TFPT scientific source excerpt Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/SPECTRAL-SEAM-RECONSTRUCTION.txt Original file SHA-256: 45c472819667e7a981f7da4246d74d31e15104335e941cde0e0893808d674654 Inclusive original lines: 1-200 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. 000001 | SPECTRAL SEAM RECONSTRUCTION FROM THE ORIGINAL OPERATOR PAIR 000002 | ========================================================== 000003 | Date: 2026-10-25 000004 | Contract: primitive-seam-operator-reconstruction-20261025 000005 | Verdict: PROVED under the explicit operator hypotheses below. 000006 | Original TFPT application: CONDITIONAL; the same-pair covariance and index 000007 | identification are not supplied by the original passages reviewed here. 000008 | 000009 | Purpose 000010 | ------- 000011 | The archived primitive seam theorem fixes a winding number and a spectral-flow 000012 | class. The electromagnetic closure later invokes a normalized seam vector as 000013 | if that theorem had already selected its line. The following theorem gives a 000014 | positive repair: construct the line and its whole circle channel from the 000015 | already supplied self-adjoint boundary operator B and seam unitary u. 000016 | 000017 | No external Hardy space, chosen rank-one bundle, new source profile, or 000018 | identification of the Calderon projection with a quantum covariance is used. 000019 | The theorem concerns one-particle operators; second quantization and the 000020 | physical interpretation of the original time remain separate interfaces. 000021 | 000022 | 1. Hypotheses on the SAME original Hilbert space 000023 | ------------------------------------------------ 000024 | Let H be a complex Hilbert space, B a self-adjoint operator on H and u a 000025 | unitary operator on H. Assume: 000026 | 000027 | (H1) Strong time covariance, for every real t, 000028 | 000029 | exp(itB) u exp(-itB) = exp(it) u. (1) 000030 | 000031 | (H2) 0 is not in the spectrum of B. 000032 | 000033 | (H3) With P_B = 1_(0,infinity)(B), the compression 000034 | 000035 | S = P_B u P_B : P_B H -> P_B H 000036 | 000037 | is Fredholm and ind(S) = -1. 000038 | 000039 | Units are chosen so that the shift in (1) is one. If the actual source uses a 000040 | positive shift omega, the rescaling B/omega must be declared and justified 000041 | using that same source; this is not a choice of a new physical clock. 000042 | 000043 | Conclusion: for exactly one epsilon in (0,1), the pair is unitarily equivalent 000044 | to 000045 | 000046 | H = L2(S1,dtheta/(2 pi)), 000047 | B = -i partial_theta + epsilon, Dom(B) = H1(S1), 000048 | u = multiplication by exp(i theta), 000049 | P_B = projection onto the Fourier modes n >= 0. (2) 000050 | 000051 | The canonical primitive seam line is 000052 | 000053 | Pi = P_B - u P_B u* = 1_(0,1](B), 000054 | rank(Pi) = 1. (3) 000055 | 000056 | Choosing a unit vector psi in this line only fixes an irrelevant common phase 000057 | for the basis u^n psi. Both Pi and every subspace in the construction are 000058 | independent of that phase. The conclusion classifies this operator class; 000059 | it does not assert that every primitive TFPT boundary must satisfy (H1)--(H3). 000060 | 000061 | 2. Domain-safe covariance and the compression 000062 | --------------------------------------------- 000063 | From (1), 000064 | 000065 | u* exp(itB) u = exp(it) exp(itB) = exp(it(B+1)). 000066 | 000067 | Uniqueness of the self-adjoint generator implies 000068 | 000069 | u Dom(B) = Dom(B), 000070 | u* B u = B+1, 000071 | B u xi = u(B+1)xi for xi in Dom(B). (4) 000072 | 000073 | Thus the spectral functional calculus gives, for every Borel set Delta, 000074 | 000075 | u E_B(Delta) u* = E_B(Delta+1). (5) 000076 | 000077 | In particular, 000078 | 000079 | u P_B u* = E_B((1,infinity)) <= P_B. (6) 000080 | 000081 | Therefore u leaves P_B H invariant and S is simply u restricted to P_B H. 000082 | It is an isometry, with 000083 | 000084 | S* S = I_(P_B H), 000085 | S S* = E_B((1,infinity))|_(P_B H). 000086 | 000087 | The defect projection in (3) is consequently an orthogonal spectral 000088 | projection. A formal commutator [B,u]=u on an unspecified set of vectors is 000089 | not a replacement for (1) or a proven equivalent common-core theorem. 000090 | 000091 | 3. The index constructs the line, with its sign fixed 000092 | ---------------------------------------------------- 000093 | An isometry has zero kernel and closed range. Hence 000094 | 000095 | ind(S) = -dim ker(S*) = -rank(Pi). 000096 | 000097 | Hypothesis (H3) yields rank(Pi)=1. Since Pi is a rank-one spectral projection 000098 | on the bounded interval (0,1], its range lies in Dom(B), and B acts there by 000099 | one scalar epsilon with 0 < epsilon <= 1. Let psi be any unit vector in it: 000100 | 000101 | B psi = epsilon psi. 000102 | 000103 | The endpoint epsilon=1 would give B u^(-1)psi=0 by (4), contradicting (H2). 000104 | Thus 0 < epsilon < 1. 000105 | 000106 | The sign in (H3) is the usual Hardy orientation: multiplication by z has 000107 | zero kernel and a one-dimensional cokernel on the nonnegative Fourier modes. 000108 | An oppositely oriented original seam must be paired with u* and the 000109 | correspondingly reversed spectral-flow convention. An orientation convention 000110 | cannot supply the missing relation between B and u. 000111 | 000112 | 4. Completeness on the original H, not an added model space 000113 | ---------------------------------------------------------- 000114 | Define e_n=u^n psi for all n in Z. Equations (4)--(5) imply 000115 | 000116 | B e_n = (n+epsilon)e_n, 000117 | E_B((n,n+1]) = u^n Pi u^(-n). (7) 000118 | 000119 | These projections have pairwise orthogonal ranges and each has rank one. 000120 | Since the half-open intervals (n,n+1] partition the real line, countable 000121 | additivity of the spectral measure gives 000122 | 000123 | sum_(n in Z) E_B((n,n+1]) = I strongly. (8) 000124 | 000125 | Consequently {e_n:n in Z} is an orthonormal basis of the ENTIRE H. There is 000126 | no unseen unitary Wold summand and no extra sector to be removed. The Fourier 000127 | map W e_n = exp(in theta) is unitary, W u W* = M_z, and the spectral theorem 000128 | gives exactly 000129 | 000130 | Dom(W B W*) 000131 | = {sum c_n exp(in theta): sum |n+epsilon|^2 |c_n|^2 < infinity} 000132 | = H1(S1), 000133 | 000134 | W B W* = -i partial_theta+epsilon. (9) 000135 | 000136 | This proves (2), including the unbounded operator domain. Compact resolvent, 000137 | a one-dimensional base manifold, a preselected scalar bundle, and purity of 000138 | an independently supplied projection were not assumed. In this particular 000139 | operator class they follow, or become unnecessary, after (H1)--(H3). 000140 | 000141 | 5. What selects the quarter offset 000142 | ----------------------------------- 000143 | If the SAME original B additionally satisfies 000144 | 000145 | exp(2 pi i B) = i I, (10) 000146 | 000147 | then exp(2 pi i epsilon)=i and 0