TFPT scientific source excerpt Original repository path: experiments/theory-contracts/compiler-current-descendant-20260919/PROOF.txt Original file SHA-256: ad45198c44b84365a862b709e8b7e49af708f2e1a58adff67dc55d7cf4fa459b 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 | UR.COMPILER.CURRENT_DESCENDANT.16 000002 | 2026-09-19 — research verdict PARTIAL 000003 | 000004 | Question 000005 | Can the correlated sixty-event process and its continuous interacting lift be 000006 | placed inside the existing A3_1 / D5_1 / E8_1 structure by a genuine operator 000007 | map, with the same phases, instead of equating representations by dimension? 000008 | 000009 | Result 000010 | There is an explicit conditional realization. Bell10=Sym²(C4) embeds as the 000011 | horizontal 10 at affine grade one of the A3_1 module L(Lambda2), with conformal 000012 | weight 3/2. Every one of the sixty phase-cleaned source events and every 000013 | continuous .15 path intertwines. The existing Z4 glue pairs this with the D5 000014 | vector10, giving a (10_D5,10_A3) subspace at weight two in the E8 vacuum module. 000015 | This is a representation and operator theorem under the named current-net 000016 | premises. It does not select the physical seam, prepare that subspace, derive 000017 | the event/record policy or identify the new flow with spacetime evolution. 000018 | 000019 | Conventions and proof layers 000020 | The sixty rays and Bell basis are rebuilt from pinned .13 source_channel.py. 000021 | P_l=z_l z_l*/4, r_l=I-2P_l, u_l=exp(-i*pi/4) r_l in SU4, U_l=Sym²(r_l). 000022 | The one-body determinant-cleaned lift u_l is the one used in contract .12. 000023 | The .15 candidate h_l=I-U_l is on Bell10; its full two-register definition is 000024 | I-r_l tensor r_l. theta is dimensionless, with endpoint tau=pi/2. 000025 | 000026 | The standard free-fermion/current-algebra realization and conformal embedding 000027 | are explicit premises from the named theory, not deductions from 8 finite 000028 | fermion modes. The checker verifies exact finite restrictions with CAR signs. 000029 | The full affine statements below are manual representation-theoretic proofs; 000030 | no Lean, Wolfram, full verification suite or local-net construction is claimed. 000031 | 000032 | 1. A concrete descendant isometry, including the CAR signs 000033 | Use four complex NS fermions in the standard U4_1 realization and factor the 000034 | U1 level-four current. Set 000035 | b_i* = psi^i_(-1/2), a_i* = psi^i_(-3/2), i=0,1,2,3. 000036 | In the ordered fermion convention a_0,...,a_3,b_0,...,b_3, define 000037 | E(e_i tensor e_i) = a_i* b_i* |0>, 000038 | E((e_i tensor e_j+e_j tensor e_i)/sqrt2) 000039 | = (a_i* b_j*+a_j* b_i*)|0>/sqrt2, i=0, (H_l^aff)^2=2 H_l^aff. 000083 | This is a bounded positive operator on the entire named affine module, not 000084 | just a truncated positive matrix. Its spectrum is contained in {0,2}. 000085 | 000086 | On the Bell descendant, E is SU4-equivariant and therefore 000087 | rho_2(u_l) E = -i E U_l, 000088 | V_l E = E U_l, 000089 | H_l^aff E = E h_l. 000090 | Functional calculus gives, for every real theta and every label l, 000091 | exp(-i theta H_l^aff) E = E exp(-i theta h_l). 000092 | The checker verifies the sixty phase-faithful restrictions exactly. Products 000093 | then preserve the full event histories and the .13 endpoint frame actions. 000094 | Algebraic transport of the associated instrument is possible in the same 000095 | corner. Neither an event-label pointer nor its physical preparation follows. 000096 | 000097 | The sign of V_l is fixed here by matching the source endpoint U_l. Replacing 000098 | V_l by -V_l is not excluded by the abstract central-character relation alone; 000099 | it would give the other unitary lift. Thus this construction transports the 000100 | already fixed .15 lift, rather than independently deriving that physical lift. 000101 | 000102 | An equivalent source-compatible U4 representation can be described by choosing 000103 | z^4=det(g) and setting R_2(g)=z² rho_2(z^-1 g). This is independent of the 000104 | fourth-root choice because rho_2(iI)=-I. On a reflection, z=exp(i*pi/4) 000105 | gives R_2(r_l)=i rho_2(u_l)=V_l. This proves the group-composition law, not just 000106 | sixty unrelated unitary assignments. It uses the charge-two extension matching 000107 | the source; other determinant twists are not selected by the SU4 sector alone. 000108 | 000109 | 4. Where the interaction term sits in the current algebra 000110 | Write q_l=P_l-I/4 and Q_l=J_0(q_l). On E(Bell10), 000111 | H_l^aff = 3I/2 + 2Q_l - 2Q_l². 000112 | Indeed on two fundamental factors N_l=P_l tensor I+I tensor P_l=Q_l+I/2, 000113 | so 2N_l(2-N_l)=I-r_l tensor r_l. This is a quadratic element of the collective 000114 | current enveloping algebra; it is outside the linear one-body tangent tested 000115 | in .15. The exact polynomial identity is checked for all sixty directions. 000116 | 000117 | This polynomial is restricted to the specified two-particle descendant image. 000118 | It must not be extrapolated to the unbounded charge spectrum of the whole 000119 | affine tower. The bounded involution formula in section3 is the valid full 000120 | module extension. The negative quadratic term by itself is not a positive 000121 | Sugawara energy functional on arbitrary states. 000122 | 000123 | 5. The E8 placement, with the compulsory companion and grade 000124 | The existing conformal embedding / simple-current extension decomposes the 000125 | E8 vacuum representation into the four paired D5_1 and A3_1 glue sectors. 000126 | A3 class2=L(Lambda2) pairs with the D5 vector10 of weight1/2. Accordingly 000127 | 10_D5 tensor E(Bell10_A3) 000128 | is a 100-dimensional subspace at conformal weight1/2+3/2=2 in the E8 vacuum 000129 | module. The diagonal Z4 glue acts trivially: (-1)*(-1)=+1. 000130 | 000131 | The two tens are different representations: a D5 vector companion and the 000132 | A3 symmetric descendant. No equality Bell10=Lambda²(W5) or W(D5) frame is used. 000133 | The map is I_D5 tensor E. On this subspace a Bell observable can be represented 000134 | by I_D5 tensor E X E*. This is unital only in its projected operator corner, 000135 | not in the full E8 Hilbert space. No arbitrary D5 vector or prepared vacuum is 000136 | silently selected. The subspace is stable under the stated D5 x A3 action, 000137 | not under every E8 current. It is not a new E8 affine primary field. 000138 | 000139 | The projector onto a fixed global energy/representation subspace is not a 000140 | proof of interval-locality. This is not yet an embedding of local observable 000141 | nets, a physical preparation process, or a source-derived readout instrument. 000142 | 000143 | 6. The exact exchange/KZ mean and why it cannot replace the source process 000144 | With tr(t_a t_b)=delta_ab/2, set Omega=sum_a t_a tensor t_a. Fierz gives 000145 | Omega=(Swap-I/4)/2. 000146 | The native second moment gives 000147 | Hbar=(1/60)sum_l h_l=(4I-Swap)/5 000148 | =3I/4-(2/5)Omega. 000149 | The level-one KZ pair residue is R_KZ=2Omega/(k+h^vee)=2Omega/5, so 000150 | Hbar=3I/4-R_KZ. 000151 | This is an operator identity with its sign and scalar shift, not only a match 000152 | of the number5. It is not the full KZ connection, a physical time equation, 000153 | or the WZW Hamiltonian. KZ differentiates in insertion positions; identifying 000154 | those coordinates or their angular continuation with theta is extra data. 000155 | 000156 | Hbar acts by3/5 on the finite symmetric10 and by1 on the antisymmetric6. 000157 | Thus on Bell10 its unitary conjugation is the identity. The actual retained- 000158 | source family from .15 is instead 000159 | C_theta=(1-u)Id+u C60, u=sin²(theta), 000160 | and is nontrivial. Its native-vs-Haar fourth-moment difference is 000161 | C_theta,native - C_theta,Haar = sin²(theta)(C60-CHaar). 000162 | The mean invariant Casimir does not contain this source-specific information. 000163 | Section3 transports the individual labelled paths, so it retains it. 000164 | 000165 | Likewise the usual affine rotation clock L0 has fixed value3/2 on the A3 000166 | image and2 on the glued E8 image. All horizontal zero-mode operations commute 000167 | with L0. The new nontrivial event flow therefore cannot be identified with 000168 | that L0 clock on this sector. This does not exclude other geometric/modular 000169 | flows or a source-induced perturbation, whose action must be supplied. 000170 | 000171 | 7. The retained source cannot be removed by a time-dependent GKLS redefinition 000172 | This is a stronger, path-specific statement than .15's homogeneous embedding 000173 | test. C_theta is invertible since its eigenvalues are1-u+u v with 000174 | v in {1,1/3,1/5,1/15}. Its unique time-local generator is 000175 | L_theta = Cdot_theta C_theta^-1. 000176 | For the same unnormalized Choi convention (trace10), the fixed vector 000177 | w=|0,1>-|1,0> is orthogonal to vec(I), with Choi-projector eigenvalues 000178 | (1/10,-1/10,-1/2,1/2). 000179 | Therefore the conditional-Choi eigenvalue of L_theta is exactly 000180 | -8u(10-7u)sin(2theta) / [5(3-2u)(5-4u)(15-14u)] <0 000181 | for every0