TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/EVEN-REFLECTION-POSITIVITY.txt Original file SHA-256: 1d19121c8292a3eba44d231502c75ff8e722fa3d530ee7a06d469ab2858b3769 Inclusive original lines: 1-170 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | EVEN REFLECTION POSITIVITY OF THE ACTUAL ORIENTED TWO-FACE GRAPH 000002 | 29 October 2026. Independent analytic proof of the concrete follow-up-4 000003 | candidate. No numerical argument, changed source or original-paper upgrade. 000004 | 000005 | RESULT 000006 | 000007 | For the already given normal-independent product collar, the canonical 000008 | graded Clifford source with covariance G(B) is reflection positive on the 000009 | ENTIRE even bounded CAR algebra of either face for the product reflection 000010 | R=face_swap tensor (S tensor gamma_F). The reflection is an antilinear, 000011 | ORDER-PRESERVING *-automorphism. The proof retains all same-face contractions 000012 | and all crossing blocks. It is not a proof that the original physical source 000013 | chooses this Clifford quantization, or that its physical OS time is fixed. 000014 | 000015 | 1. EXACT EXISTING OPERATOR AND ORIENTATION 000016 | 000017 | Reuse FINITE-COUPLED.txt without changing its coefficient or holonomy: 000018 | g=gamma_F, Z=diag(1,-1), N=[[0,-1],[1,0]], S=NZ, 000019 | B=Z tensor A-S tensor Phi, A=-i0 partial_theta+a(theta), 000020 | [A,g]=0, Phi*=Phi, {g,Phi}=0, 000021 | Gamma=Z tensor g, Ical=i0 Gamma, C0=i0 N tensor I_F. 000022 | A and B have their full periodic H1 domains. The smooth bounded backgrounds 000023 | are independent of the normal coordinate. Let H be the NEW complex Hilbert 000024 | space with multiplication by i equal to Ical, and real metric Re<.,.>_old. 000025 | Then C0 is new-antiunitary, C0^2=1, C0 B C0=-B, and B is selfadjoint. 000026 | 000027 | Put F=S tensor g. This is the ACTUAL existing product-reflection fiber 000028 | R_prod of original_insertions/PROOF.txt:179-188, not a fitted new J matrix. 000029 | There it accompanied r reversal and preserved the full coupled Dirac symbol. 000030 | Here the two labeled faces are interchanged. Direct multiplication gives 000031 | F^2=1, F Gamma=-Gamma F, F B F=-B, 000032 | F C0 F=-C0, Ical F=-C0. 000033 | Thus F is new-antiunitary and preserves the B domain. No J_F condition, 000034 | no [A,Phi]=0 and no commutation of a with Phi is required for these formulas. 000035 | 000036 | For ell>0 let, by bounded spectral calculus on the FULL B, 000037 | a_ell(B)=(1+exp(-2ell B))^{-1}, 000038 | m_ell(B)=(2cosh(ell B))^{-1}, 000039 | G=[[a_ell(B),m_ell(B)],[m_ell(B),1-a_ell(B)]]. 000040 | The orientation of the two faces is essential: 000041 | C_faces=diag(C0,-C0), R=[[0,F],[F,0]]. 000042 | One has G=G*=G^2, C_faces G C_faces=1-G, R^2=1, 000043 | R C_faces=C_faces R, R G R=G. 000044 | The last identity uses a_ell(-B)=1-a_ell(B) and m_ell(-B)=m_ell(B), 000045 | together with the face interchange. These are whole-operator identities, 000046 | not checks in a Fourier window or in selected eigenchannels. 000047 | 000048 | 2. CAR CONVENTION AND THE ACTUAL CROSSING PHASE 000049 | 000050 | Use selfdual CAR generators c(u), complex-linear in u, with 000051 | c(u)*=c(C_faces u), 000052 | {c(u),c(v)}=_new I. 000053 | The inner product is linear in its second entry. The actual graph state is 000054 | omega_G(c(u)*c(v))=_new. 000055 | It is the canonical pure quasifree state of the given projector G. Equivalently 000056 | the earlier projector-covariance uniqueness argument fixes this state and 000057 | all its words; no independently fitted Gaussian covariance is introduced. 000058 | 000059 | Because R is antiunitary and commutes with C_faces, the prescription 000060 | Theta(c(u))=c(Ru), Theta(z X)=bar(z)Theta(X), 000061 | Theta(XY)=Theta(X)Theta(Y) 000062 | extends to an isometric antilinear *-automorphism of CAR, with Theta^2=1. 000063 | The relation R G R=G implies omega_G(Theta(X))=overline(omega_G(X)). 000064 | 000065 | Let W=Fix(C0), a real Hilbert space, and j_+(f)=(f,0). For f in W, 000066 | Ff=Ical f, 000067 | since Ical Ff=-C0 f=-f. Also R j_+(f)=(0,Ff) is C_faces-real. 000068 | Define the real symmetric positive bilinear kernel on W 000069 | M(f,g)=_new=Re_old. 000070 | It is real because m_ell(B) commutes with C0. Its positivity follows from 000071 | the nonnegative function (2cosh(ell x))^{-1}; no choice of B's energy sign 000072 | or of a vacuum sea is needed. The crossing contraction is EXACTLY 000073 | omega_G(c(Rj_+f)c(j_+g)) 000074 | =_new=-i M(f,g). (1) 000075 | Here i means Ical in the new Hilbert space and the ordinary complex scalar 000076 | in the CAR algebra. Reversing the common reflection sign yields +i instead; 000077 | the even result below is unchanged. Replacing C_faces by (C0,C0) is invalid. 000078 | 000079 | 3. SAME-FACE CONTRACTIONS ARE REMOVED BY AN INVERTIBLE WICK BASIS 000080 | 000081 | Write X(f)=c(j_+f) for f in W and keep the ACTUAL same-face contraction 000082 | K(f,g)=omega_G(X(f)X(g))=_new. 000083 | For an ordered list define its Wick polynomial W_K(f_1,...,f_n) by 000084 | W_K(empty)=1, W_K(f)=X(f), 000085 | W_K(f_1,...,f_n)=X(f_1) W_K(f_2,...,f_n) 000086 | +sum_{j=2}^n (-1)^{j-1} K(f_1,f_j) 000087 | W_K(f_2,...,omit(f_j),...,f_n). 000088 | In particular W_K(f,g)=X(f)X(g)-K(f,g)I. This is triangular, with leading 000089 | term the original word and lower terms differing in length by even integers. 000090 | For every finite real test span e_1,...,e_d, the ordered W_K(e_I) for subsets 000091 | I form an invertible triangular change of the usual Clifford word basis. 000092 | Thus the even Wick polynomials span the WHOLE even face algebra in that 000093 | finite span, not a restricted subclass of normal-ordered insertions. 000094 | 000095 | The reflected same-face contractions are bar(K), by R G R=G. Antilinearity 000096 | of Theta therefore gives the reflected Wick polynomial with exactly those 000097 | conjugate contractions. In the expectation of a reflected and an unreflected 000098 | Wick polynomial all contractions internal to either group cancel. This is 000099 | the Wick inclusion-exclusion identity, also obtained by induction from the 000100 | displayed recursion and the quasifree pairing formula. 000101 | 000102 | For lists f=(f_1,...,f_n) and g=(g_1,...,g_m) the result is zero if n!=m. 000103 | For n=m, the only surviving pairings join each reflected entry to exactly 000104 | one unreflected entry. Keeping both lists in their ORIGINAL order gives 000105 | omega_G(Theta(W_K(f)) W_K(g)) 000106 | =(-1)^{n(n-1)/2} det[-i M(f_a,g_b)]_{a,b=1}^n. (2) 000107 | The prefactor is the sign of moving the second list through the first list 000108 | to make the cross pairings. For n=2 this is -det(-i M), not +det(-i M). 000109 | 000110 | 4. POSITIVITY FOR ALL EVEN LINEAR COMBINATIONS AND THE NORM CLOSURE 000111 | 000112 | If n=2m, the total phase in (2) is 000113 | (-1)^{n(n-1)/2}(-i)^n 000114 | =(-1)^{m(2m-1)}(-1)^m=1. (3) 000115 | Consequently different Wick lengths are orthogonal, and the length-2m 000116 | block is the Gram form on the exterior power Lambda^(2m) of the POSITIVE 000117 | kernel M. For a finite real basis e_1,...,e_d let M_d=(M(e_i,e_j)). 000118 | For any even face element written uniquely as 000119 | A=sum_{I even} alpha_I W_K(e_I), 000120 | equations (2)-(3) give 000121 | omega_G(Theta(A) A) 000122 | =sum_{n even} sum_{|I|=|J|=n} 000123 | bar(alpha_I) det(M_d[I,J]) alpha_J >=0. (4) 000124 | Each compound matrix Lambda^n M_d is positive semidefinite because M_d is. 000125 | The n=0 block is |alpha_empty|^2. This proves positivity for every even 000126 | polynomial, including arbitrary mixtures of degrees and their original 000127 | non-normal-ordered forms. The finite span need not reduce B or M. 000128 | 000129 | Every bounded even CAR element is a norm limit of such polynomials. Theta 000130 | is isometric and omega_G is bounded, so (4) passes to that norm limit. 000131 | Thus the result holds on the full bounded even CAR algebra of one face. 000132 | No finite-mode regulator or trace-class hypothesis on B enters the proof. 000133 | Point fields and unbounded point-split composites require their own limit 000134 | and domain control; they are not automatically covered by C*-norm closure. 000135 | 000136 | For odd n the same phase equals -i, independent of n. Hence ordinary 000137 | untwisted reflection positivity does NOT hold on the odd sector. Its 000138 | smallest decisive example is A=X(f) with f!=0: 000139 | omega_G(Theta(A)A)=-i M(f,f), 000140 | which is nonzero imaginary. Multiplying the odd reflected pairing by the 000141 | fixed twist i makes every odd exterior-power block positive as well. 000142 | The opposite overall reflection sign uses the opposite odd twist. 000143 | 000144 | 5. PRIMARY LITERATURE, DUPLICATE CHECK AND EXACT SCOPE 000145 | 000146 | The order-preserving antilinear reflection and parity twist are standard 000147 | fermionic RP conventions, not an analogy with scalar RP. Primary sources: 000148 | Jaffe and Janssens, Characterization of Reflection Positivity: Majoranas 000149 | and Spins, Commun. Math. Phys.346(2016),1021-1050, sections I.4-I.5 and II.2, 000150 | https://doi.org/10.1007/s00220-015-2545-z 000151 | Author full text consulted: 000152 | https://www.arthurjaffe.com/Assets/pdf/RP-Char-1506.04197v1.pdf 000153 | Their general twisted Clifford form includes odd sectors. The calculation 000154 | above independently establishes the exact restricted even form for THIS G. 000155 | 000156 | The current Atlas already registers the full free/finite-coupled Clifford 000157 | chart as finding:graded-boundary-clifford-typing-transports-full-scalar-and- 000158 | finite-coupled-source. The registered PROOF.txt:123-129 and the old actual 000159 | R_prod paragraph of original_insertions/PROOF.txt explicitly left this RP 000160 | step unproved. This note supplies that exact sign and all-word step. The 000161 | targeted Atlas search is not a claim of exhaustive corpus novelty. 000162 | 000163 | The result assumes the canonical Clifford source really uses G(B) as its 000164 | quantum covariance. It does not establish that identification from P1 or 000165 | from the original physical insertion algebra. It proves no general positive 000166 | physical generator, radial transfer semigroup, interacting bosonic measure, 000167 | four-dimensional chiral field theory or original physical OS reconstruction. 000168 | For a normal-dependent B(r), reflection-compatible background transport and 000169 | the corresponding covariance are additional inputs; neither this constant- 000170 | product proof nor a separate radial evolution identity supplies them.