TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/older-paper-source-recovery-20261025/ORIGINAL-CIRCLE-REVIEW.txt
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000001 | INDEPENDENT REVIEW: THE RECOVERED ORIGINAL CIRCLE OPERATOR
000002 | ==========================================================
000003 | 
000004 | 25 October 2026. Read-only source review and exact analytic check. No Atlas,
000005 | index, paper or original source was changed.
000006 | 
000007 | Verdict
000008 | -------
000009 | 
000010 | The older fullv21 paper contains a concrete one-particle circle model, not
000011 | only an abstract winding number. After one necessary orientation correction,
000012 | it supplies an exact same-space pair
000013 | 
000014 |   B_alpha=-i partial_theta on D_alpha,
000015 |   D_alpha={f in H1([0,2pi]):f(2pi)=exp(2pi i alpha)f(0)},
000016 |   u=M_z,                         0<alpha<1,              (1)
000017 | 
000018 | with the strong Ward relation, Toeplitz index -1 and canonical line
000019 | 
000020 |   Pi_alpha=|psi_0><psi_0|,
000021 |   psi_n(theta)=exp(i(n+alpha)theta).                     (2)
000022 | 
000023 | This closes the existence and internal compatibility of the one-particle
000024 | circle/shift construction inside the older declared gluing model. It does not
000025 | show that the later operational seed's B_Sigma and u_Sigma are this pair. The
000026 | paper also does not select a fixed alpha, identify B_alpha with physical OS
000027 | time, or identify (2) with the later electromagnetic transfer projector.
000028 | 
000029 | The quarter value alpha=1/4 follows only after the actual monodromy
000030 | exp(2 pi i B)=i I is supplied. Winding one and spectral flow one do not choose
000031 | that value.
000032 | 
000033 | 1. The necessary sign correction
000034 | ---------------------------------
000035 | 
000036 | The older source prints
000037 | 
000038 |   D_Gamma=i partial_theta,
000039 |   U_Gamma(theta)=exp(i theta),
000040 |   lambda_n(phi)=n+phi/(2pi).                             (3)
000041 | 
000042 | The three formulas do not have a common sign. On the twisted domain with
000043 | alpha=phi/(2pi),
000044 | 
000045 |   i partial_theta psi_n=-(n+alpha)psi_n.                 (4)
000046 | 
000047 | Thus the displayed positive spectrum in (3) belongs to
000048 | 
000049 |   B_alpha=-D_Gamma=-i partial_theta,                     (5)
000050 | 
000051 | or equivalently to the opposite orientation of the printed D_Gamma. This is
000052 | not cosmetic: retaining i partial_theta reverses the Ward and spectral-flow
000053 | signs.
000054 | 
000055 | The domain in (1) is self-adjoint. Multiplication by exp(-i alpha theta)
000056 | maps it unitarily to periodic H1, under which B_alpha becomes
000057 | -i partial_theta+alpha. Hence
000058 | 
000059 |   B_alpha psi_n=(n+alpha)psi_n,
000060 |   sigma(B_alpha)={n+alpha:n in Z}.                       (6)
000061 |