TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_time_compatibility_20261027/LOCAL-TIME-COMPATIBILITY.txt Original file SHA-256: 25336213bc323f81f8b40866cccc77bdfaf260c04cc430c08825bfc2b3638c4c Inclusive original lines: 1-150 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 | LOCAL E8 COMPLETION, ORIGINAL SOURCE TIME AND CLUSTERING 000002 | 27 October 2026 000003 | 000004 | Verdict: no completion satisfies the four requirements below simultaneously. 000005 | This is a theorem for a fixed comparison class, not a no-go for P1 or TFPT. 000006 | It replaces an unjustified appeal to uniqueness of the whole H ground space 000007 | by a concrete local correlation test. 000008 | 000009 | 1. The exact comparison class 000010 | 000011 | Retain these four requirements together: 000012 | 000013 | C1. The source D8 current embedding and its unitary level-one lattice 000014 | realization are retained. Complete by a local even unimodular lattice 000015 | extension on the same eight Cartan channels. The two possibilities are 000016 | Ls=D8 union(s+D8) and Lc=D8 union(c+D8). 000017 | 000018 | C2. The actual source time on the completed algebra remains 000019 | H=8(L0+delta.a0), delta_j=(j+a)/8-1/2, 0-||^2 tends to zero. 000030 | 000031 | The complete source bridge has not established C2 and the full observable 000032 | part of C3 from P1. C1 and the charge dictionary in C3 are also retained 000033 | comparison premises, not a claim of their primitive physical derivation. 000034 | 000035 | 2. Why the marked extension fails the clustering test 000036 | 000037 | The twice-root condition implies q/2 is a lattice vector. It belongs to Ls 000038 | but not Lc. Thus C1 and C3 select Ls independently of the target frame. 000039 | 000040 | The old all-charge calculation gives exactly two H zero states in Ls: 000041 | 000042 | Omega=|0>, psi=|p*>, p*=(1,1,1,1,-1,-1,-1,-1)/2. 000043 | 000044 | In particular |p*|^2=2, delta.p*=-1, Q(p*)=0, and L0(psi)=1. 000045 | The zero energy of psi is not a second conformal vacuum. 000046 | 000047 | With conventional level-one root modes, 000048 | 000049 | J_(p*),-1 Omega=psi, [H,J_(p*),-1]=0. 000050 | 000051 | Choose a smooth current test function supported in a proper interval whose 000052 | relevant Fourier coefficient is nonzero. The hermitian combination of the 000053 | p* and -p* current smearings is F, with nonzero. 000054 | This is an actual local current in the stipulated E8 affine net. In this 000055 | net the smooth hermitian current is essentially self-adjoint and its 000056 | unitaries X_lambda=exp(i lambda F) belong to the interval algebra. 000057 | 000058 | Since Omega is in the domain of F, 000059 | 000060 | d/dlambda |_(lambda=0)=i != 0. 000061 | 000062 | Some arbitrarily small nonzero lambda therefore gives a BOUNDED local 000063 | observable X with != 0. This removes any dependence on using 000064 | an unbounded field as the test observable. 000065 | 000066 | Let P0 be the orthogonal projection onto ker H. The spectral theorem gives 000067 | strongly exp(-tH) -> P0. Consequently 000068 | 000069 | lim_(t->infinity) [ 000070 | -||^2] 000071 | = ||P0 X Omega||^2-||^2 000072 | = ||^2 > 0. 000073 | 000074 | This contradicts C4. The original Cartan Ward values select Omega among 000075 | normal ground densities, but do not remove psi or this local matrix element. 000076 | 000077 | 3. Why switching chirality does not satisfy the same requirements 000078 | 000079 | In Lc the same H has a unique zero state Omega. Thus exp(-tH) tends strongly 000080 | to |Omega>0, removes the extra zero spectrally. 000090 | P_s is the global s-sector projector. In the fixed irreducible, multiplicity 000091 | one local order-two simple-current extension B of A_D8, an automorphism 000092 | fixing A_D8 pointwise acts on the s intertwiner with a scalar lambda_s. 000093 | The nonzero fusion channel s times s to 0 forces lambda_s^2=1. The relative 000094 | automorphism group is therefore the discrete dual Z2. 000095 | 000096 | For two strongly continuous local time evolutions with the same interval 000097 | motion, the same action on A_D8 and the same vacuum representation, their 000098 | pointwise relative automorphism is a continuous path in this Z2 starting 000099 | at the identity. It is identically the identity. A nonzero sector energy 000100 | penalty cannot furnish another such local time evolution. 000101 | 000102 | For example exp(it epsilon P_s) assigns the phase exp(it epsilon) to 000103 | vacuum-to-s vectors. This cannot obey the local s times s product rule for 000104 | all real t unless epsilon=0. A modified D8 dynamics or a different local 000105 | field construction is outside this statement. 000106 | 000107 | 5. What this settles, and what it leaves as the actual source question 000108 | 000109 | This exact comparison route is closed. Adding the marked local E8 fields 000110 | while leaving the original source time and its neutral vacuum unchanged 000111 | cannot also supply clustering on all those fields. Equal spectra, a chosen 000112 | initial density, or a global energy shift on s do not repair the route. 000113 | 000114 | The already derived even CAR/D8 observable algebra has no such transition: 000115 | every one of its operators preserves the D8 superselection sectors, so 000116 | =0. Neutral total charge is not trivial D8 sector charge. 000117 | 000118 | P1 defines its physical admissible local net by restriction of source 000119 | operators that, together with their adjoints, preserve H_adm. It does not 000120 | define that net by arbitrary global compression, nor has it supplied the 000121 | local spin field or identified its physical transfer with this H. 000122 | 000123 | The next origin calculation must therefore derive which local operators 000124 | are admissible and how the original OS transfer acts on them. In 000125 | particular it must decide whether the E8 charged fields belong to the 000126 | physical observable net with this time, or to an internal/field structure 000127 | with a different, source-derived time relation. A new relation must be 000128 | derived from the original data; it cannot be introduced just to evade the 000129 | test. The geometric E8 and previously proven clock and cover results are 000130 | unaffected in their existing scopes. 000131 | 000132 | 6. Sources and independent check 000133 | 000134 | Original all-charge minima, q/2 obstruction and zero-energy su(2): 000135 | seam-joint-ward-state-20261014/PROOF.txt, sections 2 to 5. 000136 | Primitive P1 seed and its clustering premise: 000137 | archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/ 000138 | 002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex:103-119. 000139 | Actual admissible local net definition: 000140 | same archive, 001-section-primitive-core-and-carrier-interface-handoff.tex: 000141 | 319-343. 000142 | The companion CLUSTERING-ADMISSIBLE-ROOT-FIELD-TEST.txt audits those sources. 000143 | ALL-CHARGE-GROUND-INDEPENDENT-REVIEW.txt independently verifies the spectral 000144 | clustering argument and the discrete-relative-automorphism obstruction. 000145 | 000146 | Standard primary literature for the stipulated affine net and local current 000147 | realization: Carpi, Kawahigashi, Longo and Weiner, From vertex operator 000148 | algebras to conformal nets and back, arXiv:1503.01260v4, Memoirs AMS 254 000149 | (2018), no.1213, https://arxiv.org/abs/1503.01260. 000150 | This existing mathematical construction is not a proof of a P1 scaling limit.