TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-ADMISSIBILITY-AND-TOTAL-CHIRAL-TRACE-PROOF.txt
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000001 | ORIGINAL ADMISSIBILITY AND THE WHOLE TOTAL-CHIRAL SPECTRAL READER
000002 | 31 October2026. R038/R039. Original Y/MR, sources, domains and reference retained.
000003 | 
000004 | 1. Exact original selector and source roles
000005 | Original source45 QFT002 lines31-117 explicitly defines
000006 |  Pstr=P_Seam,+ Psing PTheta,
000007 |  Ladm=Dadm^dag Dadm on Ran(Pstr),
000008 |  kappa=epsilon_adm^2/4>0,
000009 |  Padm=Pstr 1_[kappa,infinity)(Ladm).                 (1)
000010 | The positive self-adjoint realization of Ladm, its graph/form domain and
000011 | the given structural sector are used, not a new operator on another space.
000012 | The boundary001 owner lines208-261 supplies the analogous primitive split.
000013 | Its lines302-342 define PAP as a matrix-element shorthand and the local
000014 | net by restriction of those A AND A* preserving the admissible space.
000015 | Metrology001 lines105-126 instead defines Ksigma=fDgeo f+Brel, relative
000016 | Tr_rel and its reference/eta pair. These are distinct named operator calls.
000017 | 
000018 | The prior Atlas already establishes zero penalties on Ran(Padm), and the
000019 | high-pass exclusion of the SAME operator's zero mode in a prior vacuum
000020 | test. These facts are REUSED, not recorded as newly discovered. The new
000021 | test here is their effect on the WHOLE spectral functional and its residues.
000022 | No identification of Padm with normal Higgs modes, a finite chirality
000023 | half, a source state, or the printed I48/P2 follows from a dimension/name.
000024 | 
000025 | 2. The selector in(1) reduces its OWN positive operator exactly
000026 | On Hstr=Ran(Pstr), write P=1_[kappa,infinity)(Ladm). Functional calculus
000027 | gives strong commutation with every spectral function of Ladm, including
000028 | its resolvent and heat. Thus Q Ladm P=0 and
000029 |  P F(Ladm) P=F(Ladm|PH) on PH.                      (2)
000030 | Whenever the traces exist,
000031 |  Tr_Hstr(P exp(-t Ladm))=Tr_PH exp(-t Ladm|PH).      (3)
000032 | The R036 weighted/compressed heat difference is ZERO for THIS same L.
000033 | This conclusion uses strong spectral reduction, not the false claim that
000034 | P must be a pointwise parallel bundle projector. A spectral projector is
000035 | in general nonlocal and need not have that local form.
000036 | 
000037 | If Ksigma^2=Ladm under an actual original unitary/closed realization map,
000038 | (2)-(3) apply to that Ksigma^2. The original passages inspected do not
000039 | establish this operator/domain/reference map. Dadm=Dgeo or Dadm=Ksigma
000040 | is not inferred from a shared role. If Dadm=Pstr D Pstr, then generally
000041 |  Pstr D^dag D Pstr-Dadm^dag Dadm
000042 |        =Pstr D^dag (I-Pstr) D Pstr,                 (4)
000043 | not zero. A true restriction from a reducing domain or the original
000044 | off-diagonal terms are needed. No self-adjointness of Dadm was assumed
000045 | merely from the notation; D^dag D has its actual positive realization.
000046 | Equation(4) is a common-core quadratic-form/finite identity; it does not
000047 | silently identify unbounded adjoint domains after a compression.
000048 | 
000049 | Likewise Pgap kills ker(Ladm), because kappa>0. It cannot be the kernel
000050 | projector of the SAME operator on a nonzero kernel. The normal bosonic
000051 | index, its Higgs FIELD and its supplied Wedge->96 coupling remain valid;
000052 | the original map connects different operator roles rather than identifying
000053 | their kernels. This is a typed exclusion, not disappearance of the Higgs.
000054 | 
000055 | For a bounded observable in the original local net, invariance of PH under
000056 | both A and A* gives QAP=PAQ=0. Products and adjoints then restrict exactly.
000057 | It is NOT an axiom that every elementary charged field, source operation,
000058 | Ksigma or every history must commute with Padm. Original local-net
000059 | membership has to be shown for the object being used.
000060 | 
000061 | 3. Finite low-spectrum removal cannot repair Res1 versus FP0 continuously
000062 | Assume the SAME positive Ladm has compact resolvent, or more generally a
000063 | finite-dimensional spectral subspace below kappa. No eigenvalue at the
000064 | chosen endpoint is ambiguous: lambda=kappa is retained. Put N_< =
000065 | dim 1_[0,kappa)(L), n0=dim ker L, Npos_< =N_<-n0. Then exactly
000066 |  Hcut(t)=Hfull(t)-sum_(0<=lambda<kappa) exp(-t lambda). (5)
000067 | The removed finite sum is analytic at t=0. In four dimensions the heat
000068 | a0 and a2 terms are unchanged and the integrated a4 changes by -N_<.
000069 | Higher analytic heat terms may change. These counts are global spectral
000070 | data, not freely chosen local curvature or Higgs coefficients.
000071 | 
000072 | Using the same original convention excluding zero modes,
000073 |  zeta_cut(s)=zeta_full(s)-sum_(0<lambda<kappa) lambda^-s. (6)
000074 | The difference is ENTIRE in s. Every pole residue, including Res_s=1,
000075 | is unchanged, and
000076 |  FP_s=0 zeta_cut=FP_s=0 zeta_full-Npos_<.             (7)
000077 | The distinction N_< versus Npos_< is necessary: zeta_full(0)=a4_full-n0
000078 | on the ordinary smooth compact branch; the cutoff has no kernel.
000079 | On a smooth H branch without spectral crossing of kappa, N_< is constant.
000080 | If zero crossings occur without a threshold crossing, Npos_< and n0 may
000081 | change together. The UN-CUT zero-excluded zeta and Npos_< can then jump;
000082 | their jumps cancel in zeta_cut. A smooth comparison of separate zero-
000083 | excluded H jets uses a constant-nullity branch. No zero is silently lifted.
000084 | There is no continuously tunable H^2,H^4 or Einstein correction from(7).
000085 | No statement is made for infinite-rank low spectral bands, another trace,
000086 | moving threshold, or an unproved original spectral continuation.
000087 | 
000088 | In contrast, for a nonzero smooth low eigenvalue, the logarithmic
000089 | determinant loses log(lambda):
000090 |  logdet_cut=logdet_full-sum_(0<lambda<kappa) log(lambda). (8)
000091 | These terms CAN have continuous H jets. Replacing the original FP zeta(0)
000092 | by -zeta'(0) would therefore manufacture a different response. This is why
000093 | the actual original FP0 call, rather than a convenient determinant, matters.
000094 | For the original relative pair apply the SAME cutoff/reference prescription
000095 | to each operator. Their finite counts and absolute constants are retained.
000096 | 
000097 | When P=P(H) is recomputed spectrally, it moves with its operator. On a
000098 | differentiable separated finite-regulator branch, PP'P=QP'Q=0 and
000099 |  Tr(P' exp(-tL))=0,
000100 |  d Tr(P exp(-tL))/dH=-t Tr(P exp(-tL)L').             (9)
000101 | In an eigenframe, P'_ij=(p_i-p_j)L'_ij/(lambda_i-lambda_j) for opposite
000102 | spectral sectors. Higher derivatives retain these moving-projector terms.
000103 | Freezing P0 while changing L is another call. At threshold crossings a
000104 | smooth projector derivative is not asserted. Infinite traces need their
000105 | original differentiation bounds, not merely this finite formula.
000106 | 
000107 | 4. Two different halves in the existing product consumer
000108 | Use ONLY the already given conditional product branch
000109 |  D_H=-i gamma^mu nabla_mu+gamma5 tensor Phi(H),
000110 |  Phi=L(H)+DM, {GammaF,Phi}=0, nabla GammaF=0.        (10)
000111 | The original96 GammaF is fixed; all four given Y and MR are unchanged.
000112 | The graded product has Gamma_tot=gamma5 tensor GammaF. It follows directly