 | 000217 | For fixed n and fixed input cutoff P_E, only finitely many summands in (16) 000218 | act. Far from the Fermi surface both occupations agree with the sea and the 000219 | normal-ordered bilinear annihilates the vector. Every excitation in P_E uses 000220 | one of finitely many labels, and a shift by n enlarges that label window only 000221 | by |n|. Thus j^a_n P_E is a finite sum of original CAR bilinears. 000222 | 000223 | Equations (4) and (5) show more. For fixed m and E, only finitely many current 000224 | mode tuples in (3) act, and every surviving current reduces on P_E to finitely 000225 | many CAR bilinears. This observation alone does not yet define an acceptable 000226 | microscopic approximant: P_E is a comparison projection, not part of the 000227 | physical operator. 000228 | 000229 | Define instead a global finite continuum operator G_K(rho), K>=2, without 000230 | using E or P_E. Keep only 000231 | 000232 | |m|<=K for the output field mode, 000233 | |n_i|<=K^2 for every internal current index, 000234 | |r|<=K^4 in every CAR sum (16), (16a) 000235 | 000236 | and use the same sea normal ordering, coefficients, b insertions and rhohat_m. 000237 | This is a finite sum of globally defined original CAR words of fermion degree 000238 | at most sixteen. Its one-particle entries are finite Fourier sums and hence 000239 | smooth. For each fixed E, equations (4)--(5) bound every internal current 000240 | index that can act by E+K. The source-energy bound also confines all fermion 000241 | excitation labels. Consequently there is K_E such that, for K>=K_E, 000242 | 000243 | G_K(rho) P_E = F8^[K](rho) P_E. (16b) 000244 | 000245 | Thus G_K is independent of E, while P_E is used only to state eventual 000246 | agreement and convergence. 000247 | 000248 | Replace each elementary continuum CAR Fourier mode in G_K by the corresponding 000249 | actual top-row QWZ field Psi_N(g), retaining the same filled-sea ordering and 000250 | adjoint. Denote this globally defined finite microscopic Fock operator by 000251 | G_(N,K)(rho). The existing microscopic theorem gives for each fixed E,K 000252 | 000253 | ||(G_(N,K)(rho)V_N-V_N G_K(rho))P_E|| ->0, (17) 000254 | 000255 | and the analogous statement for the adjoint. The original theorem is stated 000256 | vectorwise. Since P_E is finite dimensional, convergence on a basis upgrades 000257 | it to the restricted operator norm in (17). 000258 | 000259 | Combining (14) and (17), using that V_N is a contraction, gives the transported 000260 | iterated limit 000261 | 000262 | lim_(K->infinity) limsup_(N->infinity) 000263 | ||(G_(N,K)(rho)V_N-V_N F8(rho))P_E||=0, (18) 000264 | 000265 | and the identical conclusion for the adjoint. Equation (18) is the repaired 000266 | strong microscopic smeared-composite theorem. 000267 | 000268 | No K-uniform microscopic estimate was used. Therefore (18) does not assert a 000269 | simultaneous N,K limit. The finite K approximants are Fourier-mode operators 000270 | with global circle support. Equation (18) alone does not prove exact interval 000271 | support for the microscopic approximants. 000272 | 000273 | 6. Separately mollified local route 000274 | ----------------------------------- 000275 | 000276 | The original current point-splitting prescription can also be retained 000277 | locally, but every elementary fermion must first be separately smeared. 000278 | 000279 | Choose a nonnegative smooth compactly supported mollifier eta with integral 000280 | one. Then ||eta||_1=1 and therefore |etahat(sigma r)|<=1. Let 000281 | 000282 | eta_(sigma,x)(y)=sigma^(-1) eta((y-x)/sigma), 000283 | psi_sigma(x)=a(eta_(sigma,x)). (19) 000284 | 000285 | Use translated copies of (19) at every point in the epsilon-split definition 000286 | of the currents and the octic. Let 000287 | 000288 | F8_(epsilon,sigma)(rho) 000289 | 000290 | be the result after the same centre smearing rho and the same old sea 000291 | subtractions and derivative counterterms. At finite N the subtraction may be 000292 | written with the actual microscopic filled-sea contraction; at fixed 000293 | sigma,epsilon it converges to the prescribed continuum subtraction. For each 000294 | elementary bilinear the two subtractions differ by a vanishing scalar. After 000295 | the finite Wick expansion of the octic, the complete difference is therefore 000296 | a finite sum of lower-degree CAR words whose scalar coefficients vanish as 000297 | N->infinity; it is not in general one scalar multiple of the identity. At 000298 | sigma,epsilon>0 its kernel in the at most sixteen fermion variables is smooth. 000299 | Its tensor Fourier series is absolutely summable, so it is a norm-convergent 000300 | sum of bounded finite CAR words. The fixed-regulator QWZ limit follows 000301 | termwise from the elementary theorem, with the absolute Fourier sum providing 000302 | domination. More explicitly, if c_(r_1,...,r_s) are the tensor Fourier 000303 | coefficients of one fixed-regulator kernel, then 000304 | 000305 | sum_(r_1,...,r_s) |c_(r_1,...,r_s)| < infinity. (19a) 000306 | 000307 | For a Fourier basis vector e_r, the source estimate is S_0(e_r)=1, hence 000308 | 000309 | ||Psi_N(e_r)||=||R_N e_r||<=1 000310 | 000311 | uniformly in N. Every corresponding CAR word therefore has norm at most one. 000312 | Equation (19a) is an N-independent Weierstrass majorant, so termwise QWZ 000313 | convergence passes through the infinite Fourier-word sum. Thus 000314 | 000315 | F8_(N,epsilon,sigma)(rho)V_N psi 000316 | ->V_N F8_(epsilon,sigma)(rho)psi (20) 000317 | 000318 | for psi in D_fin, and likewise for the adjoint. 000319 | 000320 | It remains to remove sigma and epsilon in the continuum. Wick expansion of a 000321 | degree-eight polynomial in bilinear currents is finite. It has the form 000322 | 000323 | sum_lambda c_lambda 000324 | :product_(l=1)^s_lambda partial^(q_lambda,l) psi^#_l:, 000325 | 000326 | s_lambda<=16, 000327 | sum_l q_lambda,l+s_lambda/2=8, (21) 000328 | 000329 | including lower fermion degrees generated by the prescribed contractions. 000330 | This is precisely why bosonic current normal ordering is not the naive 000331 | degree-sixteen fermion monomial. 000332 | 000333 | The common-counterterm hypothesis used here is exact and limited. There is 000334 | one finite universal Wick polynomial W_(epsilon,sigma)[C] in the elementary 000335 | fields, the quasifree contraction C and its prescribed divided differences. 000336 | The microscopic expression evaluates it at the actual sea contraction C_N; 000337 | the continuum expression evaluates the same polynomial, with the same 000338 | coefficients and ordering, at C. At fixed sigma,epsilon all finitely many 000339 | mollified contractions and divided differences required by W converge from 000340 | C_N to C. Expanding the finite polynomial then gives the lower-degree 000341 | vanishing terms described above. If the two sides used independently chosen 000342 | subtraction constants rather than evaluations of this same Wick polynomial, 000343 | equation (20) would not follow. The retained current point-splitting 000344 | prescription supplies precisely this common W_(epsilon,sigma)[C]; it is a 000345 | premise of the local route, not a consequence of the energy estimate. 000346 | 000347 | The subtraction terms must be combined into (21), or equivalently into 000348 | finite divided differences, before estimating them. For every fermion mode r, 000349 | 000350 | |exp(i r epsilon)|=1, 000351 | |(exp(i r epsilon)-1)/epsilon|<=|r|, (22) 000352 | 000353 | and the q-th divided difference is bounded by |r|^q. Mollifier multipliers 000354 | obey |etahat(sigma r)|<=1 and converge pointwise to one. 000355 | 000356 | On P_E each fixed total field mode has only finitely many acting fermion-mode 000357 | tuples. Equations (21) and (22) give termwise convergence as sigma->0 and then 000358 | epsilon->0. They also give a polynomial majorant uniform in small sigma and 000359 | epsilon. A conservative count is 000360 | 000361 | number of tuples: O((1+E+|m|)^(s_lambda-1)), 000362 | derivative weight: O((1+E+|m|)^(8-s_lambda/2)), 000363 | 000364 | whose combined exponent is 000365 | 000366 | s_lambda/2+7 <=15. (23) 000367 | 000368 | Smooth rho absorbs the power fifteen just as in (11). Dominated summation over 000369 | m now yields, on every P_E, 000370 | 000371 | strong-lim_(epsilon->0) strong-lim_(sigma->0) 000372 | F8_(epsilon,sigma)(rho)=F8(rho), (24) 000373 | 000374 | and the same limit for the adjoint. The target in (24) is the mode-defined 000375 | F8(rho) of (12), because the retained point-splitting counterterms are exactly 000376 | the Wick contractions that produce (21); after taking matrix elements between 000377 | finite-energy vectors, both sides have the same finite mode coefficient at 000378 | every m. Combining (20) and (24) proves the fully ordered local route 000379 | 000380 | epsilon->0 after sigma->0 after N->infinity. (25) 000381 | 000382 | For compactly supported rho and eta, the fixed-regulator microscopic fields 000383 | are supported in the corresponding (sigma+|epsilon|)-neighborhoods of the 000384 | eight original preimage arcs. This is the correct local statement. No support 000385 | claim is made for the global finite-mode approximants of section 5, and no 000386 | uniform estimate allows the three limits in (25) to be collapsed into one 000387 | simultaneous limit. 000388 | 000389 | If the old counterterms are left as separately divergent powers of epsilon 000390 | instead of first being combined into the Wick/divided-difference expression, 000391 | the bound (22) is unavailable and the local proof fails. Their algebraic 000392 | combination is therefore a load-bearing part of the theorem, not a notation 000393 | choice. 000394 | 000395 | 7. Energy-64 corollary and exact scope 000396 | -------------------------------------- 000397 | 000398 | The previous Ward result is the finite special case of the theorem. On the 000399 | continuum H<=64 space, the one-particle labels satisfy -63<=j<=64, and 000400 | Gamma(Pi_M) is the identity for N>=4096. The level-eight F8 state and