TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-ADMISSIBILITY-SOURCE-PROOF.txt
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000001 | ACTUAL ORIGINAL ADMISSIBILITY COMPLEX: WHOLE GHOST SOURCE AND GEOMETRIC TYPE
000002 | 31 October2026. R049/R050. F1 continuation after R046/R047/R048.
000003 | No new metric,normal operator,Yukawa,physical ghost measure or counterterm.
000004 | 
000005 | 1. The exact original identification being tested
000006 | Current tfpt2 section010:161-181 invokes the geometrical BRST/Hodge complex
000007 | on a compactified NORMAL two-dimensional sphere with APS data. It states
000008 | that Fredholm/Hodge decomposition discharges the step underlying R²/scalaron.
000009 | The original carrier007:95-139 instead explicitly types
000010 |  B_Ea=sigma_normal^mu nabla_mu^(a)+Phi_a
000011 | on that NORMAL sphere, and identifies the light weak field. Retain the given
000012 | light field and R026's actual Wedge to full96 coupling. Do not identify this
000013 | normal spin with physical4D spin or with the weakSU2 merely by dimension.
000014 | 
000015 | The original qft002:12-125 actually spells out a DIFFERENT full differential:
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
000061 | identically0; the grade0 coefficient gives exp(-tK). These identities are
000062 | valid without any trace-class assumption on the base.
000063 | 
000064 | If H is infinite dimensional, exp(-tK)>=exp(-t sum lambda_j)I is not trace
000065 | class. K is bounded and has at most16 spectral values. It is NOT a positive
000066 | order elliptic geometric Laplacian with compact resolvent. A convergent
000067 | bounded-operator Taylor series in t still exists; it is not the claimed
000068 | elliptic high-frequency heat trace. Appropriate externally given smearing,
000069 | relative weighting or another geometric operator must be treated explicitly.
000070 | This is not a theorem that bounded operators cannot be Fredholm. With(2),
000071 | the parity block D_Q^+:H_even->H_odd of D_Q=Q+Q^dagger is Fredholm exactly
000072 | when dim(PH)<infinity; then its index is(8-8)dim(PH)=0. If PH is infinite,
000073 | its kernel/cokernel are infinite and that Fredholm index is unavailable.
000074 | The partial supertrace0 in(5) is not a subtraction of divergent base traces.
000075 | 
000076 | At one common finite regulator and on positive K, a degree-weighted
000077 | determinant prescription, IF supplied, gives
000078 |  sum_p w_p logdet(K tensor I_Lambda^p)=
000079 |        [sum_p w_p binom(4,p)] logdetK.                 (6)
000080 | For w_p=(-1)^p p that bracket is0. The binomial multiplicity must be counted
000081 | ONCE, not both in logdet of the whole grade block and again as an external
000082 | factor. This conditional cancellation supplies no physical Faddeev-Popov
000083 | or BV measure and does not apply to unequal blocks,domains or mixed sources.
000084 | A Berezin integral is not automatically a supertrace. No determinant factor
000085 | is silently deleted from the actual full Z[0] or the bound reference.
000086 | 
000087 | 4. Constructive full original energy/source reader
000088 | The actual selfadjoint odd D_Q obeys D_Q²=K tensor I16. For z outside its
000089 | spectrum and z² outside specK its complete resolvent is
000090 |  R_z=(D_Q-z)^-1=(D_Q+z)(K-z²)^-1.                      (7)
000091 | With original degree0 basis vector |0> and |j>=c_j^dagger|0>,
000092 |  R_00=z(K-z²)^-1, R_j0=K_j(K-z²)^-1.                 (8)
000093 | Thus for z!=0 these actual source blocks recover every supplied constraint:
000094 |  K_j=z R_j0 R_00^-1.                                 (9)
000095 | Since K is bounded and z² outside specK, the inverse used here is bounded.
000096 | Its norm can depend on the supplied parameters. This is information recovery
000097 | for the original constraints, not selection of a geometric Higgs insertion.
000098 | The degree-weighted full resolvent trace is
000099 |  Tr_ghost[u^N R_z]=z(1+u)^4(K-z²)^-1.                (10)
000100 | It loses the separate K_j directions that remain in the off-diagonal blocks.
000101 | The actual source bilinear sum_I,J barJ_I R_IJ J_J retains all grades and
000102 | mixed transitions. No physical ghost path integral has been postulated.
000103 | 
000104 | For variation of the original commuting penalties with fixed projections,
000105 | deltaD_Q=sum_j(c_j^dagger+c_j)deltaK_j,
000106 | deltaK=2sum_j K_j deltaK_j. With R0=(K-z²)^-1,
000107 |  deltaR_z=-R_z deltaD_Q R_z
000108 |           =deltaD_Q R0-(D_Q+z)R0 deltaK R0.           (11)
000109 | Moving projectors instead require their actual spectral derivatives; they
000110 | must not be replaced by scalar deltaK_j. R039's own-operator filter result,
000111 | R037's full Schur/source formula and R043's whole mixed reader are retained.
000112 | 
000113 | 5. Decision at the first actual geometric identification
000114 | Equations(1)-(11) execute the ORIGINAL full penalty family. Its finite bounded
000115 | spectrum cannot be unitarily identified with the unbounded first-order
000116 | elliptic normal B_Ea on its full L2 sphere domain. A base physical selector
000117 | may be built FROM a specified geometric operator, but it is not that operator
000118 | and does not reconstruct its principal symbol or curvature coefficients.
000119 | No requirement of literal unitary equality was in the source; this excludes
000120 | only a proposed substitution, not an arbitrary valid family reduction.
000121 | 
000122 | Fredholmness of a specified normal operator gives finite-dimensional kernels
000123 | and an index; the integer alone is not the full ordinary spectral action.
000124 | The standard McKean-Singer pairing likewise distinguishes index supertrace
000125 | from the ordinary heat response. Its primary source is McKean and Singer,
000126 | Curvature and the eigenvalues of the Laplacian,JDG1(1967)43-69,
000127 | https://doi.org/10.4310/jdg/1214427880 . This general mathematics is reused,
000128 | not counted as a new theorem. The new TFPT decision concerns the actual
000129 | written four-condition differential and the claimed geometric inference.
000130 | 
000131 | Current010's mere Fredholm/Hodge step supplies neither an independent4D R²
000132 | coefficient nor its scalaron scale. The prior R046 full spinor test remains
000133 | the deciding calculation for that ordinary consumer. Original v36:1-76 was
000134 | also read: its R²/72 is exactly the incomplete curvature portion already
000135 | corrected by R046, and its f0=6(4pi)²/c3^7 matches the target by assignment.
000136 | This is a source provenance clarification, not a second curvature discovery.
000137 | No normal index,Higgs doublet,Yukawa/Flavor/Alpha result is reopened here.
000138 | 
000139 | The next original map is the genuine normal/geometric B_Ea to Brel insertion
000140 | in fDgeo f+Brel, with its field kernel,connection/symbol,normal metric,domain
000141 | and SAME relative trace/reference. Neither the four penalties nor a free
000142 | four-dimensional Hodge operator can be inserted in its place. R035 G/T and
000143 | R037 whole block reduction supply the transport once that actual original
000144 | operator is specified. Retain whole source,Z0,Ref/Eta,H0,State/Time/RP,
000145 | ordered words/memory,E8/clocks and all solved Yukawa/Flavor/Alpha outputs.
000146 | 22 exact symbolic controls pass. F1 remains in progress; no new Lean,
000147 | physical measure,RP,spacetime or scalaron closure and no ledger promotion.