TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/NS-COVER-FULL-DIRAC-RP.txt Original file SHA-256: a18c7589ac87b0ea875af7a2a46790989c80c836369b9ee001b0417dda6ab154 Inclusive original lines: 1-186 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 | FULL DIRAC GREEN POSITIVITY ON THE EXISTING NS ORIENTATION COVER 000002 | 26 October 2026 000003 | 000004 | Question and scope 000005 | 000006 | Does the actual flat Moebius comparison operator admit the required free 000007 | reflection positive Green realization after retaining its original double 000008 | cover, including every transverse mode? This tests the proposed geometric 000009 | source bridge, not a newly optimized finite model. The premise is explicitly 000010 | the charged Gaussian field with Green operator of this Dirac operator. 000011 | Neither this field quantization nor its physical identification with the 000012 | original P1 kernel is inferred from the geometric differential expression. 000013 | 000014 | 1. The cover is fixed by the original Pin lift 000015 | 000016 | Use D=-i(sigma1 partial_x+sigma2 partial_w), w in[-1,1], with boundary 000017 | sigma1 psi=b psi at both endpoints, b=+1 or-1. The actual Moebius lift is 000018 | psi(x+1,-w)=c sigma1 psi(x,w), c=+i or-i. Applying it twice gives 000019 | psi(x+2,w)=-psi(x,w). This is exactly the existing orientation cover. 000020 | Its Euclidean period2 is derived from that lift, not fitted to a desired 000021 | temperature. The length is in the source convention where the core has 000022 | length1. Angular or physical clock units require their original conversion. 000023 | 000024 | Write J=sigma1 R_w and T_a psi(x,w)=psi(x+a,w). On this NS cover the actual 000025 | deck operator A=J T_1 obeys A^2=-I and commutes with D. The original two Pin 000026 | spaces are its eigenspaces, with projectors 000027 | 000028 | P_i=(I-i A)/2, P_minus_i=(I+i A)/2. 000029 | 000030 | They are an orthogonal decomposition of one cover field space, not two 000031 | independently introduced species or positive thermal systems. 000032 | 000033 | 2. Spatial operator and the exact whole Green kernel 000034 | 000035 | Multiplying the given Dirac expression on the left gives 000036 | 000037 | L=i sigma1 D=partial_x+H, H=i sigma3 partial_w. 000038 | 000039 | The domain of H is H1([-1,1],C2) with the same endpoint conditions 000040 | sigma1 phi(+/-1)=b phi(+/-1). The boundary form is 000041 | i[phi* sigma3 psi]_-1^1. Each endpoint line is a maximal isotropic line 000042 | for sigma3, so this regular first order realization is selfadjoint and 000043 | has compact resolvent. Solving H phi=E phi gives 000044 | 000045 | spec H={0} union {+n pi/2,-n pi/2:n>=1}, 000046 | 000047 | with every eigenvalue simple. The zero vector is the constant sigma1 000048 | eigenvector v_b. Thus every original transverse branch is retained. 000049 | Also [H,J]=0. The kernel below is an identity of spectral calculus, not a 000050 | finite mode fit. For one period, away from t=0, 000051 | 000052 | S(t)=exp(-t H)(I+exp(-2H))^-1, 0 dt ds 000073 | = || integral B_s f(s) ds ||^2 >=0. 000074 | 000075 | This proves the full one particle form for every transverse mode and their 000076 | superpositions. On compact subintervals[delta,1-delta] each multiplier 000077 | decays at least exp(-delta|E|), giving the usual smoothing and controlled 000078 | mode limits. The Clifford reflection is the LINEAR sigma1 R_x in the 000079 | Dirac Green form, accompanied by the standard antilinear, charge exchanging 000080 | and word reversing reflection on the field algebra. It is not obtained by 000081 | declaring the antiunitary one particle symmetry K R_x to be this Green form. 000082 | 000083 | For the conjugate charge the second block is also positive: 000084 | 000085 | K_psi(t,s)=S(t+s)=Btilde_t* Btilde_s, 000086 | Btilde_t=exp((1-t)H)(I+exp(2H))^-1/2. 000087 | 000088 | In canonical charged variables choose the antilinear order reversing 000089 | reflection Theta psi=-barpsi composed with r_x and 000090 | Theta barpsi=-psi composed with r_x. For these source and contraction 000091 | conventions the displayed minus in K=-S(-t-s) is essential. The bare 000092 | positive Clifford matrix times slashD^-1 must not be called positive 000093 | without this source order sign. Both charge blocks now give positive 000094 | Gram determinants for normal ordered monomials. For the given free 000095 | charged Gaussian field this proves full polynomial reflection positivity. 000096 | Equivalently fill the negative H sea and use the positive 000097 | many body excitation Hamiltonian with energies |E|. Its relative thermal 000098 | partition function is 000099 | 000100 | Z_rel=2 product_(n>=1)(1+exp(-n pi))^2 < infinity. 000101 | 000102 | The factor2 is the actual zero mode. Finite spectral truncations have a 000103 | positive Gibbs density and the reflected word form is a Hilbert Schmidt 000104 | square; the convergent product and smeared kernel limit preserve positivity. 000105 | This is a full free Gaussian statement, not an interacting P1 reconstruction. 000106 | 000107 | 4. Why the two quarter sectors must be retained together 000108 | 000109 | For R=sigma1 R_x on the NS cover, R A R=-A because T_-1=-T_1 there. 000110 | Thus R P_i R=P_minus_i. The Green reflection exchanges the two original 000111 | Pin lifts. It cannot be restricted to just one lift as an endomorphism. 000112 | In contrast the antiunitary Theta_x=K R_x preserves each lift: conjugation 000113 | of the scalar i compensates A->-A. It commutes with D and preserves its 000114 | full spectral sea. These two facts are compatible; they have different 000115 | operator types and different positivity jobs. 000116 | 000117 | On the invariant transverse zero branch the exact scalar Green kernels 000118 | of partial_x are 000119 | 000120 | G_q(s)=q/(q-1) for0