TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/ORIGINAL-PROFILE-DICTIONARY.txt
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Inclusive original lines: 14-145
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000014 | [OMITTED: non-scientific conversation or execution metadata]
000015 | 
000016 | documents/archive--paper-latex--tfpt-31/sections/
000017 | 002-subsection-primitive-relative-spectral-dynamics.tex:7-28 defines
000018 | 
000019 |  Tmin_adm=(A,H,D,J,Gamma,inv,nu),
000020 |  f31(x)=int_0^infinity exp(-s x^2)dnu(s), dnu>=0.        (1)
000021 | 
000022 | Thus nu belongs to the actual supplied reduced datum in this version.
000023 | It is not a new boson configuration measure and is not derived there
000024 | from the other six tuple entries.
000025 | 
000026 | The same original:42-70 reconstructs
000027 | 
000028 |  Dref=(D+inv D inv)/2,
000029 |  Drel=(D-inv D inv)/2,
000030 |  chi=lambda1+(|B_Sigma|)^(-1).                          (2)
000031 | 
000032 | The primitive action at:80-96 is
000033 | 
000034 |  S31=Tr f31(Drel/chi)-Tr f31(Dref/chi)
000035 |                                       +(i pi/2)Delta eta. (3)
000036 | 
000037 | The original positive branch theorem:98-115 assumes positivity preserving
000038 | and reflection stable heat semigroups of Dgeo^2 for every proper time,
000039 | then transports those properties through positive superposition. It
000040 | does not prove these hypotheses from nu>=0. The well posedness passage
000041 | at:118-135 uses the original selfadjoint Fredholm APS realization and
000042 | relative bound on the same graph domain.
000043 | 
000044 | 2. The exact unsquared to squared identity
000045 | 
000046 | Define a second function name solely to disambiguate the argument:
000047 | 
000048 |  g_nu(u)=int exp(-s u)dnu(s), u>=0.
000049 | 
000050 | Then f31(x)=g_nu(x^2). The spectral theorem for the ACTUAL selfadjoint
000051 | realization K gives, with no new parameter or spectral approximation,
000052 | 
000053 |  f31(K/chi)=g_nu(K^2/chi^2)
000054 |            =int exp(-s K^2/chi^2)dnu(s).                (4)
000055 | 
000056 | At a compatible finite regulator the trace follows by finite spectral
000057 | calculus; an infinite realization uses the actual trace/relative trace
000058 | integrability. Apply(4) to BOTH original operators in(3). Therefore
000059 | 
000060 |  S31=int dnu(s){Tr exp(-s Drel^2/chi^2)
000061 |                     -Tr exp(-s Dref^2/chi^2)}
000062 |                                       +(i pi/2)Delta eta. (5)
000063 | 
000064 | For a relative trace that exists only after subtraction, keep the paired
000065 | relative trace inside the original integral. Do not separate divergent
000066 | absolute traces. The reference remains(2), with the same parameter and
000067 | domain dependence. Equation(4) is a function identity on K; it does not
000068 | replace K by Dgeo, a one particle modular generator, or a compressed
000069 | different operator merely because their symbols are similar.
000070 | 
000071 | 3. Full original radial Hessian, not a freely selected profile class
000072 | 
000073 | Use an actual affine field direction in this original chart,
000074 | 
000075 |  D_j(r)=D_j(r0)+(r-r0)V_j,       j=rel,ref,
000076 |  P_j=D_j(r0)^2/chi^2,
000077 |  A_j={D_j(r0),V_j}/chi^2,
000078 |  C_j=2V_j^2/chi^2.                                     (6)
000079 | 
000080 | Here the field direction, chi, inv and common domain are the supplied
000081 | ones. For example, the original Higgs radial family is Phi_r=r DY+DMR;
000082 | its reference derivative follows from(2), not an independent reference
000083 | choice. If chi or inv varies on the actual path, differentiate them too.
000084 | The affine formula(6) is only for the stated fixed chi, fixed inv path.
000085 | 
000086 | The exact first derivative of each heat trace is
000087 | 
000088 |  F_j'(r0)=-int s Tr(exp(-sP_j)A_j)dnu(s).
000089 | 
000090 | The complete second derivative is
000091 | 
000092 |  F_j''(r0)=int dnu(s){
000093 |  -s Tr(exp(-sP_j)C_j)
000094 |  +s^2 int_0^1 dv Tr[exp(-(1-v)sP_j)A_j
000095 |                                       exp(-v sP_j)A_j]}. (7)
000096 | 
000097 | Thus S31''=F_rel''-F_ref''+(i pi/2)(Delta eta)''. This
000098 | retains all spectral modes, actual mixed Higgs derivative terms in
000099 | {D,V}, and the complete proper time measure from the reduced datum.
000100 | It does not replace nu by a chosen delta measure, fit low moments,
000101 | truncate to a2/a4, or omit the original reference.
000102 | 
000103 | For each individual s the matrix identity is the Duhamel derivative.
000104 | Passing derivatives through the actual nu integral requires its actual
000105 | convergence. At a positive spectral gap, existence of g_nu on the
000106 | needed lower spectral neighborhood supplies the corresponding derivative
000107 | domination; at zero modes retain the required s and s^2 integrability
000108 | or use the given regulated derivative before taking limits. Moving APS
000109 | domains require the original transported operator chart and domain terms.
000110 | Regularized eta can vary smoothly without an eigen crossing; its jets
000111 | are retained, along with the actual crossing prescription.
000112 | 
000113 | 4. Source45 and the later geometric f_star are distinct declarations
000114 | 
000115 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000116 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex:
000117 | 33-37 makes spectral profiles analytic realizations rather than boundary
000118 | datum entries;:303-316 defines S_prim for ANY declared spectral profile;
000119 | :345-350 describes Bernstein as an analytic realization of a transfer
000120 | channel already fixed by the closed datum and canonical projector.
000121 | 
000122 | Its full master source at
000123 | documents/archive--tfpt-45--source_extracts--00_orientation_source/sections/
000124 | 002-section-big-picture-what-tfpt-actually-claims.tex:73-84 uses
000125 | Tr f((Drel+Aseed)/chi)-Tr f(Dref/chi)+(i pi/2)Delta eta.
000126 | If this f is the original even f31 realized by(1), equation(4) applies
000127 | to its ENTIRE operator Drel+Aseed. It does not square Drel alone while
000128 | dropping Aseed or the cross terms. For an arbitrary declared Source45
000129 | profile, a positive g_nu satisfying f(x)=g_nu(x^2) must be supplied by
000130 | that actual realization before(4) can be used as its heat representation.
000131 | 
000132 | documents/archive--tfpt-45--source_extracts--05_metrology_source/sections/
000133 | 005-section-absolute-dimensionless-metrology-from-the-bounda.tex:61-134
000134 | defines a later geometric positive measure fixed point nu_star from a
000135 | supplied RG kernel K(s,t), normalization psi, matching slice and bounds.
000136 | It then uses Tr f_star(D_Vn^2/Lambda^2) in a GEOMETRIC configuration
000137 | measure at:136-178, with a separate configuration base nu_Vn.
000138 | These proper time nu_star and configuration nu_Vn have different types.
000139 | The known literal archived RG map defect is already recorded and is
000140 | not reproduced as a new result here. The reviewed passage supplies no
000141 | numerical nu_star that could be substituted for nu in(1).
000142 | 
000143 | The source45 geometric residue construction is another named functional:
000144 | Metrology001:98-129 defines E_D=Res_(z=1)Zrel(z;sigma) and
000145 | Gamma_D4=FP_(z=0)Zrel(z;sigma), with