TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/REFLECTION-INDEPENDENT-REVIEW.txt Original file SHA-256: 2731432207346c866b7ab7eeea925f2adef902152031c62b9a7f2d0d4bb11e18 Inclusive original lines: 1-180 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 | INDEPENDENT REVIEW OF THE CHECKERBOARD REFLECTION WITNESS 000002 | ========================================================= 000003 | 000004 | 26 October 2026. Exact CAR sign, gauge scope and intrinsic Ward test. 000005 | No Atlas, original source or canonical contract was changed. 000006 | 000007 | Verdict 000008 | ======= 000009 | 000010 | The negative neutral witness in GRAPH-CUT-REFLECTION-TEST.txt is valid for 000011 | the declared free theta_3 Gaussian functional and either one fixed global 000012 | reflection S or hS. The decisive point is relative and therefore insensitive 000013 | to an overall Wick or Spin convention: the already proved same-half all-word 000014 | formula fixes the sign of the normal-ordered bilocal word. Moving only its 000015 | psi test function to the opposite Schottky half changes exactly one reflected 000016 | one-particle block from positive to negative. The charged Clifford minus, 000017 | word reversal and normal-ordering prescription remain unchanged. 000018 | 000019 | The witness is neutral under the one global U(1) present in the source. It is 000020 | not invariant under a separately postulated U(1) on each disconnected face, 000021 | and it is odd on each face although globally even. Thus an independent-face 000022 | gauge rule or facewise-even observable algebra removes the witness, but 000023 | neither rule is part of the tested source. The report scopes this exception 000024 | correctly. 000025 | 000026 | The exact intrinsic boundary metric supplies an additional narrow negative 000027 | test. On one checkerboard face the four edge lengths add to L=2, and the 000028 | pushforward of its uniform arclength generator is 000029 | 000030 | B_push=-i[a(theta) partial_theta+a'(theta)/2], 000031 | a(theta)=4 K(-1) sqrt(|sin(2 theta)|). 000032 | 000033 | Consequently 000034 | 000035 | [B_push,exp(i theta)]=a(theta) exp(i theta), 000036 | 000037 | which is not a constant spectral shift. This excludes the literal base 000038 | winding exp(i theta) as the uniform intrinsic winding operator. It does not 000039 | exclude an intrinsic uniform multiplier transported through an actual source 000040 | unitary. 000041 | 000042 | 1. Geometry and localization 000043 | ============================ 000044 | 000045 | For rho=hS, recentering t by tau=t-1/4 makes rho(tau,s)=(-tau,s). Its two 000046 | Schottky halves are A_+={0 exp(i lambda) psi, 000118 | barpsi -> exp(-i lambda) barpsi, 000119 | 000120 | the word (2) has charge zero. It therefore belongs to the global even, 000121 | gauge-invariant CAR algebra on Omega. 000122 | 000123 | If one instead declared a product gauge group for the two components, with 000124 | parameters lambda_+ and lambda_-, then (2) would acquire 000125 | 000126 | exp[-i lambda_+ + i lambda_-] 000127 | 000128 | and would be removed. The same happens if observables are required to be 000129 | even on each face separately: (2) contains one odd field on each face. These 000130 | are genuine ways to avoid the counterexample, but they change the observable 000131 | algebra. Disconnected support alone does not create either rule, and the 000132 | sources used by the tested Gaussian functional specify only the global 000133 | charged field and its global reflection. 000134 | 000135 | 4. Intrinsic length and the narrow Ward obstruction 000136 | ==================================================== 000137 | 000138 | From the actual square-torus uniformization 000139 | 000140 | dx/dz=-i c y, |c|=2 sqrt(2) K(-1), 000141 | 000142 | and x=exp(i theta), y^2=exp(4i theta)-1, the flat arclength on every open 000143 | equatorial edge is 000144 | 000145 | d ell/d theta 000146 | =1/(|c| |y|) 000147 | =1/[4 K(-1) sqrt(|sin(2 theta)|)]. (4) 000148 | 000149 | The integral of (4) between two adjacent marks is 1/2. The boundary of one 000150 | inside checker face consists of four such edges, so its intrinsic perimeter 000151 | is exactly L=2. 000152 | 000153 | Let B_ell=-i partial_ell on the uniform intrinsic half-density space. On the 000154 | core of smooth functions compactly supported away from the four vertices, 000155 | the unitary change from d ell to d theta multiplies coefficients by 000156 | sqrt(d ell/d theta). Setting 000157 | 000158 | a(theta)=d theta/d ell 000159 | =4 K(-1) sqrt(|sin(2 theta)|), 000160 | 000161 | gives 000162 | 000163 | B_push=-i[a(theta) partial_theta+a'(theta)/2]. (5) 000164 | 000165 | The half-density connection term commutes with multiplication, so 000166 | 000167 | [B_push,M_exp(i theta)] 000168 | =M_{a(theta) exp(i theta)}. (6) 000169 | 000170 | Because a vanishes at the marks and varies along every edge, (6) is not a 000171 | constant ladder relation. It excludes only the word-for-word identification 000172 | of the original base multiplier exp(i theta) with a uniform intrinsic 000173 | winding for this face boundary. A multiplier exp(2 pi i ell/L), after an 000174 | actual source identification of the cyclic intrinsic coordinate and its 000175 | Spin domain, would obey a constant relation. Equation (6) neither supplies 000176 | nor excludes that transported operator. 000177 | 000178 | 5. Status of the reviewed claim 000179 | ================================ 000180 |