TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/TRANSFER-EXTRACTION.txt Original file SHA-256: 3a1a844f8f4e9ad5642abd627ae207d83deb11d68342e8e0bab929fff7ea7f2f 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 | gives R105's required -exp(i theta)I_R in the Hamiltonian kernel, while 000202 | the other coefficients test the remaining vertex. A top coefficient 000203 | alone remains necessary, not sufficient. A different generator or time 000204 | rescaling is not inserted to force this equality. 000205 | 000206 | For an explicit check of the WHOLE density hierarchy, divide the mirror 000207 | Hamiltonian kernel by E, using its finite Grassmann inverse. Write 000208 | x_i=bar_eta_i xi_i in the native normal order. Then E=PRODUCT(1+x_i), 000209 | E^(-1)=PRODUCT(1-x_i), uv=PRODUCT x_i for this16-mode order, and 000210 | E^(-1)u=u, E^(-1)v=v. The complete normal symbol becomes 000211 | 2-PRODUCT_i(1-x_i)-PRODUCT_i x_i-exp(i theta)u-exp(-i theta)v. 000212 | Its squarefree density expansion is exactly R105's degrees0 through16, 000213 | including the top density coefficient-2. Thus(8) does not keep only the 000214 | two16-letter transitions or silently drop any of the diagonal vertices. 000215 | 000216 | With the actual admissibility/Gauss projection P, the correct zero-time 000217 | kernel is k_P, not E I_R. The actual target is 000218 | k_P star [exp(-epsilon H_R)k_epsilon] star k_P, (9) 000219 | where star is the ORIGINAL complete boundary/Berezin contraction. If 000220 | P reduces that H, this is its physical transfer. The original full gauge 000221 | and rest action remains inside k_P; no product physical Hilbert space 000222 | is assumed. Its derivative is the corresponding complete P H P kernel. 000223 | Only that physical restriction, rather than an unobserved ambient 000224 | extension, is determined by the physical source. 000225 | 000226 | For a supplied canonical factorizing gauge action Gamma(r(g)) with 000227 | det r=1 and no additional implementer phase, one can write the original 000228 | R104 group sum BEFORE taking the mirror trace: 000229 | |G|^(-|V_int|)SUM_g Gamma_R(g)exp(-epsilon H_R) 000230 | [exp(-2epsilon)exp(bar_eta r(g)xi) 000231 | +(1-exp(-2epsilon))p_theta]. (10) 000232 | This is the untraced counterpart of R104's exact group weights. It is 000233 | conditional on that ACTUAL original gauge dictionary; it is not evidence 000234 | that the unprinted finite-group embedding is such a Gamma(r). 000235 | Its interpretation as the physical transfer additionally retains the 000236 | actual reduction [P_G,H]=0 used in(9). The group sum as P_G exp(-epsilon H) 000237 | is an algebraic identity even without that reduction, but then is not 000238 | automatically a semigroup on the physical range. 000239 | For a different actual action use(9), not a guessed determinant character. 000240 | 000241 | The coherent composition law uses the original resolution of identity 000242 | int dbar_chi dchi exp(-bar_chi chi)|chi)(bar_chi|=I 000243 | with its specified Berezin order. Hence (7) sews exactly and its derivative 000244 | really generates the same flat parent. A plain Grassmann exponential of 000245 | the normal symbol is only a first-order short-step representation; one 000246 | must not replace its exact star products by ordinary commuting products 000247 | at finite epsilon. Closed thermal traces additionally carry the original 000248 | fermion sign across their temporal closing cut. They are not themselves 000249 | unamputated open kernels. 000250 | 000251 | 5. EXACT PROJECTION/SEWING DECISION, WITHOUT A NEW TOY MODEL 000252 | 000253 | For any given bounded selfadjoint H and orthogonal P, write Q=I-P and 000254 | C(t)=P exp(-tH)P on RanP. 000255 | Then directly 000256 | 000257 | C(s+t)-C(s)C(t)=P exp(-sH)Q exp(-tH)P. (11) 000258 | For s=t the right side is 000259 | (Q exp(-tH)P)* (Q exp(-tH)P)>=0. (12) 000260 | 000261 | Therefore the whole compressed family sews as a semigroup for all times 000262 | IFF RanP reduces exp(-tH), equivalently [P,H]=0 in the bounded setting. 000263 | Its first derivative is -P H P, but its second derivative is 000264 | P H^2 P=(PHP)^2+PHQHP. (13) 000265 | The leading sewing discrepancy is t^2 PHQHP. Thus a first-derivative 000266 | match is sufficient only after the actual full family has its sewing 000267 | property; it cannot justify replacing C(t) by exp(-tPHP). A Zeno product 000268 | (P exp(-tH/n)P)^n is another limiting operation unless reduction holds. 000269 | For unbounded H formulate reduction through its spectral semigroup and 000270 | keep the original common domain; the bounded derivative formula is not 000271 | extended without its domain assumptions. 000272 | 000273 | Crucial ORIGINAL correction: Boundary001:315-345 and old42Primitive 000274 | ยง003:258-286 say PaAP is only matrix-element shorthand. Their physical 000275 | local net is A restricted to H_adm for A and A* both preserving H_adm. 000276 | For bounded A this means [A,P_adm]=0. Thus restriction is multiplicative 000277 | on the actual allowed net. If the actual interaction terms/time flow 000278 | also preserve this sector, their transfer restriction is exact. The 000279 | literal net definition does not by itself prove reduction for an 000280 | arbitrarily imported different H or for |B_Sigma|. That actual time 000281 | compatibility must be taken from the source map, not declared by notation. 000282 | 000283 | In particular the original FRW operator 000284 | U_Sigma=P_adm exp(-ell|B_Sigma|)P_adm 000285 | is a real positive trace-class source object at its stated width. Merely 000286 | its displayed compression does not prove a variable-width semigroup 000287 | or identify its logarithm with the interacting H_R+2Q. One may keep its 000288 | exact full kernel, using(11), without replacing that original object. 000289 | 000290 | 6. RELATIVE NORMALIZATION IS NOT REPLICATED AT EVERY TIME SLICE 000291 | 000292 | The original Z_rel=Z_adm/Z_ref[0] is a CLOSED source quotient with a fixed 000293 | outer source-independent denominator. It is not permission to divide 000294 | each short-step transfer by that same global denominator. The latter 000295 | would multiply an extra reference factor on every composition. In the 000296 | word reconstruction(1), the actual normalized source and its Gram 000297 | pairing handle the fixed normalization consistently. In an open path 000298 | integral the original boundary/reference functor supplies the corresponding 000299 | sewing factors. Preserve it, its eta phase and actual zero-source factor. 000300 | 000301 | If an actual relative open family has a nontrivial time-dependent scalar 000302 | factor, its derivative contributes a scalar generator insertion. That 000303 | factor has to be the originally bound reference, not a fitted subtraction. 000304 | Neither normalized moments nor an arbitrary energy shift determine it. 000305 | Equality of the full original AND reference kernels under the actual 000306 | field map gives equality of their relative caller automatically, as G 000307 | already proves for its given whole coordinate transport. 000308 | 000309 | 7. WHAT THE ORIGINAL PATH HAS AND THE ONE ACTUAL NEXT EVALUATION 000310 | 000311 | Already present and preserved: full exp W source, complete moment repair, 000312 | actual OS shift reconstruction, reducing-sector net definition, complete 000313 | ordered fiber/blocking operation, and the full finite DET16 comparison 000314 | operator. The separately supplied v1013 Hamiltonian route has its own 000315 | positive transfer by construction. The original lattice contract explicitly 000316 | distinguishes this from its Euclidean overlap candidate and preserves 000317 | S_seam/top; no gate or universal continuum requirement is added here. 000318 | 000319 | The finite C6/Perron/FNW/positive-parent-domination chain is also preserved. 000320 | Its reviewed producer paragraph QFT004:137-155,282-335 names transport, 000321 | weights and selectors but does not display a short-slab coherent field 000322 | contraction or prove(8). Its own weight audit S8 already records this 000323 | precise scope; it is not replayed as a new exclusion. 000324 | 000325 | The next decisive application is therefore SINGLE and explicit: evaluate 000326 | the original complete slab derivative in(5), or the equivalent complete 000327 | shifted-word derivative(2), on the actual identified boundary CAR legs, 000328 | without discarding rest/gauge/seam/top/Ref/Eta data; compare the resulting 000329 | whole polynomial with(8) or its actual projected version(9). If the 000330 | original source produces that identity and its actual sewing, all 000331 | operator/time/source consequences follow. If it produces another whole 000332 | generator, that original generator must be assessed on its own physical 000333 | chiral/IR route; DET16 is not a universal imposed answer. 000334 | 000335 | The current reviewed passages do not evaluate this one original 000336 | infinitesimal polynomial. This is a statement about these exact source 000337 | paths, not corpus-wide absence or a new global measure/vacuum gap. The 000338 | report supplies its complete forced form and a sufficient comparison 000339 | theorem, rather than another candidate Hamiltonian or finite battery. 000340 | 000341 | 8. CONCRETE CLOSED-SOURCE EXTRACTION BY THE ORIGINAL MATRIX INSERTIONS 000342 | 000343 | An independently supplied open boundary kernel is NOT required when the 000344 | actual original finite caller is already a complete ordinary thermal 000345 | trace on the SAME physical matrix block. Its complete time-word source 000346 | then contains a direct extraction of the open generator. This conclusion 000347 | uses the actual ordinary thermal trace and time, not a BW/modular trace 000348 | or an arbitrary relative Z silently called a Gibbs trace. 000349 | 000350 | Take native matrix units E_ai=|a>0 by finite-temperature faithfulness. Therefore 000363 | for i!=j: H_ij=-C'_ij(0)/p_a, 000364 | H_ii-H_jj=-(C'_ii(0)-C'_jj(0))/p_a. (15) 000365 | The derivative of the preweighted factor cancels in these combinations. 000366 | All joint-block matrix elements are reconstructed, up to a common scalar. 000367 | If its ACTUAL unnormalized positive trace Z_beta is supplied, that scalar 000368 | is fixed as well: for the reconstructed choice H_tilde, 000369 | c=beta^(-1)log[Tr exp(-beta H_tilde)/Z_beta], H=H_tilde+cI. 000370 | An outer relative denominator or eta phase is not substituted for this 000371 | positive trace. It remains with its original caller and reference. 000372 | 000373 | Equivalently, if the original caller supplies the shorter-circle traces 000374 | too, (14) reconstructs each FULL open transfer entry at finite t by 000375 | dividing its unnormalized C_ij numerator by 000376 | (exp(-(beta-t)H))_aa=Z_(beta-t)omega_(beta-t)(P_a). 000377 | This gives the whole kernel from its original CLOSED source coefficients; 000378 | it does not add a new open-sector preparation law. 000379 | 000380 | For a reset on the sixteen-mode factor only, keep the entire rest. Insert 000381 | E_ai tensor I and E_ja tensor B, with B an actual rest test leg. Then 000382 | C_ij;B(t)=Tr_R[(exp(-(beta-t)H))_aa 000383 | (exp(-tH))_ij B]/Z_beta, 000384 | C'_ij;B(0)=-Tr_R[(exp(-beta H))_aa H_ij B]/Z_beta, 000385 | i!=j. (16) 000386 | With all original rest tests B these functionals determine H_ij because 000387 | (exp(-beta H))_aa is strictly positive on the full finite rest block. 000388 | With only B=I they determine a conditional scalar expectation, not the 000389 | rest-valued operator. Diagonal differences follow similarly; complete 000390 | rest time insertions or joint-block matrix units retain the remaining 000391 | common rest Hamiltonian. A physical projection restricts this statement 000392 | to its actually visible support. Do not invert a removed sector or turn 000393 | charged inadmissible reset legs into physical observables by declaration. 000394 | 000395 | For the ambient DET top transition choose a=0,i=F,j=0. These are the 000396 | existing composite source insertions Omega16=|0>