TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-source-character-search-20261015/source_reconstruction/PROOF.txt Original file SHA-256: 424e2f8eaf021cae575e479ccca4fcd2b8a34df1e5d5410371a86144f07ddb5d Inclusive original lines: 69-128 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. 000069 | 4. Exact diagnostic and full continuous scalar-symbol classification 000070 | 000071 | Use periodic trivialization H=L2(S1), B_alpha=N+alpha, N=-i d/dtheta, 000072 | alpha=1/4 or 3/4, and C e_n=e_n for n>=0, zero for n<0. This is equivalent 000073 | to the usual alpha-twisted boundary domain. The quarter transition is 000074 | exp(2pi i B_alpha)=exp(2pi i alpha) I and is kept throughout. 000075 | 000076 | For |a|<1 write b_a(z)=(z-a)/(1-conj(a)z). A finite rational unimodular 000077 | symbol is a phase times a quotient of finite products of these factors. 000078 | Cancel common factors first. Its winding is numerator degree minus 000079 | denominator degree. For S=C M_u C, zero leakage implies M_u 1=u belongs to 000080 | H2. Conversely analyticity and boundedness of u preserve H2 and, since 000081 | |u|=1, multiplication is an isometry there. A noncancelled denominator 000082 | factor has a pole in the disk. Hence in this rational class: 000083 | 000084 | zero leakage <=> no denominator after cancellation. 000085 | 000086 | Together with winding one this forces exactly ONE Blaschke factor, times 000087 | a constant phase. This classifies the full finite-rational-symbol family, 000088 | not just a finite root grid. Real rational roots in the executable are 000089 | exact diagnostics; the argument also covers complex roots in the disk. 000090 | 000091 | In fact this classification extends to EVERY continuous scalar unimodular 000092 | symbol u on the circle. Zero leakage implies that its negative Fourier 000093 | coefficients vanish. Analytic Fejer sums then converge uniformly to u, 000094 | so u extends to the disk algebra A(D). Since |u|=1 on the boundary, 000095 | continuity gives a zero-free collar of that boundary; all zeros are in a 000096 | compact interior subset and there are finitely many. Divide out their 000097 | Blaschke factors. The quotient v and 1/v are both in A(D), with unit 000098 | boundary modulus. The maximum principle applied to each forces |v|=1 000099 | inside; hence v is constant. By the argument principle, winding one gives 000100 | one Blaschke factor. This is an analytic theorem about the whole scalar 000101 | Hardy class; a finite executable grid is only its diagnostic check. 000102 | 000103 | An exact orientation-preserving control is u_a=z^2/b_a for real 0=1 its negative Laurent coefficient is 000108 | 000109 | [z^(-m)]u_a=(1-a^2) a^(m+1). 000110 | 000111 | Consequently ||(1-C)u_a e_0||^2=a^4(1-a^2), exactly 15/4096 at a=1/4. 000112 | The same quarter domain, C and B are retained. Therefore even a smooth, 000113 | strictly oriented winding-one representative need not satisfy the first 000114 | operator condition. This tests the inference winding -> isometry; it is 000115 | not a demonstration of all TFPT postulates in this diagnostic source. 000116 | 000117 | For a single b_a, isometry holds and its defect is one-dimensional with 000118 | normalized vector k_a(z)=sqrt(1-|a|^2)/(1-conj(a)z). Indeed b_a H2 is the 000119 | kernel of evaluation at a, whose orthogonal complement is span{k_a}. 000120 | Purity follows because a function in every b_a^n H2 has a zero of every 000121 | order at a and is identically zero. 000122 | 000123 | Nevertheless for the GIVEN original B_alpha=N+alpha, 000124 | 000125 | M_ba* B_alpha M_ba = B_alpha + (1-|a|^2)/|e^(i theta)-a|^2. 000126 | 000127 | The increment is the constant one only when a=0. A matching Wold shift 000128 | does not identify its reconstructed time with the supplied source time.