TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/MOBIUS-QUARTER-OPERATOR-DICTIONARY.txt Original file SHA-256: 0b5f5cb8a4e47bedc3c6febb00666893da1797335644d0fb058072379589bd59 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 | MOBIUS QUARTER OPERATOR DICTIONARY 000002 | =================================== 000003 | 000004 | 25 October 2026. Exact continuation of the A4 spin-lift correction. No 000005 | original source, Atlas entry, ledger, or contract index was changed. 000006 | 000007 | Question 000008 | -------- 000009 | 000010 | Does the A4-selected NS spin structure on the orientable Moebius double 000011 | already produce the two native quarter domains alpha=1/4 and alpha=3/4, 000012 | without choosing a free boundary holonomy? How does the winding-one seam map 000013 | act on those domains, and is the original per-sheet factor d=2 already the 000014 | same operator index statement? 000015 | 000016 | Verdict 000017 | ------- 000018 | 000019 | PROVED: 000020 | 000021 | The full Pin-minus Moebius monodromy is cJ, where J combines the 000022 | transverse reflection with the pinor fibre action. Its two J-eigenspaces 000023 | have longitudinal shifts 1/4 and 3/4. After passage to the orientable 000024 | double and reduction to the longitudinal core, the NS cover operator 000025 | decomposes as 000026 | 000027 | B_cov = 2 (B_1/4 direct-sum B_3/4). (1) 000028 | 000029 | The two quarters are the +i and -i characters of the A4 order-four lifted 000030 | monodromy. They are not freely chosen holonomies. Changing the sign of the 000031 | Pin lift exchanges their labels but leaves the unordered pair fixed. 000032 | 000033 | PROVED: 000034 | 000035 | In a common periodic coefficient basis the winding-one cover multiplier is 000036 | 000037 | [ 0 u_base ] 000038 | u_cov = [ ], (2) 000039 | [ I 0 ] 000040 | 000041 | so it exchanges the two quarter domains, while 000042 | 000043 | u_cov^2 = diag(u_base,u_base). (3) 000044 | 000045 | Consequently 000046 | 000047 | SF_cov(u_cov)=1, 000048 | SF_cov(u_cov^2)=2, 000049 | SF_1/4(u_base)=SF_3/4(u_base)=1. (4) 000050 | 000051 | Here u_cov is the multiplier on the seam of the orientable double. It 000052 | exchanges the two reduced longitudinal quarter subspaces. On the full 000053 | Moebius pinor space, scalar multiplication by u_cov changes c to -c at 000054 | fixed J, so it is not by itself an endomorphism of one fixed descended Pin 000055 | lift. This is the correct location for the original A3 operator, which is 000056 | defined on the cut orientable double. 000057 | 000058 | PROVED SOURCE CONSEQUENCE: 000059 | 000060 | A4 selects the full NS cover, equivalently the complete pair of quarter 000061 | character sectors. It does not select one quarter sector as the primitive 000062 | CAR channel. The primitive cover pair is (B_1/2,u_cov) with spectral flow 000063 | one. Restricting to one quarter and using u_base instead describes a block 000064 | of u_cov squared. Any identification of one quarter with the independent 000065 | P2 quarter clock therefore still requires an explicit map. 000066 | 000067 | SCOPED GAP: 000068 | 000069 | The original formula d SF=2 has exactly the same index value as either 000070 | 000071 | SF_cov(u_cov^2)=2 000072 | 000073 | or the sum of the two quarter indices. The original source does not state 000074 | the operator map which makes one of these equalities its intended 000075 | per-sheet construction. The quarter sectors are deck-character sectors, 000076 | not literal position sheets. A deck-invariant sheet trace may be changed 000077 | to this character basis, but u_cov itself is off diagonal there. Thus the 000078 | value 2 matches exactly while the original operator attribution remains 000079 | unstated. 000080 | 000081 | 1. Geometry and the two coordinates 000082 | ----------------------------------- 000083 | 000084 | Normalize the lifted core of the orientable double by 000085 | 000086 | theta in [0,2 pi], 000087 | f(theta+2 pi)=-f(theta). (5) 000088 | 000089 | The minus sign is the A4 result r^2=-I derived in 000090 | ORIGINAL-SPIN-AND-SEAM-DICTIONARY.txt. On this reduced longitudinal cover the 000091 | half-deck translation is 000092 | 000093 | (T_pi f)(theta)=f(theta+pi), 000094 | T_pi=exp(i pi B_cov), 000095 | B_cov=-i partial_theta. (6) 000096 | 000097 | The coordinate on the base core is 000098 | 000099 | x=2 theta mod 2 pi. (7) 000100 | 000101 | For a fixed x, the two values theta=x/2 and theta=x/2+pi are the two cover 000102 | sheets. Diagonalizing the full A4 lifted monodromy gives the quarter 000103 | characters below. This character transform must be distinguished from the 000104 | two position sheets themselves. In the full Moebius pinor space the 000105 | half-deck operation also contains the transverse reflection and its pinor 000106 | fibre action; T_pi in (6) is the resulting longitudinal character operator, 000107 | not a replacement for that full action. 000108 | 000109 | The original geometry and A4 wording are recorded at: 000110 | 000111 | paper-source:1fe2ddfabc74ac0e94a4686d 000112 | documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000113 | 001-section-introduction-the-problem-of-parameters.tex:165-221,225-277 000114 | SHA256 85bb11511be05385a7cadf6071dbede216b12befeb9d6767bc4460748cbec5d7 000115 | 000116 | documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000117 | 007-subsection-m-obius-matching-and-the-minimal-gluing-class.tex:1-23,75-108 000118 | SHA256 278c385040ba000fc6577b3ba4630b41b49d4486b069c2034e31c4c9bfb3ae2b 000119 | 000120 | 2. Full Pin-minus monodromy and the longitudinal reduction 000121 | ------------------------------------------------------------ 000122 | 000123 | For the flat Moebius band, write the one-step deck transformation as 000124 | 000125 | (s,w) -> (s+1,-w). 000126 | 000127 | On transverse pinor data let 000128 | 000129 | P phi(w)=phi(-w), 000130 | J=sigma_1 P, 000131 | K_j=ker(J-jI), j in {+1,-1}. (F1) 000132 | 000133 | For Pin-minus the two lifts have c=+i or c=-i, and an equivariant section 000134 | satisfies 000135 | 000136 | Psi(s+1,.)=c J Psi(s,.), 000137 | Psi(s+2,.)=-Psi(s,.). (F2) 000138 | 000139 | Thus the orientable double is NS. On K_j the one-step scalar phase is cj. 000140 | For c=+i this gives shift 1/4 on K_+ and shift 3/4 on K_-; for c=-i the two 000141 | assignments are exchanged. Hilbert-valued Fourier expansion and Parseval 000142 | give the orthogonal and complete decomposition. The same decomposition 000143 | holds on the geometric Sobolev domains because weak longitudinal and 000144 | transverse derivatives respect the two-sector coefficient characterization. 000145 | 000146 | This is precisely the full version of the scalar coefficient model below. 000147 | Equations (10)--(16) suppress K_+ and K_- after their orthogonal projections; 000148 | they do not infer the phases from the fibre matrix sigma_1 alone. A fibre 000149 | eigenvalue determines a phase only after the transverse reflection parity is 000150 | included in J. 000151 | 000152 | Primary checks: 000153 | 000154 | Anik Chakraborty, "Pin Equivariance and Local Dirac Boundary Problems on 000155 | the Flat Moebius Band", arXiv:2610.03745v1, Theorems 2.1, 3.2, 4.1 and 000156 | Section 5: 000157 | https://arxiv.org/html/2610.03745v1 000158 | 000159 | Justin Kaidi, Julio Parra-Martinez, Yuji Tachikawa, with Arun Debray, 000160 | "Topological superconductors on superstring worldsheets", SciPost Phys. 9, 000161 | 010 (2020), Section 2.2, p. 10: 000162 | https://doi.org/10.21468/SciPostPhys.9.1.010 000163 | 000164 | The older source independently states the local convention needed here: 000165 | crossing the orientation-reversing line twice gives gamma_i squared, so a 000166 | Pin-minus Moebius boundary carries the NS sector. The newer source supplies 000167 | the exact full monodromy cJ, the quarter-shift Fourier theorem, Parseval, and 000168 | the Sobolev-domain statement. Its later local boundary condition b is not 000169 | used anywhere in this dictionary. 000170 | 000171 | 3. Hilbert spaces, domains, and unitary longitudinal decomposition 000172 | ------------------------------------------------------------------- 000173 | 000174 | Use normalized measures dtheta/(2 pi) and dx/(2 pi). Let 000175 | 000176 | H_cov = {longitudinal L2 sections satisfying (5)}, 000177 | Dom(B_cov) = H1 sections satisfying (5). (8) 000178 | 000179 | Let K=L2(S1_x) with periodic Sobolev domain H1_per, and define 000180 | 000181 | D=-i partial_x, 000182 | B_a=D+a, a in {1/4,3/4}. (9) 000183 | 000184 | Define maps W_0,W_1:K -> H_cov by 000185 | 000186 | (W_0 g)(theta)=exp(i theta/2) g(2 theta), 000187 | (W_1 g)(theta)=exp(3 i theta/2)g(2 theta). (10) 000188 | 000189 | They obey the NS boundary condition because both phases change sign after 000190 | theta -> theta+2 pi. Their Fourier images are respectively 000191 | 000192 | W_0 exp(i m x) = exp(i(2m+1/2)theta), 000193 | W_1 exp(i m x) = exp(i(2m+3/2)theta). (11) 000194 | 000195 | These are the even-n and odd-n parts of the complete NS basis 000196 | exp(i(n+1/2)theta). They are orthogonal and their union is complete. 000197 | 000198 | The norm is preserved without an extra square root: 000199 | 000200 | integral_0^2pi |g(2theta)|^2 dtheta/(2pi)