TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/FINITE-COUPLED.txt
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000001 | THE ACTUAL LOCAL FINITE ODD DIRAC COEFFICIENT IN THE SAME BOUNDARY FUNCTOR
000002 | 29 October 2026
000003 | 
000004 | This continues PROOF.txt without replacing the original Yukawa coefficient.
000005 | It is an exact operator dictionary for a fixed classical background Phi(theta),
000006 | not a quantization of the bosonic fluctuation or a physical source selection.
000007 | The family is tangentially smooth and normal-independent in the fixed product
000008 | collar. A normal-dependent B(r) is outside the displayed exponential graph.
000009 | 
000010 | 1. Use the existing operator, not a fitted channel
000011 | Let gamma_F be the actual finite grading, Phi=Phi^* the actual finite Dirac
000012 | coefficient, {gamma_F,Phi}=0 pointwise. Permit smooth bounded Phi(theta) and
000013 | an even smooth Hermitian connection A=-i partial_theta+a(theta), with
000014 | [A,gamma_F]=0. Scalar a=alpha is the old special case. Retain the original
000015 | 
000016 |  B_Phi=Z tensor A-S tensor Phi,
000017 |  Z=diag(1,-1), N=[[0,-1],[1,0]], S=NZ.
000018 | 
000019 | The total grading is Gamma_tot=Z tensor gamma_F. Put
000020 | 
000021 |  Ical=i0 Gamma_tot,       C0=i0 N tensor I_F.
000022 | 
000023 | Then [B_Phi,Gamma_tot]=0, C0 B_Phi C0=-B_Phi, C0^2=I and
000024 | C0 Ical=-Ical C0. These are the same real Clifford conditions as in the
000025 | free calculation, with the TOTAL grading rather than a bare normal grading.
000026 | There is no demand that a bare right channel reduce B_Phi.
000027 | 
000028 | 2. Entire coefficient space and explicit charged Hamiltonian
000029 | Set P_F,plus=(I+gamma_F)/2 and P_F,minus=(I-gamma_F)/2. Define
000030 | 
000031 |  E v=(P_F,plus v, i0 P_F,minus v).
000032 | 
000033 | Then E^*E=I, Gamma_tot E=E, and E maps onto the entire positive total grade.
000034 | Direct multiplication gives the exact intertwiner on full periodic H1:
000035 | 
000036 |  B_Phi E=E h_Phi,
000037 |  h_Phi=A gamma_F-i0 gamma_F Phi.                         (1)
000038 | 
000039 | The second term is selfadjoint because gamma_F Phi is skewadjoint. Thus
000040 | h_Phi is selfadjoint on the H1 domain: its finite coefficient is bounded.
000041 | On H2 the square for the full smooth family is
000042 | 
000043 |  h_Phi^2=A^2+Phi^2-i0[A,Phi].                             (2)
000044 | 
000045 | For the old scalar connection, -i0[A,Phi]=-partial_theta Phi. For constant
000046 | Phi and internally scalar A this reduces to
000047 | 
000048 |  h_Phi^2=A^2+Phi^2 on Dom(A^2)=H2.
000049 | 
000050 | It is NOT identified with the old scalar A time when Phi is nonzero or
000051 | gamma_F has both signs. On the original real form Fix(C0), the generator
000052 | Ical B_Phi corresponds to i0 h_Phi under v->(v,i0v)/sqrt(2).
000053 | The selfdual completion is T(v,bar(w))=E v+C0 E w. It retains both total
000054 | grades as field and adjoint slots. Because C0 is OLD-complex-linear and
000055 | NEW-complex-antilinear, the conjugate-space type of bar(w) is essential.
000056 | It gives B_Phi=h_Phi directsum(-bar(h_Phi)) in the charged chart.
000057 | 
000058 | 3. All two-face correlations follow in this functor, with the given labels
000059 | On incoming and outgoing faces use C0 and -C0 as in PROOF.txt. The constant
000060 | unitary chart above gives the full spectral-calculus identity
000061 | 
000062 |  G(B_Phi) -> G(h_Phi) directsum(I-bar(G(h_Phi))).           (3)
000063 | 
000064 | All four crossing and diagonal blocks are transported, not just a trace,
000065 | an index or the same finite dimensional spectrum. If this Clifford graph
000066 | is the actual quantum covariance, the existing pure-projector theorem
000067 | therefore transports all CAR words and their given circle evolution.
000068 | No independent Gaussian state or chirally selected determinant is added.
000069 | The functor's physical use remains an explicit premise, as in PROOF.txt.
000070 | 
000071 | 4. Relation to existing exclusions and original charge J
000072 | Equations(1)-(3) do not undo the older full-COMPLEX-CAR constant-channel
000073 | classification. Quantizing the two normal slots independently and using
000074 | the graded real Clifford boundary functor are different statistical types.
000075 | No identification of C0 with the original finite KO6 charge J_F is claimed.
000076 | The actual 96-dimensional v252 space already includes antiparticle labels.
000077 | Their physical constraints, Pfaffian/chiral projection and any further
000078 | particle-hole identification must be checked, not counted as newly derived
000079 | physical species. No original bosonic field measure or coupled vacuum is
000080 | derived by using a classical Phi background. No physical OS Hamiltonian or four-dimensional
000081 | spacetime is inferred from this one-dimensional collar dictionary.
000082 | 
000083 | 5. Exact constructor control
000084 | check_finite_typing.py imports verification/v252_full_finite_triple.py and
000085 | uses its actual full_D and grading. The Gaussian integer Yukawa controls
000086 | and Majorana matrix are the same diagnostic constructor inputs as in the
000087 | already reviewed original_insertions check. They are not fitted or physical
000088 | mass values. Sparse exact algebra checks (1)-(2) for the entire 96-dimensional
000089 | coefficient, embedded into the full 192 normal coefficient, with symbolic
000090 | Fourier momentum. It also tests the differential-operator coefficients for
000091 | Phi(theta)=f(theta)Phi0 and scalar a(theta), retaining independent formal
000092 | jets f,f',a,a' on the same actual 96er coefficient. The analytic identities
000093 | (1)-(3) hold for the entire smooth background family by pointwise oddness,
000094 | the constant E and unitary spectral calculus of the full operators. No
000095 | commutation [A,Phi]=0 is needed for (1) or (3).
000096 | 
000097 | Original source controls reused:
000098 | verification/v252_full_finite_triple.py:86-121
000099 | original_insertions/PROOF.txt:1-200
000100 | The next derivation should test whether the original physical source uses
000101 | this canonical Clifford functor and how its actual charge J, nonlinear
000102 | bosonic fluctuations, marks and time enter it. A new free oscillator or a
000103 | newly chosen source covariance is not the proposed next step.