TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-ADMISSIBILITY-SOURCE-PROOF.txt
Original file SHA-256: a9e82303a4e420f9dfbb5fdc6c3e5b020c21debea53e751ce920aa5307838ad7
Inclusive original lines: 16-60
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.

000016 |  H_ext=H tensor Lambda(C4), Q=sum_j c_j^dagger K_j,
000017 |  K_j=sqrt(lambda_j) Pi_j, lambda_j>0,
000018 |  Pi_Sigma=(1-iota_C)/2, Pi_c=1-Psing,
000019 |  Pi_Theta=(1-QTheta)/2,
000020 |  Pi_g=1_[0,epsilon_adm²/4)(Pstr Dadm^dagger Dadm Pstr).
000021 | All four Pi_j are the GIVEN commuting orthogonal projections. Dadm's positive
000022 | realization and branch data remain supplied; no equality Dadm=Ksigma is added.
000023 | The macro backslash-iC means iota_C, NOT i times C: see original tfpt42
000024 | preamble.tex149. Retain the declared complex-linear selfadjoint involution.
000025 | This auxiliary exterior C4 is not the native Spin10 carrier exterior C5.
000026 | 
000027 | 2. Execute the WHOLE four-generator original differential
000028 | The actual exterior creation/contraction operators satisfy CAR, so
000029 |  Q²=0, Delta={Q,Q^dagger}=K tensor I16,
000030 |  K=sum_j K_j²=sum_j lambda_j Pi_j.                      (1)
000031 | This is an operator identity, not just equality of a trace. Every joint
000032 | projection pattern e in{0,1}4 has penalty sum_j lambda_j e_j. Some patterns
000033 | may be absent in the actual source, which does not change the identities.
000034 | Consequently
000035 |  0<=K<=sum_j lambda_j I,
000036 |  K|_(kerK)^perp >= min_j lambda_j I.                    (2)
000037 | All operators in(1) are bounded on H_ext. The original possibly unbounded
000038 | Dadm enters only through a bounded spectral projection. Its own domain,
000039 | boundary condition and geometric meaning have not thereby been derived.
000040 | 
000041 | Write P=1_{0}(K)=product_j(1-Pi_j)=Pstr Pgap on the base H. On the whole
000042 | positive complement define h=Q^dagger K^-1(1-P). It is bounded, with
000043 | norm at most(min lambda_j)^-1/2. It obeys
000044 |  Qh+hQ=(1-P) tensor I16.                               (3)
000045 | Thus the positive branch is acyclic. On PH all K_j,Q,Q^dagger vanish, and
000046 |  H^p(Q)=PH tensor Lambda^p(C4), p=0,...,4,
000047 |  P_kerDelta=P tensor I16.                              (4)
000048 | This proves the original base selector and clarifies its full lift.
000049 | The grade multiplicities are1,4,6,4,1. The existing base Padm is recovered
000050 | by grade0,or by explicitly forgetting the auxiliary factor for the specified
000051 | ghost-independent base observables. No compulsory global division of the
000052 | physical action or Z0 by16 is inferred. Degree-changing sources require(4).
000053 | Positive changes of ANY lambda_j leave P and(4) unchanged. They tune penalty
000054 | responses outside PH, not the retained physical source normalization.
000055 | 
000056 | 3. Partial ghost traces, full traces and index are different
000057 | For u complex,t>=0 the exact bounded partial trace is
000058 |  Tr_ghost[u^N exp(-tDelta)]=(1+u)^4 exp(-tK).             (5)
000059 | Indeed u^N=product_j[1+(u-1)n_j], NOT product_j(1+u n_j).
000060 | The ordinary ghost trace gives16 exp(-tK); the alternating ghost trace is