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000001 | CANONICAL CUT STATE AND TIME ON THE ACTUAL NS COVER
000002 | 26 October 2026
000003 | 
000004 | Question
000005 | 
000006 | The full NS cover Green field is reflection positive, but its one particle
000007 | occupation C=(1+exp(2H))^-1 is not a projection. Does the original source
000008 | problem really require a global unitary identification with the old Hardy
000009 | vacuum, or does the same geometric Dirac field already provide the local net,
000010 | state, reflection and time required by the operational seed?
000011 | 
000012 | Verdict
000013 | 
000014 | There is a positive and fully explicit answer inside the charged Gaussian
000015 | premise already used by NS-COVER-FULL-DIRAC-RP. Cut the actual period two NS
000016 | cover into its existing length one half cylinder. The Green covariance on one
000017 | cut face is exactly the marginal of the pure Cauchy graph state of that same
000018 | operator. The same Clifford reflection preserves the graph state and exchanges
000019 | the two faces. The boundary CAR inclusion preserves support in w. Its modular
000020 | logarithm is 2H, so the local RP, state and time data come from one geometric
000021 | datum.
000022 | 
000023 | This does not revise the earlier Atlas criterion, which already allowed local
000024 | net maps and enlarged purifications. It is also not a new graph projection
000025 | theorem. The general graph theorem was already proved on 15 October. The new fact is its
000026 | exact identification with the presently verified NS Green field, its actual
000027 | length, its Pin sector exchange and its reflection. A surjective one particle
000028 | unitary from this faithful face state to the old Hardy projection is unavailable.
000029 | Such a unitary is not a premise for obtaining this particular local state and
000030 | time package.
000031 | 
000032 | The construction does not yet identify this package with the old quarter Hardy
000033 | source. Channel rotation at finite aspect ratio gives an annulus amplitude with
000034 | boundary states. It does not project onto the Hardy vacuum. An exact zero mode
000035 | and partition function test below shows the difference. Thus the result closes
000036 | the internal state, time and reflection consistency of the actual NS cover,
000037 | while the local operator map carrying the old W8 field into this package remains
000038 | the next source question. The compact H dynamics is exactly periodic. This
000039 | rules out real time mixing, but the archived clustering clause does not state
000040 | that it means mixing for tau_H. It therefore remains a typing question, not a
000041 | failure of the seed axiom.
000042 | 
000043 | 1. Original data and earlier theorem
000044 | 
000045 | The current operator is
000046 | 
000047 |   D=-i(sigma1 partial_x+sigma2 partial_w),
000048 |   L=i sigma1 D=partial_x+H,
000049 |   H=i sigma3 partial_w,
000050 | 
000051 | on w in[-1,1], with sigma1 phi(+/-1)=b phi(+/-1). The Pin lift gives
000052 | psi(x+2,w)=-psi(x,w), hence an actual Euclidean period two. The transverse
000053 | operator H is selfadjoint and has simple spectrum
000054 | 
000055 |   spec H={n pi/2:n in Z}.
000056 | 
000057 | In particular it has one genuine zero mode. The already proved Gaussian Green
000058 | kernel is
000059 | 
000060 |   S(t)=exp(-tH)(1+exp(-2H))^-1,                 0<t<2,
000061 |   S(t)=-exp(-(t+2)H)(1+exp(-2H))^-1,          -2<t<0.
000062 | 
000063 | This field and its charged CAR statistics are explicit premises of the proof.
000064 | They are not consequences of the bare differential expression.
000065 | 
000066 | The archived primitive source defines an operational seed as
000067 | 
000068 |   (A_loc,tau_t,Theta,omega,[u_Sigma],D_coll)
000069 | 
000070 | and maps such a seed to one sided boundary data. The direction is Seedop to
000071 | SBord_adm. Therefore the boundary tuple alone still does not manufacture a
000072 | CAR algebra or a Gaussian state. Once the present Gaussian premise is retained,
000073 | however, the following identification is canonical.
000074 | 
000075 | 2. The actual half cover is the earlier two face graph
000076 | 
000077 | Cut the period two x circle at x=0 and x=1. On this existing half cylinder the
000078 | homogeneous equation L psi=0 has transfer
000079 | 
000080 |   T=exp(-H).
000081 | 
000082 | Its closed solution graph has orthogonal projection
000083 | 
000084 |   G = [ A  X ],     A=(1+exp(-2H))^-1,
000085 |       [ X  C ]      X=(2 cosh H)^-1,
000086 |                      C=(1+exp(2H))^-1.
000087 | 
000088 | This is the 15 October two face theorem specialized to the actual H and the
000089 | already fixed length one. No collar length, temperature or purification was
000090 | chosen after the fact.
000091 | 
000092 | The exact Green boundary limits identify the face labels:
000093 | 
000094 |   S(0+)=A,
000095 |   -S(0-)=C.
000096 | 
000097 | Thus the equal time occupation of the charged field on the second labeled face
000098 | is precisely the covariance C appearing in the NS cover proof. The identity is
000099 | spectral and holds for every transverse mode, including the zero mode C=1/2.
000100 | Since G^2=G, the doubled face state is pure. Its restriction to the second face
000101 | is faithful.
000102 | 
000103 | This is a geometrically fixed purification. It is not an arbitrary Araki Wyss
000104 | choice: its cross covariance X and its two marginals are forced by the actual
000105 | solution graph of L on the retained half cover.
000106 | 
000107 | 3. The same reflection and support
000108 | 
000109 | Let R0=sigma1 on the transverse Hilbert space. It preserves the endpoint
000110 | domain and R0 H R0=-H. On the two cut faces set
000111 | 
000112 |   R_cut = [ 0   R0 ].
000113 |           [ R0  0  ]
000114 | 
000115 | Functional calculus gives R0 A R0=C and R0 X R0=X, hence
000116 | 
000117 |   R_cut G R_cut=G.
000118 | 
000119 | This is the boundary action of the same Clifford x reflection used in the
000120 | positive Green form. It exchanges the two cut faces. On the full NS cover it
000121 | also exchanges the c=+i and c=-i Pin character sectors. This explains why the
000122 | positive construction requires their complete pair and cannot be restricted
000123 | to one quarter sector.
000124 | 
000125 | For every proper interval I in the w interval there is a literal local
000126 | inclusion
000127 | 
000128 |   CAR(L2(I,C2)) -> CAR(H_face0 direct_sum H_face1),
000129 |   a(f)           -> a(0 direct_sum f).
000130 | 
000131 | It preserves support exactly. R_cut sends it to the corresponding algebra on
000132 | the other face without moving the w support. The cross block X is nonlocal as
000133 | a correlation kernel, but it does not change the support of the algebra map.
000134 | For the first order H, real time has finite propagation with the given local
000135 | endpoint reflection. This is the local net content required here. It is not a
000136 | claim that every local interval is invariant under time evolution.
000137 | 
000138 | The restricted quasifree state is KMS for
000139 | 
000140 |   a(f) -> a(exp(itH)f)
000141 | 
000142 | at inverse temperature two, because
000143 | 
000144 |   log((1-C)/C)=2H.
000145 | 
000146 | Consequently the same cut datum supplies state, cyclic standard form,
000147 | reflection and time. This is stronger than merely observing a thermal list of
000148 | occupations and weaker than demanding equality with a different global vacuum
000149 | representation.
000150 | 
000151 | There is one exact time typing test. Since spec H=(pi/2)Z,
000152 | 
000153 |   exp(i 4H)=I.
000154 | 
000155 | For example, take normalized f with proper local support and
000156 | N_f=a*(f)a(f). Faithfulness gives 0<omega(N_f)<1, so its variance is positive,
000157 | while tau_(4k)(N_f)=N_f. Its connected real time correlation therefore cannot
000158 | mix. This does not decide the archived clustering axiom. That text calls the
000159 | net Euclidean and says only that omega is clustering on the admissible local
000160 | algebra; it does not identify clustering with the Lorentz real time tau_H.
000161 | Euclidean semigroup clustering, spatial clustering and real time mixing are
000162 | different statements. The calculation merely prevents using periodic tau_H
000163 | mixing as an unspoken proof of the required clause. The zero mode C=1/2 also
000164 | means that a unique vacuum interpretation would need an explicit sector rule,
000165 | but a general KMS state does not require such a rule.
000166 | 
000167 | 4. What the old sources do and do not select
000168 | 
000169 | The A4 and Moebius dictionary proves that the longitudinal NS operator is
000170 | 
000171 |   B_cov = 2(B_1/4 direct_sum B_3/4).
000172 | 
000173 | It derives the two quarter characters from the full Pin monodromy and does not
000174 | choose them as two independent source worlds. This is compatible with the
000175 | present result: the linear Green reflection exchanges those sectors.
000176 | 
000177 | It does not identify the old longitudinal Hardy vacuum with the face KMS state.
000178 | Indeed the two channel Hamiltonians have different exact spectral types:
000179 | 
000180 |   spec H = (pi/2) Z,                       ker H has dimension one,
000181 |   spec B_cov = (2Z+1/2) union (2Z+3/2),   ker B_cov is zero.
000182 | 
000183 | No modewise unitary time intertwiner can map one to the other. This is the
000184 | expected open versus closed channel distinction, not a failure of the common
000185 | Euclidean Dirac field.
000186 | 
000187 | The original operational category also keeps tau_t and omega as input. Its
000188 | collar completion theorem runs from an already reflection positive local seed
000189 | to boundary data. It therefore accepts the present NS Gaussian data only after
000190 | all seed axioms and the P1 identification are supplied; it neither selects the
000191 | package nor turns the old quarter vacuum into it. The present compact package supplies the local net,
000192 | RP state, reflection and KMS time legs. Whether it satisfies the archived
000193 | clustering clause depends on the clause's intended Euclidean flow, which the
000194 | quoted definition does not specify.
000195 | 
000196 | 5. The finite channel rotation test
000197 | 
000198 | The relative open channel partition function proved for this exact cover is
000199 | 
000200 |   Z_rel=2 product_(n>=1)(1+exp(-n pi))^2.