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000001 | INDEPENDENT REVIEW: NORMAL GREEN TRACE TO THE CALDERON GRAPH
000002 | 30 October 2026
000003 | 
000004 | Question
000005 | 
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000011 | 
000012 | Verdict
000013 | 
000014 | The ORIGINAL homogeneous collar already fixes the closed Poisson range
000015 | 
000016 |   graph(exp(-ell h)).
000017 | 
000018 | Together with its positive boundary metric, this range fixes the orthogonal
000019 | Calderon projector, independently of any chosen compatible closed Green
000020 | realization.  No new antiperiodic selection gate is needed for that existing
000021 | G contract.  For a selfadjoint tangential operator h the result is
000022 | 
000023 |   G = [[A,X],[X,1-A]],
000024 |   A=(1+exp(-2 ell h))^-1,
000025 |   X=exp(-ell h)(1+exp(-2 ell h))^-1.
000026 | 
000027 | The antiperiodic circle of length 2 ell supplies one stronger realization:
000028 | the boundary trace of its raw Green Poisson formula is already this orthogonal
000029 | G.  This genuine Green-to-Calderon trace identity uses the jump, the sign of
000030 | the negative Green branch and both boundary limits.  It is not the naked
000031 | equality of a bulk two-point function with a boundary projector.
000032 | 
000033 | The antiperiodic closure is actual data in the already constructed elliptic
000034 | NS source and in the period-two NS cover.  There the raw Green trace identity
000035 | is an original source identity inside the explicitly retained free charged
000036 | Gaussian premise.  It is not forced for every primitive P1 Green realization
000037 | by a local Spin(2) lift with square -I alone.  The transport from the abstract
000038 | P1 datum to this particular elliptic NS source is needed only if one wants to
000039 | identify their BULK Schwinger kernels or quantizations.  It is not a
000040 | prerequisite for the original orthogonal Calderon projector obtained from the
000041 | Poisson range and boundary metric.  This review does not identify h with the
000042 | old longitudinal quarter generator, and it supplies no new state, physical
000043 | time or P1 quantization rule.
000044 | 
000045 | 1. Exact scalar identity
000046 | 
000047 | Fix ell>0 and one spectral value lambda of h.  Put
000048 | 
000049 |   q=exp(-ell lambda),
000050 |   a=1/(1+q^2),
000051 |   x=q/(1+q^2),
000052 |   c=q^2/(1+q^2)=1-a.
000053 | 
000054 | On the antiperiodic circle of length 2 ell the inverse of partial_s+lambda has
000055 | the two branches
000056 | 
000057 |   S(tau)= exp(-tau lambda)a,                    0<tau<2 ell,
000058 |   S(tau)=-exp(-(tau+2 ell)lambda)a,          -2 ell<tau<0.
000059 | 
000060 | They satisfy
000061 | 
000062 |   S(0+)-S(0-)=a+q^2 a=1,
000063 |   S(tau+2 ell)=-S(tau).
000064 | 
000065 | For 0<s<ell, the Poisson field produced from an arbitrary boundary source
000066 | pair is
000067 | 
000068 |   psi(s)=S(s)f_0-S(s-ell)f_ell.                         (1)
000069 | 
000070 | The interior limits of (1) are
000071 | 
000072 |   psi(0+)   =a f_0+x f_ell,
000073 |   psi(ell-) =x f_0+c f_ell.
000074 | 
000075 | Thus the normal trace of the Green Poisson operator is
000076 | 
000077 |   g_lambda=[[a,x],[x,c]].                               (2)
000078 | 
000079 | Directly,
000080 | 
000081 |   a^2+x^2=a,
000082 |   x^2+c^2=c,
000083 |   ax+xc=x.
000084 | 
000085 | Therefore g_lambda*=g_lambda and g_lambda^2=g_lambda.  Its range is the
000086 | one-dimensional graph {(u,q u)}.  This proves both the signs and the
000087 | normalization without equating the Green kernel itself to g_lambda.
000088 | 
000089 | 2. Full selfadjoint operator
000090 | 
000091 | Let h be an arbitrary selfadjoint operator.  The transfer
000092 | 
000093 |   T=exp(-ell h)
000094 | 
000095 | can be unbounded when h is unbounded below, but it is a closed positive
000096 | operator.  The scalar functions
000097 | 
000098 |   a(lambda)=1/(1+exp(-2 ell lambda)),
000099 |   x(lambda)=exp(-ell lambda)/(1+exp(-2 ell lambda)),
000100 |   c(lambda)=1-a(lambda)
000101 | 
000102 | are bounded.  Borel functional calculus therefore defines bounded operators
000103 | A, X and C=1-A on the whole boundary Hilbert space.  The scalar identities of
000104 | section 1 hold almost everywhere for the spectral measure of h, hence
000105 | 
000106 |   G=[[A,X],[X,C]]=G*=G^2.                              (3)
000107 | 
000108 | The standard closed-graph formula gives the same result:
000109 | 
000110 |   P_graph(T)=
000111 |     [[(1+T*T)^-1,          (1+T*T)^-1 T*],
000112 |      [T(1+T*T)^-1, T(1+T*T)^-1 T*]].                  (4)
000113 | 
000114 | Since T=T*, all four entries of (4) are A,X,X,C.  Thus (3) is the
000115 | orthogonal Calderon projector for the homogeneous collar, including the full
000116 | unbounded spectral case.  Boundary limits are understood in the usual trace
000117 | domain of the Poisson solution; the resulting projector is bounded.
000118 | 
000119 | 3. Why antiperiodicity matters, and what does not depend on it
000120 | 
000121 | For comparison, suppose a periodic circle of length 2 ell is chosen and
000122 | 1-exp(-2 ell h) has a bounded inverse.  Put
000123 | 
000124 |   A_p=(1-exp(-2 ell h))^-1,
000125 |   X_p=exp(-ell h)A_p.
000126 | 
000127 | The periodic Green branches have no minus sign across the closure.  Applying
000128 | the same two-boundary trace to (1) gives
000129 | 
000130 |   C_p=[[A_p,-X_p],[X_p,-exp(-2 ell h)A_p]].             (5)
000131 | 
000132 | Scalar functional calculus gives C_p^2=C_p, and its range is again graph(T),
000133 | but generally C_p is not selfadjoint.  Its orthogonal range projection is the
000134 | operator G in (3).  A zero mode of h makes the periodic inverse singular,
000135 | whereas the antiperiodic inverse and (3) remain regular with A=C=X=1/2 on that
000136 | mode.
000137 | 
000138 | Consequently the graph subspace and its orthogonal projector are intrinsic to
000139 | the given homogeneous collar transfer and its positive boundary metric.  The
000140 | statement that the RAW CLOSED Green jump matrix produces this orthogonal
000141 | projector directly does use the antiperiodic spin closure.  Local face reality
000142 | and the square of a local lift do not on their own choose that closure, but no
000143 | such choice is needed to orthogonally project onto the already fixed Poisson
000144 | range.
000145 | 
000146 | 4. Status in the actual sources
000147 | 
000148 | The existing elliptic source fixes the relevant global data.  On its NS--NS
000149 | spin line, partial_t+B is antiperiodic in both directions.  Its spectral Green
000150 | kernel has precisely the two branches above, and its half-annulus Cauchy data
000151 | have transfer exp(-W B).  ELLIPTIC-GREEN-RP proves that the time-zero Green
000152 | covariance is the labeled face compression of that same finite-width graph.
000153 | CANONICAL-CUT-STATE-TIME-BRIDGE specializes the same calculation to the actual
000154 | period-two NS cover, where the Pin lift gives psi(x+2,w)=-psi(x,w), the cut is
000155 | at x=0,1, and h=H.  In that source the antiperiodicity is therefore not a new