TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/PIN-WINDING-DESCENT.txt Original file SHA-256: 55205d7226e878958e7c2ee7c925f291a4d2248bde83407bf4324a4e16fa4c26 Inclusive original lines: 1-107 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 | WINDING TIMES CHARGE CONJUGATION ON A FIXED PIN DOMAIN 000002 | ===================================================== 000003 | 26 October 2026 000004 | Claim MATH.ORIGINAL-SEAM.PIN-WINDING-DESCENT.01 000005 | 000006 | The next actual candidate after the recovered quarter comparison is 000007 | 000008 | A=exp(i pi x) sigma1 K = u_cov C. 000009 | 000010 | Use precisely the flat Moebius family in Chakraborty, arXiv:2610.03745v1: 000011 | D=-i(sigma1 partial_x+sigma2 partial_w), 000012 | Psi(x+1,-w)=c sigma1 Psi(x,w), c=plus or minus i, 000013 | sigma1 Psi(x,plus/minus1)=b Psi(x,plus/minus1), b=plus/minus1. 000014 | This is a supplied comparison family, not a derivation of all P1 geometry. 000015 | 000016 | 1. Positive fixed-domain result 000017 | -------------------------------- 000018 | The already given C=sigma1 K sends(c,b) to(conj(c),b)=(-c,b). 000019 | The cover winding exp(i pi x) changes that lift sign once more. Their 000020 | composition therefore preserves the SINGLE fixed(c,b) Sobolev domain. 000021 | A is antiunitary, A²=I, and it preserves support in the cover chart. 000022 | Thus the two old operations compose to a fixed-domain descent map. 000023 | 000024 | 2. The first operator identity determines its actual scope 000025 | ---------------------------------------------------------- 000026 | Differentiate the actual phase instead of treating it as a constant: 000027 | 000028 | D u_cov=u_cov D+pi sigma1 u_cov, 000029 | D C=-C D, 000030 | D A=-A D+pi sigma1 A, 000031 | A D A=-D+pi sigma1. (1) 000032 | 000033 | The nonzero matrix term is essential. A is neither a symmetry nor an exact 000034 | particle-hole symmetry of the original D. For example, 000035 | A exp(-t|D|) A=exp(-t|-D+pi sigma1|), 000036 | which is not the same positive semigroup. The transverse derivative makes 000037 | sigma1 fail to commute with D, so a scalar phase cannot remove the term. 000038 | 000039 | The centered operator D_cen=D-(pi/2)sigma1 does anticommute with A. Its 000040 | centering is an additional flat connection. Multiplication by 000041 | exp(-i pi x/2) relates it to a different multiplier c'=-ic=plus/minus1. 000042 | That changes the displayed Pin/flat data and is not an equality for the 000043 | given fixed Pin-minus realization. 000044 | 000045 | 3. Full two-dimensional sea versus reduced circle 000046 | ------------------------------------------------- 000047 | For the GIVEN full D negative sea, a particle-hole Fock lift needs the 000048 | polarization mismatch to be Hilbert-Schmidt. The exact spectrum in the 000049 | primary paper supplies, for each transverse n>=1, blocks 000050 | 000051 | H=bq sigma1+mu sigma3, mu=n pi/2, 000052 | q=2pi(k+theta_n), theta_n=1/4 or3/4. 000053 | 000054 | By (1), the mismatch has the same Hilbert-Schmidt norm as the difference 000055 | of the positive projectors of H and H-pi sigma1. Put x=bq. Exactly, 000056 | 000057 | ||DeltaP||_HS² 000058 | =1-[x(x-pi)+mu²]/[sqrt(x²+mu²)sqrt((x-pi)²+mu²)]. (2) 000059 | 000060 | For |x|<=mu and mu>=2pi, the determinant identity 000061 | 000062 | (x²+mu²)((x-pi)²+mu²)-[x(x-pi)+mu²]²=pi²mu² 000063 | 000064 | rationalizes (2). Since E0²<=2mu² and E1²<=13mu²/4, 000065 | 000066 | ||DeltaP||_HS² >= pi²/(13mu²)=4/(13n²). 000067 | 000068 | For every n>=4, the longitudinal grid of spacing2pi contains at least 000069 | n/4 frequencies in |q|<=mu. Each transverse level therefore contributes 000070 | at least1/(13n). The full Hilbert-Schmidt sum diverges. This is an analytic 000071 | lower bound, not a numerical extrapolation of a finite truncation. 000072 | 000073 | Consequently A has no particle-hole Fock implementation for this canonical 000074 | FULL TWO-DIMENSIONAL sea. The transverse zero branch is different: its 000075 | quarter circle mismatch has only the known finite rank-one defect. 000076 | The old reduced-circle Fock result remains valid. A general original P1 000077 | kernel is not excluded by this calculation on the stated comparison class. 000078 | 000079 | 4. Calderon and geometric reflection require their own maps 000080 | ----------------------------------------------------------- 000081 | For a local bulk D solution, (1) gives D(A Psi)=pi sigma1 A Psi. Thus A is 000082 | not a bulk D solution intertwiner. This fact alone does NOT decide whether 000083 | the induced boundary map preserves the Calderon range: another D solution 000084 | could extend the same transformed boundary trace. The separate identity 000085 | A_boundary Ran(P_C)=Ran(P_C) has not been proved or disproved here. 000086 | 000087 | A also moves no base point; it is pointwise conjugation and multiplication. 000088 | Its involution/domain property therefore does not identify it with the 000089 | original OS collar reflection Theta. State, time, actual boundary Cauchy 000090 | data and common collar/reference transport remain to be matched. 000091 | 000092 | 5. Checks and primary sources 000093 | ------------------------------- 000094 | check_pin_winding_descent.py verifies the Pauli/plane-wave identity, fixed 000095 | Pin multiplier, boundary eigenlines, exact projector determinant identity 000096 | and250 quarter-grid controls. The infinite conclusion is proved by the 000097 | lower bound above. Normal and python -OO outputs are byte-identical. 000098 | DESCENT-CANDIDATE-REVIEW.txt independently reviews the same candidate; its 000099 | Calderon statement has the precise boundary-range limitation of section4. 000100 | 000101 | Primary source actually read: 000102 | Anik Chakraborty, Pin Equivariance and Local Dirac Boundary Problems on the 000103 | Flat Moebius Band, arXiv:2610.03745v1, sections3,6,8.2-8.4: 000104 | https://arxiv.org/html/2610.03745v1 000105 | Old exact source dictionary: 000106 | canonical-seam-fermionization-20261025/MOBIUS-QUARTER-OPERATOR-DICTIONARY.txt. 000107 | No physical T gate or TFPT.TOE.COMPLETE.01 is closed by this assessment.