TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/FULL-COUPLED-SEAM-INDEX.txt
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000001 | THE ACTUAL WHOLE COUPLED SOURCE AND THE ORIGINAL SEAM MATCHING
000002 | Independent analytical continuation, 29 October 2026.
000003 | Existing registered sources and scalar proof remain unchanged.
000004 | 
000005 | 1. Fixed original data and the exact scope of the matching
000006 | Reuse F5-HARDY-SEAM-OPERATOR-REVIEW.txt and the sources cited there.
000007 | The original fullv21 introduction:225-247 specifies FOUR boundary pieces,
000008 | APS on C1,C2, and the seam clutch U_Gamma=e^(i theta) on Gamma_+,Gamma_-.
000009 | It does not identify that seam pair with the two F5 collar faces.
000010 | Subject to that specific endpoint assignment, use its unitary clutch
000011 |   x(ell)=U x(0), U=M_(exp(i theta)) tensor I_F.
000012 | Keep the previously fixed orientation U e_n=e_(n+1). Reversed clutch gives
000013 | the opposite primitive index. No scalar channel or three copies are selected.
000014 | 
000015 | The actual existing charged coefficient is
000016 |   h=A gamma+V, V=-i gamma Phi, A=-i partial_theta+alpha,
000017 |   Phi=Phi*, {gamma,Phi}=0, gamma=gamma_F.
000018 | Here Phi is CONSTANT and internally A and U are scalar. Thus V=V*,
000019 | V^2=Phi^2, {A gamma,V}=0, and U*VU=V. These are used below.
000020 | This is the constant constructor class already tested in EXACT-CHANNELS.txt,
000021 | not all Higgs backgrounds or all possible charged seam holonomies.
000022 | References: graded_boundary_functor/FINITE-COUPLED.txt:10-56;
000023 | verification/v252_full_finite_triple.py:86-121.
000024 | 
000025 | For R=exp(-ell h), C=(I+R^2)^(-1/2), the SAME complete Cauchy graph has
000026 |   V_h=(C,R C),       F_U V_h=(R-U)C.
000027 | Both coefficients are bounded, with no truncation or arbitrary Cauchy ODE.
000028 | This is relative boundary mismatch, not compression of U on either face.
000029 | The existing C0, charged/conjugate slots, graph state, reflection and circle
000030 | action are retained. No equivalence with the original P1 source is added.
000031 | 
000032 | 2. Full principal symbol and the actual ordinary index
000033 | Let d_+=dim F_+, d_-=dim F_- for the finite gamma grading. At Phi=0 the
000034 | whole bounded mismatch is the direct sum of the scalar positive and negative
000035 | gamma blocks. Its compact reference is
000036 |   (P_- -U P_+) tensor I_(F_+)
000037 |        directsum (P_+ -U P_-) tensor I_(F_-),
000038 | where P_+ selects n>=0 and P_-=I-P_+ for 0<alpha<1.
000039 | The first scalar reference is injective with one cokernel, index -1.
000040 | The second is surjective with one kernel, index +1.
000041 | 
000042 | This remains the reference for constant Phi. Indeed
000043 |   h_n=p_n gamma+V,       h_n^2=p_n^2+Phi^2,
000044 | and its positive/negative spectral projections differ from those of
000045 | p_n gamma by O(1/|p_n|). The two bounded graph coefficients converge to
000046 | the same projections with the same O(1/|p_n|) error. Consequently their
000047 | difference from this reference is compact, in fact Hilbert-Schmidt in one
000048 | circle dimension. The massless error was trace class; no trace-class claim
000049 | is made for the general coupled error.
000050 | Therefore, on the WHOLE charged coefficient,
000051 |   ind((R-U)C)=d_- -d_+=-Tr_F(gamma).                  (1)
000052 | The actual v252 grading has d_+=d_-=48, hence the whole ordinary index is0.
000053 | 
000054 | This uses the full principal symbol rather than assigning the scalar -1
000055 | to selected physical copies. Coupling mixes F_+ and F_-: gamma does NOT
000056 | commute with h when Phi!=0. A difference of separate chiral indices is
000057 | then not an additional gamma-graded Fredholm invariant of this operator.
000058 | The selfdual completion is not an additional set of independent physical
000059 | fields. Vanishing whole integer index does not trivialize every determinant
000060 | line character and does not refute the old one-sided spin Hardy theorem.
000061 | 
000062 | 3. Stronger exact defect theorem for this SAME constant coupling
000063 | Use K_Phi=partial_r+A gamma+V with the above unitary clutch, initially on
000064 | smooth matching fields and then on its closed H1 realization. Its adjoint
000065 | is -partial_r+A gamma+V with the SAME unitary clutch: integration by parts
000066 | gives y(ell)=U y(0), since U is unitary.
000067 | 
000068 | Expansion, Hermiticity and anticommutation give
000069 |   ||K_Phi x||^2=||K_0 x||^2+||Phi x||^2,            (2)
000070 |   ||K_Phi* x||^2=||K_0* x||^2+||Phi x||^2.         (3)
000071 | The derivative cross term is exactly
000072 |   [<x(r),V x(r)>]_(0)^ell=0
000073 | by U*VU=V. The tangential cross term is zero by {A gamma,V}=0.
000074 | No fixed-domain APS projection is being varied in this argument.
000075 | 
000076 | For K_0, gamma+ has no kernel and one adjoint kernel per internal vector;
000077 | gamma- has one kernel and no adjoint kernel. The gamma- kernel is the
000078 | smooth Gaussian clutch solution
000079 |   f(r,theta)=sum_n c_n exp(r p_n) e_n(theta),
000080 |   c_n=c_0 exp[-ell(n(n+1)/2+alpha n)].
000081 | The same scalar f is the adjoint-kernel function on gamma+.
000082 | Equations(2)-(3) therefore imply exactly
000083 |   dim ker K_Phi =dim(ker Phi intersect F_-),
000084 |   dim ker K_Phi*=dim(ker Phi intersect F_+).          (4)
000085 | 
000086 | These are also the kernel and cokernel dimensions of the bounded matching
000087 | map in section1. One can justify the domain passage without an arbitrary
000088 | L2 evolution assumption: the compact-reference remainder above gains one
000089 | Fourier Sobolev order. Its explicit shift reference has a parametrix
000090 | preserving every Sobolev order, up to finite smooth modes. Bootstrap makes
000091 | all matching kernels and adjoint matching kernels smooth. R x=U x then
000092 | gives smooth endpoint traces and the corresponding H1 interior solutions.
000093 | Conversely each H1 clutch solution supplies these graph data. For the
000094 | adjoint matching equation, pointwise Fourier equality R_n y_n=y_(n+1)
000095 | also proves y lies in Dom R; it is the adjoint boundary consumer.
000096 | 
000097 | In particular, invertible Phi gives NO kernel and NO cokernel. On the
000098 | closed differential realization (2)-(3) give the inverse bound
000099 |   ||K_Phi^(-1)|| <= 1/s_min(Phi).
000100 | This is not asserted as a bound on a differently normalized boundary map.
000101 | Hermitian odd Phi on equal-dimensional grades has equal kernel dimensions
000102 | on the two grades, so the surviving defects occur in opposite pairs.
000103 | 
000104 | 4. Apply the already verified WHOLE v252 controls, without another run
000105 | Atlas finding:whole-quarter-channels-identified-by-exact-internal-kernel,
000106 | research-source:02a45b61c44bd61d15ef2453, EXACT-CHANNELS.txt:1-44;
000107 | research-source:054e763500be2aab6476a2e4, exact-source-channels.json:1-44.
000108 | Its existing diagonal control has positive kernel0, hence negative kernel0:
000109 | the actual whole coupled seam matching has defect(ker,coker)=(0,0).
000110 | Its rank-deficient controls have positive kernels2 and1, respectively;
000111 | equation(4) gives whole defects(2,2) and(1,1), all with integer index0.
000112 | These are specified constructor controls, not derived physical masses.
000113 | 
000114 | The actual neutral positive submodule is nu_R^c tensor C3. Its criterion
000115 |   ker [[MR],[entrywise_bar(Ynu)]]
000116 | is retained exactly. It is not the adjoint of Ynu. No phase or scaling
000117 | chooses three neutral solutions when this actual matrix has trivial kernel.
000118 | In the actual positive/negative neutrino order, the off-diagonal Phi block is
000119 |   Q_nu=[[Ynu,0],[MR*,Ynu^T]],   Phi_nu=[[0,Q_nu],[Q_nu*,0]].
000120 | Here * denotes adjoint. This explicitly shows which original Yukawa and
000121 | Majorana insertions lift the two grades; it does not add a new interaction.
000122 | 
000123 | 5. The actual primitive NS clutch, insertion and time tests
000124 | BOUNDARY-SOURCE.txt:114-194 gives the full NS operator and
000125 |   u_cov=[[0,u_base],[I,0]], u_cov^2=diag(u_base,u_base).
000126 | Thus the original primitive winding exchanges the two quarter sectors.
000127 | It has one primitive scalar contribution in each internal gamma direction,
000128 | not two independently selected quarter contributions. The same proof in
000129 | the NS Fourier basis p_n=n+1/2 gives (1)-(4). Physical normalization of A
000130 | rescales ell and the Gaussian, not their dimensions or integer index.
000131 | 
000132 | For a constant original odd insertion deltaPhi,
000133 |   delta K=-i gamma deltaPhi.
000134 | Between a surviving normalized gamma- Gaussian kernel and gamma+ adjoint
000135 | kernel, its first-order finite defect matrix is precisely
000136 |   -i P_(kerPhi intersect F_+) deltaPhi P_(kerPhi intersect F_-).
000137 | This is a decisive actual insertion test. A deltaPhi pairing these null
000138 | directions removes opposite defects; a freely chosen scalar multiplicity
000139 | does not survive this test automatically.
000140 | 
000141 | For the whole quarter h, [h,U]=U gamma and [h,gamma]=2i Phi, so the old
000142 | scalar covariance U_t=exp(it)U does not apply on the whole field space.
000143 | The already given time transports the matching FAMILY by
000144 |   U_t=exp(it h) U exp(-it h).
000145 | It need not preserve the fixed original seam domain. In the primitive NS
000146 | basis the same identities hold with its actual Fourier normalization.
000147 | 
000148 | 6. Outcome and retained first original assignment
000149 | The whole constant coupled source has an exact automatically decidable
000150 | seam defect, given by its already computable internal kernel, and ordinary
000151 | index0 for the actual96-dimensional v252 coefficient. The complete given
000152 | source is used; no channel, state, physical family or scalar coefficient
000153 | has been fitted. The old scalar Fredholm proof remains a valid block result.
000154 | This extends the OLD full-kernel recognition result to the original relative
000155 | seam boundary consumer. It does not repeat the older full-COMPLEX-CAR
000156 | vector-current cancellation as a new exclusion of the graded source.
000157 | 
000158 | The true original comparison still needs the calibrated assignment of
000159 | Gamma_+,Gamma_- to these two collar traces and the actual spin/cable domain
000160 | map. A charged noncentral U, nonconstant Phi or connection not commuting
000161 | with Phi changes the hypotheses of (2)-(4); no cancellation of their cross
000162 | terms has been assumed. No whole P1 exclusion, physical mass selection,
000163 | three-family derivation or new absolute source normalization follows.