TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/session_coverage/supplements/PRIMITIVE-DIRECT-RECONSTRUCTION.txt Original file SHA-256: 308e8e741e1999dcedf422ca1b063150b1da00f83ced7221851565c4f49653f9 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 | P1/P2 FUNDAMENTAL RESTART: DIRECT CYCLIC RECONSTRUCTION WITHOUT A FREE CAR ORIGIN 000002 | ================================================================================== 000003 | 26 October 2026 000004 | 000005 | Verdict 000006 | ------- 000007 | 000008 | There is a real reduction of the origin problem. 000009 | 000010 | 1. A full free CAR source is not needed to reconstruct the P1 Hilbert space, 000011 | state and positive collar time. The stronger operational seed already 000012 | contains a local Euclidean algebra, its time translations, reflection and 000013 | reflection positive state. The Osterwalder Schrader quotient reconstructs 000014 | the cyclic Hilbert space and its positive time semigroup directly. 000015 | 000016 | 2. A global unitary from a preselected Fock space onto the whole P1 Hilbert 000017 | space is not needed. A state, reflection and time preserving local star 000018 | homomorphism on the cyclic subalgebra generated by the marked E8 fields is 000019 | sufficient. Equality of its OS word kernel then produces the unitary 000020 | intertwiner on that cyclic closure automatically. 000021 | 000022 | 3. A global Cartan basis and five physical CAR modes are not needed to obtain 000023 | the internal P2 packet. For the already derived rank five carrier 000024 | E=E3 direct sum E2, the half spin module Lambda_even(E) and the Clifford 000025 | action are basis free consequences of the natural E plus E* pairing. 000026 | This does not make those five internal slots physical spacetime fermions. 000027 | 000028 | The earliest still missing operator is therefore much smaller than the former 000029 | "P1 equals a complete free CAR theory" demand. It is one localized map of the 000030 | marked extension generator, together with the already fixed base current net, 000031 | into the actual operational seed algebra. The original source does not yet 000032 | write this map. Its primitive spectral action also does not define a field 000033 | dependent D_rel[Phi] from which the map could be differentiated. 000034 | 000035 | This is a scoped origin advance. It neither identifies the present NS cover 000036 | Green function with P1 nor derives a physical E8 field from c3 alone. 000037 | 000038 | 000039 | 1. Source layers that must not be collapsed 000040 | -------------------------------------------- 000041 | 000042 | The original text contains two materially different objects. 000043 | 000044 | The numerical and spectral boundary primitive kernel is 000045 | 000046 | Tker^partial = 000047 | (A,H,D,J,Gamma,tau_double,iC,P_prim,[u_Sigma],c3), 000048 | [u_Sigma]=1, c3=1/(8 pi). 000049 | 000050 | See 000051 | 000052 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000053 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex, 000054 | lines 335 to 354. 000055 | 000056 | Earlier in the same source, however, the operational seed is already 000057 | 000058 | S = (A_loc,tau_t,Theta,omega,[u_Sigma],D_coll). 000059 | 000060 | Its definition says that A_loc is a quasilocal Euclidean observable net, 000061 | omega is reflection positive and clustering, tau_t is the time action and 000062 | D_coll is a reflection regular first order collar generator. See the same 000063 | file, lines 103 to 121. The operational collar completion then runs in the 000064 | forward direction 000065 | 000066 | C_seed : Seed_op -> SBord_adm, 000067 | 000068 | and its proof architecture explicitly invokes OS reconstruction, lines 123 to 000069 | 140. Physical operationality is a separate axiom at lines 245 to 248. 000070 | 000071 | This changes the correct logical question. Starting from the operational 000072 | seed, one need not derive an algebra, state and time from c3 or from a scalar 000073 | Calderon kernel. Those data are inputs. Starting only from the reduced 000074 | primitive boundary labels, one cannot reverse C_seed unless an inverse or a 000075 | faithfulness theorem is proved. The source states neither. 000076 | 000077 | The local insertion definition confirms the distinction. It names a 000078 | Euclidean source algebra A_E(O) generated by local field insertions, but it 000079 | does not name those generators or relate them to the P2 carrier or the marked 000080 | C4 sectors: 000081 | 000082 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000083 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex, 000084 | lines 303 to 343. 000085 | 000086 | Consequently the direct route is available from the strong operational seed, 000087 | while the claim that the small primitive labels determine that seed remains 000088 | unproved. 000089 | 000090 | 000091 | 2. The first genuine P1 operator relation 000092 | ------------------------------------------ 000093 | 000094 | Let A_+ be the positive time part of the actual operational seed algebra. In 000095 | the source convention the OS form is 000096 | 000097 | <[F],[G]>_OS = omega(Theta(F) G), 000098 | 000099 | with the adjoint included in Theta as declared there. In the convention in 000100 | which reflection and adjoint are written separately it is 000101 | 000102 | <[F],[G]>_OS = omega(Theta(F*) G). (2.1) 000103 | 000104 | Reflection positivity says that this is positive semidefinite. Quotient its 000105 | null space N and complete: 000106 | 000107 | H_OS = closure(A_+ / N), Omega = [1]. (2.2) 000108 | 000109 | The actual Euclidean translation has the operator relation 000110 | 000111 | T_s [F] = [tau_s(F)], T_(s+t)=T_s T_t. (2.3) 000112 | 000113 | Under the regularity and null stability conditions used in OS reconstruction, 000114 | T_s is a strongly continuous symmetric contraction semigroup and hence 000115 | 000116 | T_s = exp(-s H_OS), H_OS >= 0. (2.4) 000117 | 000118 | Equations (2.1) to (2.4) are the first operator relations furnished by the 000119 | P1 operational seed itself. They are independent of a CAR presentation, 000120 | quasifree state, Cartan basis or global Fock vacuum. 000121 | 000122 | This is precisely the type of theorem proved for a Euclidean C star net by 000123 | Schlingemann, From euclidean field theory to quantum field theory, 000124 | arXiv:hep-th/9802035. Lemma 3.1 constructs the contraction semigroup and its 000125 | positive generator from a regular reflection positive functional. Theorem 000126 | 3.4 reconstructs positive energy Poincare covariance under the stronger 000127 | Euclidean covariance hypotheses. Theorem 3.7 reconstructs the local net when 000128 | the time zero condition is available. The original Osterwalder Schrader 000129 | sources are Commun. Math. Phys. 31 (1973) 83 to 112 and 42 (1975) 281 to 305. 000130 | The distinction matters here: the present operational tuple is enough for the 000131 | collar Hilbert space and time semigroup asserted by its own theorem, while a 000132 | complete Minkowski Haag Kastler net still requires the stated covariance, 000133 | regularity and time zero hypotheses. 000134 | 000135 | The new whole NS orientation cover calculation gives a concrete realization 000136 | of (2.1), not a replacement for it. Its exact factorization 000137 | 000138 | K(t,s) = B_t* B_s 000139 | 000140 | identifies the OS quotient with the closure of the ranges of B_t and proves 000141 | positivity for both charge blocks and all transverse modes. It still does 000142 | not say that its free Dirac field is the local insertion algebra of P1. 000143 | 000144 | 000145 | 3. The cyclic embedding lemma removes the global unitary demand 000146 | --------------------------------------------------------------- 000147 | 000148 | Let B_8 be the positive time word algebra of the already reconstructed marked 000149 | E8 vacuum source in the exact Ext_H(U) class, with state omega_8, reflection 000150 | Theta_8 and Euclidean time alpha_s. The current Atlas finding 000151 | 000152 | finding:marked-c4-glue-unique-local-e8-vacuum-source 000153 | 000154 | records that this exact class has one local unitary VOA and one normalized 000155 | local Q system class, represented by V_E8. Its products, vacuum, conformal 000156 | vector and local correlation functions are fixed inside that class. The 000157 | same finding also keeps explicit that primitive P1 has not been proved to 000158 | supply the class. Primary source in the repository: 000159 | 000160 | experiments/theory-contracts/e-series-source-reconstruction-20261024/ 000161 | THEOREMS.txt, lines 1 to 196 and 503 to 517. 000162 | 000163 | Suppose there is a localized star homomorphism on the needed cyclic algebra 000164 | 000165 | iota : B_8 -> A_loc (3.1) 000166 | 000167 | such that for all positive time words X,Y 000168 | 000169 | iota(XY)=iota(X)iota(Y), iota(X*)=iota(X)*, (3.2) 000170 | iota(Theta_8 X)=Theta iota(X), (3.3) 000171 | iota(alpha_s X)=tau_s iota(X), (3.4) 000172 | omega(iota(X))=omega_8(X). (3.5) 000173 | 000174 | Then 000175 | 000176 | V_0 : [X]_8 -> [iota(X)]_P1 (3.6) 000177 | 000178 | is well defined and isometric, because (3.2), (3.3) and (3.5) give 000179 | 000180 | _P1 000181 | = omega(Theta(iota(X)) iota(Y)) 000182 | = omega_8(Theta_8(X) Y) 000183 | = <[X],[Y]>_8. (3.7) 000184 | 000185 | Thus null vectors map to null vectors. Completion extends V_0 uniquely to a 000186 | unitary from the E8 cyclic Hilbert space onto the closure of iota(B_8)Omega. 000187 | Equation (3.4) gives 000188 | 000189 | V exp(-sL_0) = T_s V (3.8) 000190 | 000191 | on this cyclic subspace. No independently selected global unitary is an 000192 | input. Surjectivity onto the whole P1 Hilbert space is unnecessary unless 000193 | one additionally claims that the E8 fields exhaust every P1 observable. 000194 | 000195 | This also avoids demanding a free CAR origin. CAR is one realization of the 000196 | target chiral net. The actual bridge can be stated entirely in terms of local 000197 | products, adjoints, reflection, state and time. 000198 | 000199 | 000200 | 4. The smallest decisive origin test