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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/GRAPH-CUT-REFLECTION-TEST.txt
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000001 | GRAPH CUT REFLECTION TEST FOR THE ACTUAL EQUATOR
000002 | ================================================
000003 | 
000004 | 26 October 2026. Independent analytic review. No Atlas, original source, or
000005 | canonical contract was changed.
000006 | 
000007 | Question
000008 | ========
000009 | 
000010 | Let
000011 | 
000012 |   E=C/(Z+iZ),
000013 |   Omega=p^(-1)({|x|<1}),
000014 | 
000015 | for the fixed elliptic double E:y^2=x^4-1 with its Q invariant NS spin line
000016 | alpha. The boundary of Omega is the full equatorial graph
000017 | 
000018 |   p^(-1)({|x|=1})=Fix(S) union Fix(hS),
000019 | 
000020 | where, in z=t+i s coordinates,
000021 | 
000022 |   S z=bar z+i/2,
000023 |   hS z=-bar z+1/2.
000024 | 
000025 | The open set Omega consists of two opposite checkerboard faces. Does either
000026 | one of the fixed global reflections S or hS make the already constructed
000027 | theta_3 free Gaussian functional reflection positive on the CAR algebra of
000028 | their union?
000029 | 
000030 | Verdict
000031 | =======
000032 | 
000033 | No. Each reflection is positive on one of its two annular Schottky halves,
000034 | exactly as proved in ELLIPTIC-GREEN-RP.txt. The checkerboard union Omega is
000035 | not contained in either half. It contains one face in each of the two
000036 | opposite annuli.
000037 | 
000038 | This is not only a mismatch of pictures. In the full gauge invariant CAR
000039 | algebra of the disconnected union there is an explicit neutral bilocal Wick
000040 | word whose reflected norm is strictly negative. The witness uses the actual
000041 | theta_3 Szego Green kernel and the already fixed charged Clifford minus. It
000042 | has compact support in the interiors of the two checkerboard faces, so no
000043 | choice of graph vertex condition, boundary trace measure, or branch point
000044 | domain enters the calculation.
000045 | 
000046 | The conclusion applies to the named free Gaussian functional and either
000047 | single global reflection S or hS. It does not exclude a separately supplied
000048 | piecewise reflection with compatible vertex data, nor a restriction to the
000049 | smaller algebra generated by operators that are even on each face
000050 | separately. Neither extra structure is supplied by the current equatorial
000051 | pullback.
000052 | 
000053 | 1. The checkerboard is not a Schottky half
000054 | ==========================================
000055 | 
000056 | Use rho=hS and recenter the first coordinate by
000057 | 
000058 |   tau=t-1/4.
000059 | 
000060 | Then
000061 | 
000062 |   rho(tau,s)=(-tau,s),
000063 | 
000064 | and its fixed circles are tau=0 and tau=1/2 modulo one. Its two Schottky
000065 | halves are the annuli
000066 | 
000067 |   A_+={0<tau<1/2},
000068 |   A_-={-1/2<tau<0}.
000069 | 
000070 | The four edges of Fix(S) cut each annulus into two rectangular disks. Since
000071 | rho covers x -> 1/bar x, it sends Omega to p^(-1)({|x|>1}). The open disk
000072 | {|x|<1} contains no branch value, so its inverse image has two components
000073 | exchanged by the deck map h. In the centered coordinates h sends tau to
000074 | -tau and hence exchanges A_+ with A_-. Therefore the two components of
000075 | Omega are opposite checkerboard disks, one contained in A_+ and one
000076 | contained in A_-. In particular Omega is not a subset of A_+ or A_-.
000077 | 
000078 | The same statement for S follows by the quarter rotation Q, which exchanges
000079 | the vertical and horizontal presentations. This is the exact geometric
000080 | difference from the previous annulus theorem: that theorem chose one whole
000081 | component of E\Fix(rho), whereas the actual graph side takes one disk from
000082 | each component.
000083 | 
000084 | 2. The actual theta_3 kernel and its two signs
000085 | ===============================================
000086 | 
000087 | On the translated rho chart the NS Green operator is the same translation
000088 | invariant kernel as in ELLIPTIC-GREEN-RP.txt. With
000089 | 
000090 |   B=-i partial_s,
000091 |   spec(B)=2 pi (Z+1/2),
000092 | 
000093 | write S_E(a) for the Green kernel of partial_tau+B. It is exactly
000094 | 
000095 |   S_E(w)=i/(2 pi)
000096 |     theta_3(w|i) theta_1'(0|i)
000097 |     /[theta_3(0|i) theta_1(w|i)],
000098 | 
000099 | with the already proved Fourier branches
000100 | 
000101 |   S_E(a)=exp(-aB)(1+exp(-B))^(-1),       0<a<1,
000102 |   S_E(-a)=-exp((a-1)B)(1+exp(-B))^(-1), 0<a<1.
000103 | 
000104 | For 0<tau,u<1/2, the fixed charged reflection convention gives
000105 | 
000106 |   K_bar(tau,u)=-S_E(-tau-u)=X_tau^* X_u,
000107 |   K_psi(tau,u)= S_E( tau+u)=Y_tau^* Y_u,
000108 | 
000109 | where
000110 | 
000111 |   X_tau=exp(tau B)(1+exp(B))^(-1/2),
000112 |   Y_tau=exp(-tau B)(1+exp(-B))^(-1/2).
000113 | 
000114 | This is the previous positive annulus factorization. On the opposite half,
000115 | put tau=-r and u=-v with 0<r,v<1/2. The same global reflection and the same
000116 | Clifford sign give
000117 | 
000118 |   K_psi(-r,-v)=S_E(-r-v)=-X_r^* X_v.                  (1)
000119 | 
000120 | Thus the reflected one particle form of this charge block is strictly
000121 | negative on suitable nonzero test functions in A_-. A constant Real spin
000122 | linearization phase cannot alter the relative sign between the two halves.
000123 | For a neutral word its two charge phases cancel in any case.
000124 | 
000125 | This use of S_E is typed narrowly. GLOBAL-SPIN-KERNEL.txt proves the full
000126 | holomorphic alpha Szego kernel and its rank two pushforward, but does not by
000127 | itself define a positive equal time CAR covariance. The present test uses
000128 | the declared free Gaussian Green functional of ELLIPTIC-GREEN-RP.txt, where
000129 | the theta kernel, CAR covariance, word reversal, and Clifford sign were
000130 | already fixed together.
000131 | 
000132 | 3. A compactly supported neutral negative witness
000133 | =================================================
000134 | 
000135 | Choose real smooth product bumps f_+ and g_- with
000136 | 
000137 |   supp(f_+) compactly contained in Omega intersect A_+,
000138 |   supp(g_-) compactly contained in Omega intersect A_-.
000139 | 
000140 | They may be chosen so that the factorized vectors below are nonzero. For
000141 | example, take nonnegative time bumps and any nonzero spatial bump supported
000142 | inside the corresponding checker face. At least one NS Fourier coefficient
000143 | of the spatial bump is nonzero, and its time Laplace integral is then
000144 | nonzero. Equations above give
000145 | 
000146 |   q_bar(f_+)=|| integral X_tau f_+(tau) d tau ||^2 >0,
000147 | 
000148 |   q_psi(g_-)=-|| integral X_r g_-(-r) d r ||^2 <0.    (2)
000149 | 
000150 | Now take the normal ordered bilocal word
000151 | 
000152 |   F=:barpsi(f_+) psi(g_-): .                           (3)
000153 | 
000154 | It has total U(1) charge zero and is even. Its support is contained in the
000155 | two component open set Omega. Gaussian Wick expansion with the same order
000156 | reversing reflection used in the annulus theorem gives the one by one
000157 | exterior Gram factors
000158 | 
000159 |   omega(Theta_rho(F) F)=q_bar(f_+) q_psi(g_-)<0.       (4)
000160 | 
000161 | There is no unlisted sign in (4): with both supports in A_+, the identical
000162 | word convention is precisely the product of the two positive charged Gram
000163 | forms used in the proof of all word positivity. Moving only the psi test
000164 | function to A_- changes its factor by the exact NS sign (1).
000165 | 
000166 | Equation (4) is the smallest decisive test in the full gauge invariant CAR
000167 | algebra of the disconnected checkerboard union. It is stronger than a
000168 | charged one field counterexample and does not appeal to a freely chosen
000169 | projector or covariance.
000170 | 
000171 | 4. The second original reflection and the exact scope
000172 | =====================================================
000173 | 
000174 | The quarter map Q exchanges S and hS and preserves the unordered alpha
000175 | kernel. Rotating f_+,g_- by Q therefore gives the same witness for S. Hence
000176 | neither fixed original reflection supplies reflection positivity on all of
000177 | Omega, although each supplies it on its own annular Schottky half.
000178 | 
000179 | The obstruction is already interior. The induced measure on the graph,
000180 | closed trace spaces, and self adjoint branch vertex conditions cannot repair