TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/LOCAL-CURRENT-ERASURE-AND-ORDERING.txt Original file SHA-256: 4c3397bb7c0bc75c790b0cca29f5a38abfdff9a564ca5179203ef5a801c554de Inclusive original lines: 4-95 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. 000004 | 1. Question and original-source boundary 000005 | 000006 | Does the already actual native erasure a_a=bar(u_a)(1) itself give a local 000007 | boundary intervention, and what does its complete original current field do? 000008 | No algebra, source coupling, detector or projection is fitted here. 000009 | 000010 | The original P1 source, _archive/tfpt-45/source_extracts/ 000011 | 01_boundary_kernel_source.tex:302–343, defines the relative spectral action 000012 | from supplied D_rel,D_ref,chi,eta and a declared spectral profile. It then 000013 | defines local field insertions by test-function support. Its physical local 000014 | net is restriction to a subspace invariant under A AND A*. It explicitly 000015 | does not define locality through a global compression P_adm A P_adm. 000016 | These definitions do not identify the native five-code with H_adm and do 000017 | not supply a map from primitive D_rel variations to the E8 current fields. 000018 | This calculation tests the actual current-side map, not an absence claim 000019 | about every other part of P1 or the whole TFPT corpus. 000020 | 000021 | The accepted source is the existing E8_1 current realization and the 000022 | literal original J-positive Cartan-Fock embedding, with its existing 000023 | J-equivariant root/Cartan isometry. No root, family omission, R_neut branch 000024 | or improved stress field is needed. These remain premises, not a new raw 000025 | P1/P2 derivation. Native4, quartic5 and actual erasure20 are the existing 000026 | objects, not physical spinor/Higgs assignments or the charged E8 F20 block. 000027 | 000028 | Primary mathematical cross-checks: Tuite–Zuevsky arXiv:1106.6149v2, 000029 | Sec.3, Heisenberg modes and normal ordered descendant fields; and 000030 | Carpi–Kawahigashi–Longo–Weiner arXiv:1503.01260v4, introductory local-net 000031 | definition and Sec.6 on supported smeared fields. They justify the usual 000032 | given-current interpretation, not TFPT's missing physical-source map. 000033 | https://arxiv.org/pdf/1106.6149 000034 | https://arxiv.org/pdf/1503.01260 000035 | 000036 | 2. Fixed source conventions and complete OPE 000037 | 000038 | u_a=(e_(2a)-i e_(2a+1))/sqrt2, a=0,...,3; 000039 | u_a.u_b=0, bar(u_a).u_b=delta_ab. 000040 | [h_v(m),h_w(n)]=m delta_(m+n,0)(v.w) I. 000041 | h_v(n)*=h_bar(v)(-n); h_v(0)=0 on this neutral Fock sector. 000042 | Use original normalized quartics f_i, not a fitted five-dimensional space. 000043 | They obey =delta_ab delta_ij. 000044 | Write F_i=Y(f_i,z), g_ia=partial_a f_i, G_ia=Y(g_ia,z). 000045 | 000046 | The local source current j_bar(u_a)(z)=sum_n bar(u_a)(n) z^(-n-1) 000047 | acts on the actual code as 000048 | 000049 | j_bar(u_a)(z) f_i 000050 | =z^-2 g_ia + sum_(m>=1) z^(m-1) bar(u_a)(-m) f_i. 000051 | 000052 | Positive modes n>=2 kill these mode-one quartics, and the neutral zero mode 000053 | vanishes. Thus there is no simple pole and no higher singular term. Wick's 000054 | identity gives the full field OPE, not just its singular coefficient: 000055 | 000056 | j_bar(u_a)(z) F_i(0) 000057 | =G_ia(0)/z^2 000058 | +sum_(k>=0) z^k/k! :partial^k j_bar(u_a)(0) F_i(0):. 000059 | 000060 | This uses the fixed source product. Its double-pole coefficient is exactly 000061 | the existing erasure20 intertwiner. Its first regular state is exactly 000062 | t_ia=bar(u_a)(-1) f_i. The previously derived primary carry is therefore 000063 | already present in this actual local OPE, rather than an arbitrary newly 000064 | chosen response field: 000065 | 000066 | r_ia=t_ia-(7/85)L_-2 g_ia+(3/170)L_-1^2 g_ia, 000067 | L0 r_ia=5 r_ia, L_m r_ia=0 (m>0), Gram(r)=(78/85) I20. 000068 | 000069 | The denominator85 and numerator78 are fixed by the original c8,h3 000070 | descendant Gram, not matching parameters. The entire OPE is native 000071 | equivariant; its g,t,r maps all carry rho5 tensor conjugate(rho4). 000072 | The earlier carry proof contains the literal event checks. This addendum 000073 | does not repeat those finite checks as new independent evidence. 000074 | 000075 | 3. Local field versus global erasure mode 000076 | 000077 | In the same plane convention, with j_u_b(w)=sum_n u_b(n)w^(-n-1), 000078 | 000079 | [a_a,j_u_b(w)]=delta_ab I, a_a=bar(u_a)(1). 000080 | 000081 | On the circle the covariant weight-one current is 000082 | mathscr(j)_u_b(theta)=e^(i theta) j_u_b(e^(i theta)); its commutator 000083 | is delta_ab e^(i theta) I. For every proper interval I there is a smooth 000084 | test function supported in its open complement whose corresponding 000085 | integral is nonzero. Thus a_a fails a necessary local commutation test in 000086 | every such I. It is not an operator localized/affiliated there in the 000087 | given current net. The statement is on the common finite-energy core; 000088 | one may also use the complementary bounded Weyl operators to witness it. 000089 | 000090 | Extracting the z^-2 coefficient is a global Fourier/contour operation. 000091 | It is mathematically legitimate but not an already derived local P1 000092 | intervention. This clarifies the meaning of 'actual erasure': actual 000093 | native code map, not independently proved local physical operation. 000094 | Global boundary observables and other allowed P1 operations are not 000095 | ruled out by this test.