TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/NS-CONTINUUM-WINDING-LIMIT.txt Original file SHA-256: 4a61f9604bb51f0d13419606d7d8c64fa2cfc840b417a7297403db7d6706f916 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 | NS CONTINUUM WINDING LIMIT 000002 | ========================== 000003 | 000004 | Question 000005 | -------- 000006 | 000007 | Does the positive finite NS CAR source of v519/v524/v526 carry the original 000008 | fullv21 winding operator u=M_z, and can its finite ordinary determinant converge 000009 | to the old polarized factor 1-z without deleting a UV contribution by hand? 000010 | 000011 | Verdict 000012 | ------- 000013 | 000014 | There is an exact, parameter free one-particle map. At NS twist a=1/2, sample 000015 | the old twisted Fourier basis on the N=2M sites theta_j=2 pi j/N. The original 000016 | operator u=M_z becomes the diagonal site phase 000017 | 000018 | (u_N f)_j=exp(i theta_j) f_j. (1) 000019 | 000020 | In the sampled Fourier basis it is the cyclic mode shift. The v519 covariance 000021 | Q_N=(I+i C_N)/2 is exactly the projection onto the M modes n=0,...,M-1. 000022 | Thus the same finite CAR space contains both the v519 state and the sampled old 000023 | u. No identification of the Euclidean site translation S_1 with u is used. 000024 | 000025 | The exact finite pair has two crossings. For the negative-sea convention 000026 | C_N^-=I-Q_N, which matches the old C=1_{(-infinity,0)}(B_{1/2}), 000027 | 000028 | u_N C_N^- u_N^* - C_N^- 000029 | = |psi_0> Pi_0 strongly, Tr D_N=0, 000043 | ||D_N-Pi_0||_1=1. (4) 000044 | 000045 | Consequently the finite determinants in (3) cannot be passed to the limit by 000046 | Fredholm determinant continuity. On the original infinite polarized circle the 000047 | ordered relative operator is defined directly and has determinant 1-z. This is 000048 | a different limit topology and a different operator category from ordinary 000049 | finite determinants. The UV factor may not be removed from the finite OS model. 000050 | It is absent only after returning first to the original infinite polarization, 000051 | where there is no cyclic alias and the relative perturbation is trace class. 000052 | 000053 | The existing v519/v524/v526 OS prescription does not select that relative 000054 | Fredholm limit. It retains the complete finite circle, including the wrap mode, 000055 | and defines no normal ordering, spectral cut determinant, or UV subtraction for 000056 | (3). If one forms the natural ordinary finite determinant from its supplied 000057 | covariance and the sampled u_N, the result is the finite cancellation 1. The 000058 | factor 1-z is selected by the separately supplied fullv21 polarized continuum 000059 | pair (B_{1/2},u) and its directed relative Fredholm determinant. Combining the 000060 | two sources gives a canonical local strong-limit map, but not an OS derivation 000061 | of the continuum determinant prescription. 000062 | 000063 | Status: PROVED for the one-particle sampling map, the two finite crossings, the 000064 | determinants and the limit topology. SOURCE NEGATIVE for an OS rule choosing the 000065 | continuum relative determinant. No free kernel, state, generator or cutoff 000066 | counterterm is introduced. 000067 | 000068 | Atlas and reviewed sources 000069 | -------------------------- 000070 | 000071 | Atlas audit frame: revision b97bc83c0fac45ea6bf03cf7, read only. 000072 | 000073 | 1. Original circle and winding pair 000074 | 000075 | paper-source:1fe2ddfabc74ac0e94a4686d 000076 | documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000077 | 001-section-introduction-the-problem-of-parameters.tex:198-251 000078 | sha256 85bb11511be05385a7cadf6071dbede216b12befeb9d6767bc4460748cbec5d7 000079 | 000080 | The source gives the seam circle, D_Gamma=i d/dtheta, the multiplication 000081 | operator U_Gamma(theta)=e^{i theta}, the twisted family and one spectral-flow 000082 | crossing. With the already documented common sign repair B=-D_Gamma, the 000083 | displayed eigenvalues are B_a psi_n=(n+a)psi_n and u psi_n=psi_{n+1}. 000084 | DIRECT-DEFECT-CLOSURE.txt:20-35 records the repaired convention and proves 000085 | u C u^*=C+Pi_0 for 0 L^2(S^1) satisfying simultaneously 000188 | 000189 | J_N u_N = u J_N (11) 000190 | 000191 | with u=M_z. Indeed, the finite unitary u_N has an eigenvector. Equation (11) 000192 | would map it to an L^2 eigenvector of multiplication by z. Multiplication by z 000193 | has no nonzero L^2 eigenvector, because each level set z=lambda has measure 000194 | zero. Thus an exact finite cyclic subsystem of the bilateral winding shift does 000195 | not exist. 000196 | 000197 | There is nevertheless a canonical asymptotic embedding. Map every sampled 000198 | psi_n in (5) to the old continuum psi_n. It intertwines the covariance exactly 000199 | on H_N. The sampled u_N differs from the compressed old u only on the top mode: 000200 |