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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ELLIPTIC-GREEN-RP.txt
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000001 | ELLIPTIC NS GREEN REFLECTION POSITIVITY AND THE ANNULUS GAUSSIAN STATE
000002 | 26 October 2026
000003 | 
000004 | Question and verdict
000005 | ====================
000006 | 
000007 | Take the actual square elliptic curve E=C/(Z+iZ), its Q-invariant NS--NS spin
000008 | line alpha and the dividing real form
000009 | 
000010 |   rho(z)=-bar(z).
000011 | 
000012 | On the half 0<t<1/2, z=t+i s, use
000013 | 
000014 |   L=partial_t+B,  B=-i partial_s,
000015 | 
000016 | with antiperiodic conditions in t and s. The full free charged Gaussian Green
000017 | functional of L is reflection positive for both charge blocks and hence for
000018 | all polynomial CAR words supported in the open half. Its time-zero covariance
000019 | is exactly the second-face compression
000020 | 
000021 |   C=(1+exp(B))^-1
000022 | 
000023 | of the Calderon graph already derived on this same elliptic annulus. Thus the
000024 | Green KMS state is the labeled boundary marginal of that graph, while the
000025 | full two-face graph is its canonical purification. They are one free Gaussian
000026 | source construction rather than two one-particle formulas with matching
000027 | spectra.
000028 | 
000029 | This is a theorem for the explicitly named free Gaussian CAR functional. It
000030 | does not turn a two-point identity into an interacting field theorem and does
000031 | not identify this state with the primitive physical P1 kernel.
000032 | 
000033 | 1. The actual elliptic Green kernel
000034 | ===================================
000035 | 
000036 | Let
000037 | 
000038 |   e_n(s)=exp(2pi i(n+1/2)s),
000039 |   B e_n=lambda_n e_n,
000040 |   lambda_n=2pi(n+1/2).
000041 | 
000042 | There is no zero mode. For 0<t<1 the inverse of L is
000043 | 
000044 |   S(t)=exp(-tB)(1+exp(-B))^-1,                           (1)
000045 | 
000046 | and for -1<t<0 it is
000047 | 
000048 |   S(t)=-exp(-(t+1)B)(1+exp(-B))^-1.                     (2)
000049 | 
000050 | The two branches have jump I at zero and obey S(t+1)=-S(t). They therefore
000051 | give the inverse on the actual temporal NS domain. Although B has both signs,
000052 | each displayed product is bounded on its stated open interval.
000053 | 
000054 | This spectral Green operator is exactly the elliptic Szego kernel, including
000055 | normalization. Put w=(s-s')+i t and use Jacobi theta functions with periods
000056 | 1 and i. For 0<t<1,
000057 | 
000058 |   S_E(w)=i/(2pi)
000059 |          theta_3(w|i) theta_1'(0|i)
000060 |          /[theta_3(0|i) theta_1(w|i)].                  (3)
000061 | 
000062 | Indeed, (3) has residue i/(2pi w), the fundamental singularity for
000063 | 
000064 |   L=-2i partial_bar_w,
000065 | 
000066 | and is antiperiodic under w->w+1 and w->w+i. The difference between (3) and
000067 | the Fourier kernel of (1) is a holomorphic section of alpha. Since
000068 | H0(E,alpha)=0, the difference vanishes. This proves the theta identity
000069 | without fitting a coefficient or a thermal parameter.
000070 | 
000071 | 2. Reflection positivity for both charges
000072 | ==========================================
000073 | 
000074 | The Schottky reflection is
000075 | 
000076 |   rho(t,s)=(-t,s).
000077 | 
000078 | For 0<t,u<1/2, set a=t+u. The first reflected charged kernel is
000079 | 
000080 |   K_bar(t,u)=-S(-a)
000081 |             =exp(aB)(1+exp(B))^-1
000082 |             =X_t^* X_u,                                (4)
000083 | 
000084 |   X_t=exp(tB)(1+exp(B))^-1/2.
000085 | 
000086 | The conjugate charge block is
000087 | 
000088 |   K_psi(t,u)=S(a)
000089 |             =exp(-aB)(1+exp(-B))^-1
000090 |             =Y_t^* Y_u,                                (5)
000091 | 
000092 |   Y_t=exp(-tB)(1+exp(-B))^-1/2.
000093 | 
000094 | Both X_t and Y_t are bounded for 0<t<1/2. Spectral calculus therefore gives
000095 | 
000096 |   integral <f(t),K_bar(t,u)f(u)> dt du
000097 |      =||integral X_u f(u)du||^2 >=0,
000098 | 
000099 | and the analogous identity with Y for the second charge. The statement holds
000100 | for every spatial mode and arbitrary superpositions, not only for a finite
000101 | mode sample.
000102 | 
000103 | In the charged field convention used in the previous full-cover proof, define
000104 | the anti-linear order-reversing reflection first on unsmeared generators by
000105 | 
000106 |   Theta psi(t,s)    =-barpsi(-t,s),
000107 |   Theta barpsi(t,s)=-psi(-t,s).                         (6)
000108 | 
000109 | The constant minus is the reduced normal-Clifford and source-order phase. It
000110 | is essential for the first sign in (4), and Theta^2=1. Moving this minus into
000111 | the opposite source-order convention is equivalent; omitting it while
000112 | keeping the displayed contractions is not.
000113 | 
000114 | For a gauge-invariant free Gaussian functional, Wick expansion of reflected
000115 | normal-ordered words gives exterior Gram determinants of (4) and (5).
000116 | Consequently every finite polynomial F in the positive-half CAR algebra
000117 | satisfies
000118 | 
000119 |   omega(Theta(F)F)>=0.                                  (7)
000120 | 
000121 | Equation (7) is the requested all-word result for this free Gaussian source.
000122 | It makes no claim for an interacting deformation with the same two-point
000123 | function.
000124 | 
000125 | 3. The same state as the annulus Cauchy graph
000126 | ==============================================
000127 | 
000128 | For the strip 0<=t<=W with W=1/2, the Cauchy relation for L is
000129 | 
000130 |   f_1=exp(-WB)f_0.
000131 | 
000132 | Its graph projector has second-face compression
000133 | 
000134 |   C_A=(1+exp(2WB))^-1=(1+exp(B))^-1.                    (8)
000135 | 
000136 | This is precisely the covariance in (4). The first-face compression is
000137 | I-C_A, precisely the covariance in (5). The off-diagonal graph block gives
000138 | the canonical purification of these two charge and face marginals.
000139 | 
000140 | Thus, after applying the ordinary CAR quasifree functor, the torus Green
000141 | functional has exactly the labeled boundary marginal of the annulus Cauchy
000142 | graph and the same reflected word functional on that marginal algebra. The
000143 | full graph is the canonical two-face purification fixed by the same solution
000144 | operator. No additional Araki-Wyss purification or chosen temperature is
000145 | inserted. The inverse temperature is the geometric beta_s=1 for B,
000146 | equivalently beta_angle=2pi for B_cov=B/(2pi).
000147 | 
000148 | 4. Winding domains and the relative source
000149 | ===========================================
000150 | 
000151 | The original winding multiplier
000152 | 
000153 |   u(s)=exp(2pi i s)
000154 | 
000155 | preserves H1_NS and obeys on that domain
000156 | 
000157 |   B u=u(B+2pi).                                         (9)
000158 | 
000159 | Because Theta is anti-linear and rho leaves s fixed,
000160 | 
000161 |   Theta u Theta^-1=u^-1.                                (10)
000162 | 
000163 | Thus the actual OS word reflection reverses the winding, as required. The
000164 | state is not claimed to be invariant under u. Rather, (9) gives its spectral
000165 | flow and the already proved relative covariance determinant
000166 | 
000167 |   det_F[(1-z u C_A u^*)(1-z C_A)^-1]=1-z,
000168 | 
000169 | for |z|<1. This determinant remains a relative Gaussian source response. It
000170 | is not automatically the physical EM or Quillen functional.
000171 | 
000172 | 5. Quarter clock transport between the two cuts
000173 | ================================================
000174 | 
000175 | The geometric quarter map Qz=iz does not translate one chosen boundary. It
000176 | maps the vertical half A_+ to the horizontal half A_-. In their natural
000177 | coordinates,
000178 | 
000179 |   z=t+i s  on A_+,
000180 |   z=s'+i t' on A_-,
000181 | 
000182 | one has
000183 | 
000184 |   t'=t,  s'=-s.                                         (11)
000185 | 
000186 | Therefore its fixed spin lift U_Q satisfies
000187 | 
000188 |   B_- U_Q=-U_Q B_+,
000189 |   U_Q C_+ U_Q^-1=I-C_-,
000190 |   U_Q u_+=u_-^-1 U_Q.                                  (12)