TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/CLUSTERING-ADMISSIBLE-SECTOR.txt Original file SHA-256: 349e5a149d34078da67775e56f25e3aebf2c55111798a86d58322aa61cd380bc 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 | CLUSTERING ON THE ORIGINAL COLOR-CENTER ADMISSIBLE SECTOR 000002 | 28 October 2026 000003 | 000004 | VERDICT 000005 | 000006 | The old second-ground-state obstruction is real on the stipulated full E8 000007 | lattice field module, but it does not survive the color-center selector already 000008 | present in the original full admissibility complex. 000009 | 000010 | For both quarter marks alpha=1/4 and alpha=3/4, the unchanged affine time 000011 | 000012 | H8 = 8 K_alpha, 000013 | K_alpha = L0 + delta_alpha.H0, 000014 | delta_(alpha,j) = (j+alpha)/8 - 1/2, 000015 | 000016 | has exactly the two lattice ground charges 0 and 000017 | 000018 | p_* = (1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2). 000019 | 000020 | The fixed rational Weyl dictionary sends p_* to e1+e4 in the actual first-five 000021 | carrier slots. With the original 3+2 mark this root lies in (3,2)_(1/6), has 000022 | color triality one, and therefore has Z_c eigenvalue omega. Consequently 000023 | 000024 | P_sing p_* = (1+omega+omega^2)p_*/3 = 0, 000025 | P_sing Omega = Omega. 000026 | 000027 | No Hamiltonian correction is used. Since H8 and Z_c are commuting Cartan 000028 | actions, the center-neutral subspace is H8 invariant. On that subspace the 000029 | H8 kernel is the single vacuum line. The exact first positive gaps are 3/2 000030 | for alpha=1/4 and 2 for alpha=3/4. 000031 | 000032 | This removes the earlier ground-doublet argument as an independent obstruction 000033 | to clustering of the later center-neutral observable restriction. It does not 000034 | remove the obstruction on the full graded E8 field net, and it does not by 000035 | itself identify H8 with the physical OS transfer Hamiltonian. 000036 | 000037 | 000038 | 1. WHAT WAS ALREADY KNOWN 000039 | 000040 | The quotient-locality contract proved by coordinatewise convexity that the 000041 | stipulated E8 extension with unchanged K_alpha has precisely two ground 000042 | charges, 0 and p_*. It also checked the fixed rational equality 000043 | 000044 | M p_* = e1+e4 000045 | 000046 | and the vanishing of the transported source U(1) charge on that root. The 000047 | contract explicitly scoped its exclusion to the unchanged charged source and 000048 | left a selected local observable completion open. 000049 | 000050 | The later carrier-action contract then supplied the missing charge type: 000051 | 000052 | E_(e1+e4) = epsilon_1 epsilon_4, 000053 | [Y,E_(e1+e4)] = E_(e1+e4)/6, 000054 | 000055 | and the six roots e_i+e_a, i=1,2,3 and a=4,5, form (3,2)_(1/6). It already 000056 | warned that neutrality under the transported source U(1) is not P2 000057 | hypercharge or color neutrality. 000058 | 000059 | The original full admissibility complex, which was not applied in the old 000060 | full-field ground comparison, defines 000061 | 000062 | P_sing = (1+Z_c+Z_c^2)/3 000063 | 000064 | after the carrier theorem has supplied the color factor. The source calls 000065 | this the color-center singlet projector. It is a center-neutrality projector, 000066 | not the Haar projector onto a full connected SU(3) singlet. 000067 | 000068 | 000069 | 2. THE SMALLEST DECISIVE ACTION TEST 000070 | 000071 | Use the actual ordered carrier frame E=C^3 direct-sum C^2. The scaled 000072 | hypercharge mark is 000073 | 000074 | X=6Y=(-2,-2,-2,3,3,0,0,0). 000075 | 000076 | For q=M p_*=e1+e4, 000077 | 000078 | X.q = 1, Y(q)=1/6, 000079 | Q_source.q = 0. 000080 | 000081 | The SU(3) center exponent in this frame is the coweight pairing 000082 | 000083 | t(q) = q1+q2-2q3 mod 3. 000084 | 000085 | It equals one on e1+e4. Thus Z_c acts by omega. Reversing the convention for 000086 | the fundamental would replace omega by omega^2 and leaves the projector result 000087 | unchanged. The vacuum has t=0 and survives. 000088 | 000089 | On the two-dimensional full ground space the exact actions are therefore 000090 | 000091 | Omega |p_*> 000092 | H8 0 0 000093 | Z_c 1 omega 000094 | P_sing 1 0. 000095 | 000096 | This also explains the observable statement without treating P_sing as a 000097 | global compression. For a center-neutral field A, so that [A,Z_c]=0, 000098 | 000099 | 000100 | = 000101 | = omega^{-1} 000102 | = 0. 000103 | 000104 | The same vanishing follows directly from the original local-net definition for 000105 | any A for which A and A* preserve the center-neutral subspace, because A Omega 000106 | then lies in a subspace orthogonal to p_*. The colored root current may remain 000107 | an intersector field in the graded field net. It is not an operator of the 000108 | restricted center-neutral local observable net because it sends the vacuum out 000109 | of the admissible subspace. 000110 | 000111 | 000112 | 3. COMPLETE ZERO AND GAP CLASSIFICATION 000113 | 000114 | For a lattice charge p and oscillator level N, 000115 | 000116 | K_alpha(p,N) = N + ||p||^2/2 + delta_alpha.p. 000117 | 000118 | Because |delta_(alpha,j)|<1/2, every nonzero integer coordinate contributes 000119 | 000120 | n^2/2 + delta_j n = n(n/2+delta_j) > 0. 000121 | 000122 | Hence the integer D8 coset has the unique minimum p=0. On the half-integral 000123 | coset, strict coordinatewise convexity uniquely chooses +1/2 on the first four 000124 | coordinates and -1/2 on the last four. This is p_*, it satisfies the D8 parity 000125 | condition, and its energy is zero. Positive oscillator level adds at least one 000126 | to K_alpha. These facts prove that 0 and p_* are all lattice ground charges; 000127 | the root census is not being used as a cutoff proof of that statement. 000128 | 000129 | For the first positive center-neutral energy there is a short global bound. 000130 | For both quarters, 000131 | 000132 | ||delta_alpha||^2 = 85/128 < 49/64, 000133 | ||delta_alpha|| < 7/8. 000134 | 000135 | The E8 lattice is even. Every nonzero vector that is not one of its 240 roots 000136 | has norm at least two. Cauchy-Schwarz therefore gives 000137 | 000138 | K_alpha(p,0) 000139 | >= ||p||^2/2 - ||delta_alpha|| ||p|| 000140 | > 2 - 7/4 = 1/4 000141 | 000142 | for every non-root. The exact census of the 112 integer and 128 half-integral 000143 | roots is therefore complete for the gap at or below 1/4. Applying the same 000144 | fixed M and the center coweight to all 240 roots gives 000145 | 000146 | alpha dim ker K dim ker K on P_sing gap K gap H8 000147 | 1/4 2 1 3/16 3/2 000148 | 3/4 2 1 1/4 2. 000149 | 000150 | The checker records the first minimizing charge and its transported carrier 000151 | weight in each case. It also checks all 240 center characters exactly over 000152 | the rationals. 000153 | 000154 | 000155 | 4. WHAT THIS CHANGES 000156 | 000157 | The previous statement 000158 | 000159 | "the same H8 gives a second ground state in the E8 extension" 000160 | 000161 | remains correct for the full lattice field module. The stronger inference 000162 | 000163 | "therefore the physical admissible observable sector has a degenerate 000164 | vacuum and cannot cluster" 000165 | 000166 | does not follow. On the branch where the original P_sing is operative, the 000167 | only extra zero state has nontrivial color triality and is removed. The 000168 | unchanged H8 already has a one-dimensional center-neutral kernel and a positive 000169 | spectral gap. 000170 | 000171 | This is a real scope correction to the old obstruction, not a new source or a 000172 | fitted energy offset. Other parts of the old unchanged-source verdict remain 000173 | separate, including the missing raw P1 field map, the actual quarter-source 000174 | operator realization, and the half-Cartan/current locality questions. 000175 | 000176 | 000177 | 5. PRECISE LIMITS 000178 | 000179 | The theorem proved here is: 000180 | 000181 | On the stipulated E8 lattice extension, with its fixed M dictionary and 000182 | the original later color-center action, ker(H8|Ran P_sing)=C Omega and the 000183 | displayed positive gaps hold. 000184 | 000185 | Four additional identifications are still needed before calling this a 000186 | physical clustering theorem. 000187 | 000188 | 1. P_sing is not part of primitive P1. The original source introduces it only 000189 | after the carrier theorem has identified the color factor. The result is 000190 | therefore about the closed admissible branch, not the primitive source by 000191 | itself. 000192 | 000193 | 2. Intersecting with P_prim and P_Theta cannot recreate p_*, but the present 000194 | calculation does not independently prove that both projectors retain the 000195 | vacuum line. Unique physical vacuum means one-dimensional only after that 000196 | retention is checked. 000197 | 000198 | 3. The original physical local net is defined by restriction to operators for 000199 | which both A and A* preserve H_adm. This calculation identifies the center 000200 | selection on the ground states. It does not construct the full invariant