TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/REFERENCE-FIBER-EVALUATION.txt
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Inclusive original lines: 181-348
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000181 | 
000182 | Thus the fermionic reference contribution has actually been evaluated
000183 | for these original branches without selecting new inputs. The entire
000184 | remaining original prefactor is displayed, not fitted or discarded.
000185 | The per mode prefactor notation in(8)-(12) is for a supplied mode fiber.
000186 | When forming the entire circle product, a global original scalar/body
000187 | factor is retained ONCE. It is not copied into each Fourier mode unless
000188 | the original caller actually factorizes it in that way.
000189 | 
000190 | 5. Complete normalized fiber density and source words
000191 | 
000192 | For each fixed bosonic background and nonzero scalar branch, normalization
000193 | of the full fixed fermion fiber cancels its actual scalar:
000194 | 
000195 |  rho_F,j=Gamma(exp(-t h_j))/det(I+exp(-t h_j)).          (13)
000196 | 
000197 | This is a positive normalized density for finite Hermitian h_j, including
000198 | its signed old occupation spectrum. In the original excitation chart it
000199 | is unitarily the positive heat state
000200 | 
000201 |  rho_exc,j=Gamma(exp(-t|h_j|))/det(I+exp(-t|h_j|)).       (14)
000202 | 
000203 | For S its signed covariance is
000204 | 
000205 |  C_R,S=(I+exp(tp gamma))^-1,
000206 | 
000207 | and for T it is C_R,T=(I+exp(t h_A))^-1. Taking the actual t=2ell from
000208 | SEED-FIELD:64-78 gives exactly the original Cref reader. All original
000209 | ordered fiber words follow by the given finite CAR/Berezin pairing. For
000210 | instance an ordered normally paired word has determinant entries
000211 | <f_i,C_R,j g_k> in its actual convention. Other orderings retain the
000212 | original CAR signs and old field/excitation map.
000213 | 
000214 | The ordinary fermion trace closes the coherent source with its actual
000215 | antiperiodic endpoint, not a periodic supertrace. In particular(8)-(10)
000216 | are ordinary exterior traces. A selfdual physical source keeps its actual
000217 | Pfaffian/reality contract instead of treating every doubled slot as an
000218 | independent determinant species a second time.
000219 | 
000220 | The cancellation statement for the T branch concerns the null source or
000221 | the same background comparison in BOTH given fermion kernels. The actual
000222 | original relative generating functional has Z_ref[0] in its denominator.
000223 | External sources inserted only in the numerator do not disappear:
000224 | equal zero source kernels still give their nontrivial normalized fermion
000225 | word functional. No Z_rel[J]=1 assertion is made.
000226 | 
000227 | 6. General mixed circle backgrounds and whole source scope
000228 | 
000229 | Equation(2), the coherent formula(5) and the finite exterior identities
000230 | do not require [A,Phi]=0. In a general smooth tangential gauge/Higgs
000231 | background use the ENTIRE h with
000232 | 
000233 |  h^2=A^2+Phi^2-i[A,Phi],
000234 | 
000235 | including the actual derivative and gauge couplings. Equations(7)-(12)
000236 | are the original constant scalar p mode specialization; balanced48/48
000237 | alone does not make an arbitrary gauge connection A equal to pI.
000238 | 
000239 | For smooth bounded original coefficients on the circle, the already
000240 | established trace class result Tr exp(-t|h|)<infinity gives the complete
000241 | intrinsic excitation determinant det_F(I+exp(-t|h|)). For the S flat
000242 | circle branch it is product_n(1+exp(-t|n+alpha|))^96. No divergent
000243 | infinite Esea or raw exterior determinant is assigned an absolute value.
000244 | The original finite regulator scalar and its full relative/eta limit
000245 | must be carried as in SEED-FIELD:172-179.
000246 | 
000247 | In particular Gamma(exp(-t h)) is not a trace class object on the old
000248 | empty Fock representation when the signed circle h has an infinite
000249 | negative tail. The infinite expression here is instead the intrinsic
000250 | given sea/excitation representation of Gamma(exp(-t|h|)). Comparing its
000251 | old charged fields, vacua or reference families requires their actual
000252 | original sea/relative implementability and phase/domain hypotheses;
000253 | finite regulator unitary particle/hole equivalence is not asserted as
000254 | an automatic unitary equivalence of infinite empty Fock representations.
000255 | The determinant and summability statements concern the stated FIXED
000256 | BACKGROUND circle spectrum. They establish no new4D quantization,
000257 | shared dynamical boson measure, or conversion of collar t into a
000258 | physical thermodynamic inverse temperature.
000259 | 
000260 | This calculation integrates the entire actual FERMION reference fiber.
000261 | It does not perform the original dynamical boson/gauge/geometry integral,
000262 | compute its remaining a_R, or imply that whole Z_ref[0] equals1. The full
000263 | reference and source transport, original Hodge/gauge quotient, reference
000264 | normalization and holographic IR alternative remain in force. The time
000265 | used here is the original conditional collar/exterior parameter unless
000266 | its established source time map is explicitly invoked.
000267 | 
000268 | No new S/T lift, coupling, numerical background, field measure, or physical
000269 | model has been selected. Registered earlier notes and repository sources
000270 | are unchanged.
000271 | 
000272 | 7. Positive exact application: full original r=0 reference force
000273 | 
000274 | Use the same original constant circle branch p=n+alpha and the actual
000275 | radial family Phi_r=r DY+DM. The profile here is not a chosen cutoff:
000276 | it is forced by the evaluated positive fermion heat trace(10),
000277 | 
000278 |  g_p(x)=log(1+exp(-t sqrt(p^2+x))),
000279 |  log Z_heat,A,p(r)=Tr96 g_p(Phi_r^2),
000280 |  log Z_heat,R,S,p=96 g_p(0).                            (15)
000281 | 
000282 | Keep the actual nonzero original quarter modes, so p!=0. The complete
000283 | mixed Yukawa/Majorana functional calculus has no square root cusp at
000284 | r=0. Its derivative is exactly
000285 | 
000286 |  g_p'(x)=-t/[2sqrt(p^2+x)(1+exp(t sqrt(p^2+x)))].        (16)
000287 | 
000288 | Let
000289 | 
000290 |  A_ch=Tr3(3Yu*Yu+3Yd*Yd+Ye*Ye).
000291 | 
000292 | The existing ORIGINAL-HIGGS-FIBER-PROOF.txt:section4 (R028) proves
000293 | the coefficient for log(z+Phi^2), with the noncommuting Majorana term
000294 | retained. Its finite spectral functional calculus continuation gives
000295 | for an admissible smooth spectral profile g (analytic on a neighborhood
000296 | of the finite spectrum suffices for the contour derivation),
000297 | 
000298 |  partial_u Tr96 g(Phi_r^2)|_(u=r^2=0)
000299 |    =4g'(0)A_ch+4Tr3[g'(MR MR*)Ynu*Ynu].                (17)
000300 | 
000301 | One can obtain(17) directly from that proof's resolvent identity and
000302 | the contour representation of g; it also follows by integrating the
000303 | resolvent expression for log(z+Phi^2). No simultaneous diagonalization
000304 | of MR and Ynu, or deletion of their cross term, is used. The original
000305 | complex symmetric MR hypothesis is retained. Formula(17) extends
000306 | continuously across singular MR for the present smooth g_p.
000307 | 
000308 | Therefore the actual S reference fermion response is
000309 | 
000310 |  d_r^2 log[Z_heat,A,p(r)/Z_heat,R,S,p]|_(r=0)
000311 |   =8{g_p'(0)A_ch
000312 |                    +Tr3[g_p'(MR MR*)Ynu*Ynu]}.         (18)
000313 | 
000314 | Every original Y and MR enters with its actual multiplicity. The
000315 | quantity is strictly negative if any original Yukawa is nonzero, since
000316 | g_p' is negative on the entire positive spectrum. The curvature of the
000317 | corresponding negative log fermion ratio has the opposite sign. This
000318 | is a conditional fermion source response, not a physical Higgs mass
000319 | or an evaluation of the full boson measure.
000320 | 
000321 | There is also an explicit Majorana value for the reference comparison
000322 | itself at r=0. The actual DM^2 has90 zero eigenvalues, while each of the
000323 | three eigenvalues of MR MR* occurs twice. Thus
000324 | 
000325 |  log[Z_heat,A,p(0)/Z_heat,R,S,p]
000326 |         =2Tr3[g_p(MR MR*)-g_p(0)I3].                   (19)
000327 | 
000328 | In particular the S branch at H=0 is not an identical reference anchor
000329 | when MR!=0. The whole original comparison additionally retains the
000330 | actual a_A/a_R,S at this point. On the T branch h_R,T=h_A for the
000331 | entire even family, so(18)-(19) cancel in the same fermion comparison,
000332 | while its whole original a_A/a_R,T is still retained.
000333 | 
000334 | For the full original circle, both(18) and(19) sum absolutely over
000335 | n. In particular
000336 | 
000337 |  |g_p'(x)|<=t exp(-t|p|)/(2|p|), x>=0,
000338 | 
000339 | and the original quarter shift bounds |p| away from0. The bound times
000340 | the finite original Yukawa traces is summable; a corresponding mean
000341 | value bound applies to(19). This is the complete specified circle
000342 | fermion response, not a finite mode sample replacing a limit.
000343 | 
000344 | For the WHOLE source append the actual original derivative of
000345 | log(a_A/a_R,S) or log(a_A/a_R,T) in the given field/source convention.
000346 | Those remaining original factors have not been set to1 or evaluated
000347 | by the sea normalization. The reference at zero external source is
000348 | still distinct from inserting identical external sources in both legs.