TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/INDUCED-HARDY-DEFECT-BMU-FINITE-READER.txt Original file SHA-256: 209e29b96e4f43d7a4211f3a0a8ccaa0e4607db9991947b622ee174fceb510d3 Inclusive original lines: 18-102 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. 000018 | 1. The actual first bundle dictionary, not a dimension count 000019 | Read ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt:40-54,85-92,119-146 and 000020 | FUCHS-FRAME-INDEPENDENT-TRACE-REVIEW.txt:37-91. Its actual first column is 000021 | g=h0 (p1/(ab),z p2/b,z p3)^t, 000022 | with h0^2=(w+1)/(w+omega). The other columns are its two c translates. 000023 | On the actual V4 cover the right action is 000024 | diag(epsilon0,epsilon2,epsilon1), product epsilon=1, 000025 | and the cubic deck acts by Pc, the actual cyclic column permutation. 000026 | Hence its three horizontal characters are the orbit of the SPECIFIC 000027 | nontrivial V4 character chi0 defined by L0^q. These are exactly the matrices 000028 | of Ind_(V4)^(A4) chi0. The displayed invertible original G is the intertwiner: 000029 | F_v117 isomorphic pi_*L0^q, 000030 | pi^*F_v117 isomorphic L0^q+L1^q+L2^q. 000031 | The original Q identity G(Qy)=U G(y)KQ retains the quarter signs; no new 000032 | constant family identification is fitted. Choosing L0 is the already marked 000033 | column, with the other two choices its actual c-related induction charts. 000034 | 000035 | The canonical ramified S0 pullback and half-form trace then give, over the 000036 | open original disk (unramified there), the spin version 000037 | pi_*(S0 tensorL0^q) isomorphic S_base tensorF_v117. 000038 | Consequently its M_z quotient has exactly the three fibers 000039 | Kdef=direct_sum_(a=0..2) (S0 tensorL0^q)|q_a. 000040 | This is the original old three-cable quotient, not an added copy of Fq. 000041 | Pulling back the entire rank-three bundle to all three sheets gives rank9; 000042 | its genuine C3 descent or the above induction returns the same rank3. 000043 | It is not legitimate to retain all nine and then call them three families. 000044 | 000045 | 2. Exact sea/Hardy projection and the finite defect 000046 | Use the source domain already fixed by its original interior continuation: 000047 | H1_NS and D_H={G c:c in H1_NS}, with H=G^(-*)G^-1. Its NS Hardy half 000048 | has coefficients z^n sqrt(dz), n>=0. This is the EXISTING original Hardy 000049 | premise, not a new state selected from the graph covariance alone. 000050 | Let P_H be its orthogonal projection and U=M_z. Then 000051 | Delta=P_H-U P_H U*=E0 E0*, rank_family Delta=3. 000052 | Here the exact defect embedding, with the standard boundary 1/(2pi) norm, is 000053 | E0 xi(theta)=exp(i theta/2) G(e^(i theta)) xi/sqrt(2pi), 000054 | where xi is the original horizontal three-column coefficient. Equivalently 000055 | for v in F_v117|0, Psi=G G0^-1 and H0=G0^(-*)G0^-1, 000056 | E0 v(theta)=exp(i theta/2) Psi(e^(i theta))v/sqrt(2pi), 000057 | ||E0 v||_H^2=v*H0 v. 000058 | No extra scalar norm N was introduced. The already fixed spin square, 000059 | coefficient metric, signed continuation and ramification weights are used. 000060 | 000061 | For the ACTUAL whole graph put 000062 | V_h=(sqrt(A),sqrt(I-A))^t, A=(I+exp(-2ell h))^-1, 000063 | G=V_h V_h*, Q_G=V_h P_H V_h*, Uhat=V_h U V_h* on RanG. 000064 | Then V_h*V_h=I and therefore exactly 000065 | Delta_G=Q_G-Uhat Q_G Uhat*=V_h Delta V_h*=E E*, 000066 | E=V_h E0. (1) 000067 | It is rank3 in the family factor, with identity on the other retained 000068 | slots; it is NOT rank3 on the entire 96-component source. This holds 000069 | without assuming [h,P_H]=0 or [h,U]=0. It does not make P_H the full 000070 | coupled-Hamiltonian positive-energy sea. Its provenance is the named 000071 | original kinetic/Hardy half. The whole given state is unchanged. 000072 | 000073 | 3. Smallest countercheck against substituting the wrong seam operation 000074 | For the scalar graph G_n=v_n v_n*, 000075 | v_n=(1,e^(-ell(n+alpha)))/sqrt(1+e^(-2ell(n+alpha))), 000076 | W=diag(U,U) has (WGW*)_n=G_(n-1). Every difference block has trace zero 000077 | and is nonzero for finite ell>0. Infinitely many such nonzero blocks give 000078 | infinite rank, even though their exponential decay makes the difference 000079 | trace class and its total trace zero. It is not Delta_G in (1). 000080 | Likewise z is holomorphically invertible on the full annulus; that whole 000081 | Laurent operation is bilateral and has index0. The rank3/index-3 result 000082 | belongs to the named one-sided Hardy space. Finally Uhat differs from 000083 | both the physical two-endpoint diagonal shift and the already proved 000084 | relative matching [-U,I]V_h. Their distinction is retained, not repaired 000085 | by relabeling all three operations as the same shift. 000086 | 000087 | 4. The finite Bmu matrix, with actual endpoints and all slots retained 000088 | Use the ALREADY GIVEN B=G_q J*, J=Swap tensor j, 000089 | j=i gamma_F F_R R_x, 000090 | and let M_mu denote the original K-valued Hom insertion AFTER its actual 000091 | specified tangent/Clifford contraction, on the two labeled faces. Define 000092 | its finite matrix as the canonical form, not an unproved L2 product: 000093 | M_rs==. (2) 000094 | The equality uses G_q E_r=E_r; no alternate covariance or response algebra 000095 | has been chosen. In the given p Hermitian metric and half-density pairing, 000096 | M_rs=sum_i integral [(J E_r)_i(x)]* (M_mu E_s)_i(x) dx. 000097 | For a componentwise insertion m_mu this is explicitly 000098 | sum_i integral [j(E_r)_(1-i)(x)]* m_mu(x)(E_s)_i(x) dx. (3) 000099 | The j term evaluates the q field at R_x x with its actual original spin, 000100 | quarter carry, F_R and endpoint frame. It is not a free pointwise matrix B. 000101 | The original L1 ramification behavior is integrable against these smooth 000102 | weighted source test columns. No new smearing or cutoff is selected.