TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/two_face_source_20261029/PROOF.txt
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000043 | Reuse the exact NS Green inverse and graph, not a fitted numerical kernel:
000044 |   S(t)=exp(-tB)(I+exp(-beta B))^-1, 0<t<beta,
000045 |   S(W)=(2cosh(WB))^-1=X,
000046 |   C=(I+exp(2WB))^-1,
000047 |   P=[[I-C,X],[X,C]], P=P*=P^2.
000048 | These all-mode identities are the old Green and graph results. The midpoint
000049 | is not missing from the already given free source. HALF-PERIOD-GNS-SEWING.txt
000050 | already realizes its full charged CAR double in the source's own HS space:
000051 |   Omega=sqrt(rho), L_A X=A X, R_A X=X A,
000052 |   Gamma X=Pi X Pi, a_R=L_a, a_L=-Gamma R_a,
000053 |   a_L*=Gamma R_(a*), J_HS X=X*.
000054 | No auxiliary physical particle or freely chosen state is introduced.
000055 | 
000056 | The actual source null relations in a spectral mode t_lambda=exp(-W lambda)
000057 | are (a_R-t_lambda a_L)Omega=0 and
000058 | (a_L*+t_lambda a_R*)Omega=0. CAR then fixes all four covariance blocks above.
000059 | The old projector-state theorem determines every CAR word. These null
000060 | relations hold in the given free realization; their origin for actual
000061 | primitive P1 boundary fields remains to be shown, not postulated here.
000062 | 
000063 | 3. Opposite current orientation follows from the existing right action
000064 | 
000065 | Let W(f) be an already defined bounded neutral current Weyl on the given
000066 | normal source representation. Define bounded partner operations on HS by
000067 |   W_R(f)=L_(W(f)), W_L(f)=R_(W(f)*).
000068 | They commute across the two sides. Since R_A R_B=R_(B A), the original
000069 |   W(f)W(g)=exp(-i sigma(f,g)/2) W(f+g)
000070 | gives exactly
000071 |   W_L(f)W_L(g)=exp(+i sigma(f,g)/2) W_L(f+g).
000072 | The opposite Weyl sign is thus DERIVED on the canonical partner algebra.
000073 | It is not an additional chosen current orientation or a new physical source.
000074 | This does not prove that this algebra is the geometrical P1 second face.
000075 | 
000076 | Directly, J_HS W_R(f) J_HS=W_L(f). This action is conjugate-linear and
000077 | MULTIPLICATIVE. It is not the anti-multiplicative OS Euclidean word rule.
000078 | For finite mode bilinears N(A)=sum A_ij a_i* a_j, Hermitian A, the same
000079 | graded CAR construction gives
000080 |   n_L(A)=tr(A) I-R_(N(A)),
000081 |   j_L(A)=-R_(j(A)), J_HS j_R(A) J_HS=-j_L(A),
000082 | where each current has its actual source mean subtracted. In particular
000083 |   J_HS n_R,j J_HS=I-n_L,j.
000084 | Bare exchange of particle occupations misses this hole contribution.
000085 | Infinite divergent traces are not used: the bounded Weyl formulas above
000086 | define the all-mode result directly with the retained source phases.
000087 | 
000088 | 4. Joint bounded responses and time reflection, without a new Gaussian axiom
000089 | 
000090 | For ANY existing normalized normal density rho, the two-side response is
000091 |   <Omega,W_L(f) W_R(g)Omega>
000092 |     =Tr(sqrt(rho) W(f)* sqrt(rho) W(g)).                (1)
000093 | For a finite list f_i this is the positive HS Gram of
000094 |   V_i=rho^(1/4) W(f_i) rho^(1/4).
000095 | Schatten Holder gives V_i in HS. This is a source-dependent full joint
000096 | functional, not a fit from one-face spectra. It does not imply maximal
000097 | odd-field cross coherence for an interacting source; the old unsaturated
000098 | interaction control and its conclusion remain unchanged.
000099 | 
000100 | There is also an all-mode bounded time-reflection construction on the same
000101 | given source. Let H>=0 be its source Hamiltonian with Z=Tr exp(-beta H)<infinity
000102 | and W=beta/2. For 0<t1<...<tn<W and bounded neutral insertions A1,...,An put
000103 |   K_F=exp(-(W-tn)H) An exp(-(tn-t(n-1))H) ...
000104 |          ... A2 exp(-(t2-t1)H) A1 exp(-t1 H).
000105 | All intermediate heat factors are retained. Schatten Holder gives
000106 |   ||K_F||_2 <= sqrt(Z) product_j ||A_j||.
000107 | Indeed each positive heat length d has Schatten exponent beta/d, and the
000108 | sum of their reciprocal exponents is W/beta=1/2.
000109 | 
000110 | The thermal Euclidean functional, with the reflected word correctly TIME
000111 | ORDERED and each reflected insertion adjointed, is
000112 |   S(Theta F,G)=Tr(K_F* K_G)/Z.                         (2)
000113 | It is reflection positive for any finite linear combination of histories.
000114 | No bare unordered product at opposite times is substituted for this form.
000115 | Equation(2) extends to bounded current Weyls already present in the source
000116 | von Neumann algebra. On the original polynomial free source it matches the
000117 | two charged Green Gram factors; uniform bounded strong-star approximation
000118 | and the positive heat factors extend the normal functional to those Weyls.
000119 | No norm-C* membership of the Weyls or new unsmeared point fields is claimed.
000120 | Only the time-zero insertions A_j are assumed bounded. Their formal imaginary
000121 | time conjugates need not be bounded operators; the heat products define the
000122 | ordered functional without making that invalid domain claim. The bounded
000123 | neutral theorem adds no unproved rule for arbitrary odd Schwinger words.
000124 | 
000125 | Thus the proper normal-time OS reflection is constructed for bounded neutral
000126 | histories of the GIVEN thermal source. It is compatible with the old full
000127 | Green/graded-CAR realization. It does not identify this geometric normal
000128 | time with physical P1 time, or this source with the full shifted Hardy/K_alpha
000129 | or local E8 vacuum. Those are different retained source types.
000130 | 
000131 | 5. Automatic check on the actual existing path
000132 | 
000133 | check_two_face_source.py calls the OLD source_lift, graph_projection and
000134 | double_data. It uses two actual positive NS modes lambda=pi,3pi of the old
000135 | square annulus. With z=exp(-pi/12), W=1/2, beta=1, the exact symbolic