TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/REFERENCE-FIBER-EVALUATION.txt
Original file SHA-256: b816066a532162b0b6dbe2f49203c484ab1f91662b290a759f3e9223963c4ee5
Inclusive original lines: 272-348
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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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.