TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/ORIGINAL-STATE-MAP.txt Original file SHA-256: 0fb21f2038204bdef0314d3c1bcad8a20a9ac4deb1ef6f0f2ef3e444bac44905 Inclusive original lines: 1-200 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 | J: THE ORIGINAL SOURCE STATE, VACUUM AND MODULAR TRANSPORT 000002 | 3 November2026. relational_logic independent source review. 000003 | Entry plan90, Atlas457bcebe946f1d5ac3b6f403. G/H/I retained. 000004 | No canonical changes. No new source model or clock branch. 000005 | 000006 | DIRECT RESULT 000007 | 000008 | There is no independent vacuum choice AFTER a complete original P1 source 000009 | has been supplied and transported by the actual full G carrier chart. 000010 | Its whole normalized word functional carries its OS vector [1], positive 000011 | time, reflection, and, on the same standard local pair, all Tomita modular 000012 | data. A valid BW identification transports with that same pair. This is 000013 | an application of an existing source-preserving theorem, not a new BW 000014 | theorem or a claim that arbitrary P1 objects share one state. 000015 | 000016 | Moreover, the Atlas already contains a CONSTRUCTIVE LOCAL VACUUM/BW 000017 | solution on the actual original neutral Q0 current source. It uses the 000018 | SAME original sea Omega, derives a positive parabolic generator with 000019 | unique kernel C Omega, and constructs the full affine group and local 000020 | standard inclusion. It must be counted as solved in that scope, not 000021 | replaced by the earlier finite v438/v455 witness or called an unknown 000022 | neutral modular existence problem. Its actual old-source checker passed 000023 | unchanged on this review. 000024 | 000025 | The precise remaining distinction is SOURCE TYPE, not another freely 000026 | chosen vacuum measure. The original neutral vacuum is not the elliptic 000027 | Gibbs state, nor automatically the whole coupled S_adm source. A full 000028 | Euclidean covariance theorem for the SAME complete S/Z makes its vacuum 000029 | invariance a reconstruction consequence. A passive chart transport alone 000030 | does not assert a new symmetry of an unchanged background. 000031 | 000032 | 1. WHAT P1 ALREADY SUPPLIES 000033 | 000034 | Source45 Boundary002:100–123 defines an operational seed 000035 | S=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll), 000036 | where A_loc is a quasilocal Euclidean observable net, omega is RP and 000037 | clustering with finite relative action to the declared reference, and 000038 | D_coll is a reflection-regular first-order elliptic collar generator. 000039 | Morphisms preserve ALL these data. Thus omega, time tau_t and positive-time 000040 | reflection Theta are genuine original data. Requiring another arbitrary 000041 | state/quantization law just because one later changes the full carrier 000042 | frame would duplicate those data. 000043 | 000044 | Boundary002:125–143 constructs the one-sided OS collar functor from this 000045 | source. Its direction is Seedop -> SBord_adm. The subsequent defect 000046 | essentialization/minimality:145–242 acts on the induced essential bordism. 000047 | It is not a displayed inverse theorem recovering all source words from 000048 | only finite defect numbers. This distinction does not invalidate the 000049 | declared full seed as the starting axiom. It prevents silently replacing 000050 | it by a thinner c3/geometric package and then demanding a new state. 000051 | 000052 | The original common action/OS/1PI chain already exists; the current task 000053 | must use its SAME complete source rather than infer source equality from 000054 | a shared Dirac eigenvalue or finite covariance. Root and joint_origin 000055 | review its full source covariance and full-moment OS step separately. 000056 | 000057 | 2. THE ACTUAL NEUTRAL SOURCE HAS ITS LOCAL VACUUM AND BW PAIR 000058 | 000059 | Read in full: neutral_moebius_source_20261029/PROOF.txt:1–193 and independent 000060 | review:1–46, together with the ORIGINAL stress/current map:33–115 and 000061 | source-quarter relation:166–180. The original current identity is 000062 | Lcal_n^alpha=L_n^0+b J_n+(b^2/2)delta_n0, b=alpha-1/2. 000063 | The two quarter stress families are compared on the SAME CAR generators 000064 | by the actual state-preserving cover arrow; their difference reconstructs 000065 | the current. The other cover arrow crosses the sea and is not silently 000066 | declared vacuum preserving. The actual local support and unitary product 000067 | formula identify the local paired-stress algebra with A_J(I). 000068 | 000069 | On the original neutral cyclic space 000070 | H0=closure span{J_-n1 ... J_-nk Omega}, Q=0, 000071 | the existing circle excitation is H_alpha=L0^0. The same original Omega 000072 | is annihilated by L0^0,L1^0,L-1^0. Within the displayed affine improvement 000073 | class, these vacuum conditions force the subtraction b J_n+b^2 delta_n0/2; 000074 | they are not fitted to a target source. 000075 | 000076 | The derived operators are 000077 | P=L0^0-(L1^0+L-1^0)/2, D=(L1^0-L-1^0)/(2i). 000078 | The full normalized Laguerre basis takes its original one-current P to 000079 | multiplication x/2 in L2(R+,x exp(-x)dx). Completeness and the Carleman 000080 | core give its selfadjoint closure, spectrum[0,infinity), and zero kernel. 000081 | Second quantization on THIS already identified neutral current Fock space 000082 | gives P>=0 and ker P=C Omega. Its explicit unitary dilation is 000083 | V_t f(x)=r exp((1-r)x/2)f(rx), r=exp(-t), 000084 | and V_t P V_t*=exp(-t)P. Direct differentiation and change of variables 000085 | verify the generator, unitarity and complete group law without a finite 000086 | matrix limit guess. 000087 | 000088 | For M=A_J(0,infinity), N=A_J(a,infinity), a>0, the actual vacuum current 000089 | net supplies a standard half-sided inclusion. Its modular action is 000090 | Delta_M^(it)=exp(i2pi tD), 000091 | Delta_M^(it)exp(isP)Delta_M^(-it)=exp(i exp(-2pi t)sP). 000092 | The inclusion side is t<=0 for the source's N, and 000093 | log Delta_N-log Delta_M=2pi aP on the actual common core/closure. 000094 | All signs agree with the explicit dilation. This is a genuine local net 000095 | and standard vector statement, not just a commutator of finite matrices. 000096 | 000097 | The actual original quarter on Q0 is R=exp(-i pi L0^0/2). It carries the 000098 | four parabolic generators AND their moved cuts/algebras/modular groups. 000099 | It does not claim four half-sided preservers of one fixed cut. Opposite 000100 | parabolic averages recover the original neutral circle energy; P is not 000101 | set equal to that gapped circle energy. 000102 | 000103 | PRIMARY APPLICATION CHECK: Tanimoto1807.11723v2 sections3.1–3.2 provides 000104 | the current net, essential selfadjointness and Sugawara representation. 000105 | GLW hep-th/9703129v2 section1, equations3–4 and Lemma1.3 supplies the 000106 | conformal local-net BW/standardity result. These primary passages were 000107 | opened in this review. Their hypotheses fit the specifically identified 000108 | Q0 current vacuum net, not an arbitrary interacting4D source. 000109 | https://arxiv.org/pdf/1807.11723 000110 | https://arxiv.org/pdf/hep-th/9703129 000111 | 000112 | ACTUAL CURRENT REPLAY, 3 November2026: 000113 | /opt/homebrew/bin/python3 -B 000114 | experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000115 | neutral_moebius_source_20261029/check_neutral_moebius.py 000116 | exit0, PASS_SCOPED;48 literal original CAR compensated-stress identities, 000117 | 7 Laguerre recurrences,49 Gram entries. No original verifier main or 000118 | new harness was used; -B prevents cache writes. These checks validate 000119 | the first source identification/recurrences. The all-mode and local-net 000120 | statements were analytically reviewed above, not inferred from the count. 000121 | The unchanged output explicitly retains P1_source_map_closed=false and 000122 | full_TFPT_solution=false. 000123 | 000124 | 3. THE WHOLE G SOURCE TRANSPORT NOW CARRIES ITS VACUUM AS WELL 000125 | 000126 | G REFERENCE-SOURCE-COMPLETION:86–126 proves for the actual fixed full 000127 | carrier U=(E,NE), extended by identity on the remaining original fields, 000128 | Z_adm^U[J_U]=Z_adm[J], Z_ref^U[0]=Z_ref[0], 000129 | Z_rel^U[J_U]=Z_rel[J]. 000130 | Both normal grades, actual reference, all Berezin/Pfaffian orientations, 000131 | eta, bosonic weights and all source legs remain. Hence the normalized 000132 | FULL moments agree as well. No Z_rel[0]=1 replacement was made. 000133 | 000134 | Let Phi denote this induced WHOLE local word * isomorphism, not a 000135 | nonreducing P_F compression. Its actual transported reflection and time 000136 | satisfy Theta_U Phi=Phi Theta and tau_U,t Phi=Phi tau_t. With the original 000137 | graded reflected kernel K, full moment equality gives 000138 | K_U(Phi F,Phi G)=K(F,G). 000139 | Thus 000140 | W[F]_OS=[Phi F]_OS,U 000141 | is well defined and isometric; the inverse full map makes it unitary. 000142 | The constant word is preserved, so 000143 | W Omega=Omega_U, Omega=[1], Omega_U=[1]. (1) 000144 | The original time intertwines on all words, then on the reconstructed 000145 | semigroups/domains. Thus H_U=W H W*, including H>=0 and H Omega=0 when 000146 | the actual vacuum OS hypotheses apply. Reflection/support and operations 000147 | are transported, not replaced by a differently marked clock or collar cut. 000148 | 000149 | For the same represented local standard pair (M,Omega), put 000150 | M_U=W M W*. On the Tomita core M Omega, 000151 | S_U W(A Omega)=S_U(Phi(A)Omega_U) 000152 | =Phi(A)^*Omega_U=W A^*Omega=W S(A Omega). 000153 | Closing this equality and using uniqueness of polar decomposition gives 000154 | Delta_U=W Delta W*, J_U=W J W*. (2) 000155 | Consequently ANY already valid BW relation for this actual pair is 000156 | transported exactly, with its original boost parameter and normalization. 000157 | No independent modular state, modular Hamiltonian or beta is chosen. 000158 | 000159 | This theorem was already recorded in the17September modular audit 000160 | RESULTS:135–154. It is reused. The concrete new APPLICATION is that G 000161 | supplies the full actual source-preserving carrier chart to which it 000162 | applies. One should not continue calling the vacuum/state on that chart 000163 | an additional independent input. It does not establish an isomorphism 000164 | between two different unverified sources or a one-slot compressed reader. 000165 | 000166 | 4. SOURCE COVARIANCE FIXES THE VACUUM REPRESENTATION, NOT AN EXTRA STATE 000167 | 000168 | If the SAME complete source satisfies the original full Euclidean 000169 | covariance and vacuum OS hypotheses, its reconstructed vacuum is the 000170 | constant class. The geometric actions preserve the constant insertion; 000171 | the reconstructed spacetime symmetry therefore fixes Omega. This removes 000172 | an independent post-OS 'choose a rotation-invariant vacuum' step in THAT 000173 | branch. No Gibbs state is inserted to obtain it. 000174 | 000175 | For group elements preserving the positive-time domain and reflection, 000176 | the isometry on classes is direct. General Euclidean rotations mixing 000177 | time and space use the actual OS continuation theorem; they must not all 000178 | be represented naively by [F]->[alpha_g F] on a fixed positive halfspace. 000179 | The constant word lies in the common reconstruction domains and remains 000180 | fixed. Joint_origin supplies the complete OS reconstruction review. 000181 | 000182 | The source-family identity Z_(g.xi)[g.J]=Z_xi[J] transports Omega_xi to 000183 | Omega_(g.xi). It implies invariance in ONE fixed source only when its 000184 | actual backgrounds/reference/marks are invariant under that g. Passive 000185 | carrier transport and physical symmetry are different statements. Root 000186 | is checking this full original S/Z covariance, with its actual fields, 000187 | measure, reference and normalization; this report does not substitute 000188 | a principal-symbol symmetry for that equality. 000189 | 000190 | With the resulting actual Wightman/conformal vacuum net, the applicable 000191 | BW theorem identifies its local modular flow geometrically. BGL1993 000192 | section1 explicitly assumes a covariant positive-energy representation 000193 | with invariant cyclic vacuum, and Theorem2.3 derives the modular identity. 000194 | https://arxiv.org/pdf/funct-an/9302008 000195 | Its original hypotheses are carried by a proved full-source map; a 000196 | commuting finite logodds matrix is not a replacement for that local pair. 000197 | 000198 | 5. ACTUAL DIFFERENT-STATE/TIME TESTS THAT MUST NOT BE OVERWRITTEN 000199 | 000200 | The neutral current theorem also gives an exact SAME-SOURCE diagnostic.