TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/SEED-FIELD.txt Original file SHA-256: db9d551297ad797c51b8fef682db85f13a08cbb289592e238289b17e59b66a0b Inclusive original lines: 27-147 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. 000027 | 1. EXACT NATIVE OPERATOR AND FIELD LABELS 000028 | 000029 | Keep the full actual finite coefficient, its grading and smooth fields: 000030 | 000031 | gamma=gamma_F, Phi(theta)=Phi(theta)^*, {gamma,Phi}=0, 000032 | A=-i partial_theta+a(theta), [A,gamma]=0, 000033 | B=Z tensor A-S tensor Phi, N=[[0,-1],[1,0]], S=NZ, 000034 | Gamma_tot=Z tensor gamma, Ical=i Gamma_tot, C0=i N tensor I_F. 000035 | 000036 | The original product collar is normal-independent. Phi may vary smoothly 000037 | with theta and need not commute with A. The original periodic H1 domain, 000038 | including its actual holonomy frame, is retained. No quarter holonomy or 000039 | length is newly fitted. On the positive total grade use the given map 000040 | 000041 | E v=(P_+ v,i P_- v), P_+=(1+gamma)/2, P_-=(1-gamma)/2. 000042 | 000043 | Direct action on the full H1 domain gives 000044 | 000045 | B E=E h, h=A gamma-i gamma Phi, 000046 | h^2=A^2+Phi^2-i[A,Phi] on H2. (1) 000047 | 000048 | This is the ENTIRE original coefficient space, including the 96-dimensional 000049 | Yukawa/Majorana coefficient when that original is used. It is not an 000050 | identification of those 96 labels with 16 many-body modes. C0 is not J_F. 000051 | The full ordinary frame U=(E,NE) also carries D and D_ref and their source 000052 | legs, as proved in G; neither normal degree is discarded. 000053 | 000054 | With oriented faces define 000055 | 000056 | T_faces(v0,v1,bar(w0),bar(w1)) 000057 | =(E v0+C0 E w0, E v1-C0 E w1). (2) 000058 | 000059 | The opposite C0 sign on the second face is part of the actual orientation. 000060 | Using (C0,C0) on the two faces would change the CAR source. 000061 | 000062 | 2. COMPLETE STATE, ALL FIELDS AND THE SAME TIME 000063 | 000064 | For the actual collar length ell>0, beta=2ell, set 000065 | 000066 | C=(1+exp(2ell h))^-1, 000067 | X=(2cosh(ell h))^-1, 000068 | P_h=[[1-C,X],[X,C]]. (3) 000069 | 000070 | This is the full graph projection, including both crossings. If it is the 000071 | actual quantum covariance of the given Clifford source, CAR and positivity 000072 | fix its state and every word. Quasifreeness is then not an independent fit. 000073 | 000074 | In the sea-relative excitation representation of THIS h let 000075 | 000076 | H=sum_j |lambda_j| n_j^exc, 000077 | Z_beta=product_j(1+exp(-beta |lambda_j|)), 000078 | rho=exp(-beta H)/Z_beta. (4) 000079 | 000080 | Smooth bounded Phi on the circle gives Tr exp(-beta |h|)0. Finite zero modes 000082 | give finite multiplicity and occupation 1/2 in (3); they cause no divergent 000083 | sea subtraction. The infinite-dimensional H is defined by (4), not by a 000084 | difference of two divergent absolute energies. 000085 | 000086 | The a in the following field map is the OLD signed charged field, expressed 000087 | in these particle/hole excitation coordinates. Explicitly, with spectral 000088 | Q_+=1_[0,infinity)(h), Q_-=1_(-infinity,0)(h), 000089 | a_old(f)=b(Q_+ f)+d^*(overline(Q_- f)). 000090 | These Q signs are spectral signs of h, not the grading signs in E. Thus 000091 | occupation is C(h) and creators transform by exp(it h). Replacing a_old by 000092 | a bare excitation annihilator would incorrectly give C(|h|) and the wrong 000093 | time on negative h modes. The a below always denotes a_old. 000094 | 000095 | On HS(K) put Omega=sqrt(rho), L_A Y=A Y, R_A Y=Y A and 000096 | Gamma Y=Pi Y Pi, with Pi the genuine excitation parity of the whole source. 000097 | The actual anticommuting field families are 000098 | 000099 | a_R=L_a, a_L=-Gamma R_a, a_L^*=Gamma R_(a^*). 000100 | 000101 | The native complex-linear selfdual field map is exactly 000102 | 000103 | iota[c(T_faces(v0,v1,bar(w0),bar(w1)))] 000104 | =a_R^*(v0)-a_L^*(v1)+a_R(w0)-a_L(w1). (5) 000105 | 000106 | Creators are linear in their tests; annihilators are antilinear. This map 000107 | has covariance (3) on all bounded CAR words. The GNS map from the graph 000108 | source to (5) is onto the existing thermal HS representation; it adds no 000109 | independent source copy or new physical species. 000110 | 000111 | The given geometric group and this source group obey 000112 | 000113 | iota[c(exp(t Ical B)u)]=U_t iota[c(u)] U_t^*, 000114 | U_t Y=exp(itH)Y exp(-itH), (6) 000115 | 000116 | with the SAME t. In particular both creator faces transform by exp(it h) 000117 | and the conjugate slots by exp(-it bar(h)). The HS generator L_H-R_H is 000118 | distinct from positive H; its energy differences have both signs. Neither 000119 | operator is simply the classical Dirac D. Equation (6) is already proved 000120 | inside the actual graph source, not a new conditional time reconstruction. 000121 | 000122 | 3. ACTUAL PRODUCT REFLECTION AND WHOLE POSITIVE WORD FORM 000123 | 000124 | The original product-reflection fiber is F=S tensor gamma. With face 000125 | interchange R_geo=[[0,F],[F,0]] one has R_geo P R_geo=P and 000126 | Ical F=-C0. In (5), 000127 | 000128 | Theta_geo(a_R^*(v))=+i a_L(v), 000129 | Theta_geo(a_L^*(v))=-i a_R(v). 000130 | 000131 | It is an antilinear ORDER-PRESERVING algebra map. On the HS source it is 000132 | implemented by 000133 | 000134 | J_R=Q_Gamma^* J_HS, Q_Gamma=P_even-i P_odd, J_HS Y=Y^*. 000135 | 000136 | For every bounded even A_i,A_j its exact reflected form is 000137 | 000138 | Tr(sqrt(rho) A_i^* sqrt(rho) A_j) 000139 | =_HS. (7) 000140 | 000141 | The independent Wick proof preserves the real crossing -i M. At word 000142 | length n its cross block is (-1)^(n(n-1)/2) det(-i M). This is positive 000143 | for every even n, and the Wick basis is an invertible change of the whole 000144 | finite Clifford word basis. Norm closure gives the entire bounded even 000145 | face algebra, not only Wick monomials. Odd words require the fixed i twist; 000146 | ordinary untwisted odd RP is not claimed. Grassmann reversal and the actual 000147 | OS word map remain correctly typed; J_R alone does not define them.