TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-COLOR-SECTORS-PROOF.txt Original file SHA-256: 9cd6d5a71c47a68af94b4720a2066bbcbd74d18969877c3fe12f63ec787b9df6 Inclusive original lines: 1-198 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 | R069. Execute the original color selector on the whole existing E8 field source. 000002 | 1 November2026, Sitzung037. Scoped source/field result; physical F1 open. 000003 | 000004 | 1. Actual next question and changed premise 000005 | R068 excluded an equivariant fundamental-quark SLOT embedding into the SAME 000006 | center-neutral STATE space. It did not exclude charged field operators. 000007 | Now evaluate the ORIGINAL color-center selector on the already stipulated 000008 | full E8 lattice/current source, instead of guessing another Fock projector. 000009 | Retain the actual D5+A3 target chart, color in its first three carrier slots, 000010 | the old Ward root, the weak coefficient block and every existing Y/Flavor/ 000011 | Alpha, clock, source/state/time/reflection and exterior-memory result. 000012 | 000013 | Use L=E8=D8 union(D8+(1/2)^8), with roots of squared norm2. In this FIXED 000014 | original target frame, a=e1-e2,b=e2-e3 generate COLOR A2. This is not the 000015 | family A2 of a D5-containing E6 GUT cut. The actual original center is 000016 | Zc e^p = exp(2pi i t(p)/3)e^p, t(p)=p1+p2-2p3. 000017 | It fixes the Cartan oscillators. P0=(I+Zc+Zc^2)/3 is the original P_sing 000018 | on this stipulated source branch. We do not identify it with primitive 000019 | P1, physical OS Hilbert space, Pprim/PTheta/Padm, P_F or the96CAR consumer. 000020 | Original local-net definition requires restriction to an A,A*-invariant 000021 | state sector, not replacement of the whole field product by factorwise 000022 | PAP. That precise original distinction is implemented below. 000023 | 000024 | Generic E6+A2 glue and the old native-J obstruction are REUSED results. 000025 | The new step fixes the actual COLOR character and executes its entire 000026 | source/field action, the given Ward insertion and the inherited grading. 000027 | No free group, projector, scalar weight, family omission or Y is chosen. 000028 | 000029 | 2. The whole neutral lattice is fixed, not just a finite root census 000030 | The functional t is integer on ALL of L, including its half-integral coset. 000031 | It is surjective modulo3 since t(e1+e4)=1. Therefore L0=ker(t mod3) has 000032 | index3 in L. Let K be the orthogonal E6 root lattice to a,b. Its roots 000033 | have p1=p2=p3. There are40 integer roots on e4..e8 and32 half roots, 000034 | 72 in total, rank6 and determinant3. The COLOR A2 has6 roots, rank2, 000035 | determinant3. Both lie in L0: t(a)=0 and t(b)=3, not literally0 on b. 000036 | Their orthogonal direct sum has determinant9, hence index3 in E8. 000037 | Equal index and inclusion prove 000038 | L0 = K direct-sum A2_color 000039 | for EVERY lattice vector, not only the norm2 shell. The checker obtains 000040 | the same exact Hermite basis independently from the original roots. 000041 | 000042 | The original full lattice field space is M(1) tensor C[L]. Since Zc 000043 | fixes ALL oscillators, its fixed part is exactly 000044 | V0=V_L^Z3=V_L0=V_E6(perp color) tensor V_A2(color). 000045 | The two other cosets give V1,V2. At conformal weight1 the splitting is 000046 | 248 = (78,1)+(1,8)+(27,3)+(27bar,3bar), 000047 | with sector dimensions86,81,81 including the8 original Cartans in V0. 000048 | The decomposition is not inferred from dimensions alone: projection onto 000049 | the actual color plane gives three fundamental color weights of norm2/3, 000050 | each with the same27 perpendicular weights of norm4/3; the adjoint root 000051 | actions and lattice vertex products preserve these weights. The ground 000052 | weights of the two modules are2/3+1/3=1, not1/2. 000053 | 000054 | This color-centralizing E6 does NOT contain the original first-five D5: 000055 | their intersection has only4 roots, not40. The original24 weak mediator 000056 | roots +-e_a+-e_b, a=4,5 and b=6,7,8, are all in this E6. The creation 000057 | weak12 and annihilation weak12 are distinct prescribed components of the 000058 | same existing(10,6) block. The original bar5 coefficients t3,t4 have 000059 | Y=-1/2, their conjugate H hasY=+1/2. No particular family6 direction, 000060 | 24 physical Higgs fields, new Yukawa or Higgs measure is selected here. 000061 | e4+e6 is only a component witness, not a new physical H choice. 000062 | 000063 | The character is an ALL-MODE lattice/oscillator identity. Complete finite 000064 | coefficient controls give grades0..3: 000065 | V0: 1,86,1370,11586; 000066 | V1: 0,81,1377,11583; V2: the same; 000067 | sum:1,248,4124,34752, matching the given whole E8 source. 000068 | These finite coefficients check the identity; its proof is the full 000069 | character action on M(1) tensor C[L], not a truncation argument. 000070 | 000071 | 3. Three meanings of neutrality must remain different 000072 | P0 imposes Z3 triality only. It RETAINS all8 color-adjoint currents. 000073 | The full global SU3-invariant algebra has no color-adjoint currents at 000074 | weight1, but retains the COLOR Sugawara stress tensor at weight2. 000075 | The affine color commutant is the separate E6 current/lattice factor. 000076 | Thus center fixed point, global SU3 fixed point and affine commutant 000077 | are three different objects. The center and global fixed points retain 000078 | the full original conformal vector of central charge8. The affine 000079 | commutant has the complementary stress tensor of central charge6. 000080 | A global Haar restriction does not lower the full stress central charge 000081 | to below6. No full color-singlet Gauss law is imposed by this calculation. 000082 | 000083 | 4. The original charged field has its actual intersector action 000084 | For every homogeneous lattice field Y_q and every allowed coefficient, 000085 | Pr Y_q P_s=0 unless r=s+t(q) mod3. 000086 | Oscillator descendants have the same character. Products carry the sum 000087 | of their field charges, irrespective of their intermediate charges. 000088 | Hence a balanced word maps V0 into V0 and its adjoint does too. This is 000089 | an exact grading statement, not a dynamically selected history or a new 000090 | time law. All original OPE coefficients, cocycles and intermediate paths 000091 | remain those of the full E8 source; P0 has not been inserted between fields. 000092 | 000093 | The ALREADY GIVEN Ward representative lambdaA=e3+e6 has t=-2=1 mod3. 000094 | It therefore takes V0 to V1; its adjoint takes V1 back to V0. A weak 000095 | mediator component e4+e6 has t0 and preserves the triality sector. 000096 | These are actual marked field actions, not identifications of all81 000097 | module vectors with physical quarks. The known neutral wedge/Yukawa 000098 | charge combination 3 times1 times3bar is preserved. Charge consistency 000099 | alone is NOT a new derivation of the original Yukawas or of a family 000100 | matrix from a single E8 structure constant. 000101 | 000102 | A minimal exact level1 normalization control uses v=J_lambdaA,-1 Omega, 000103 | with the existing unit-normalized lattice root state. Its norm is1: 000104 | =1. 000105 | Endpoint restriction P0 retains this balanced word. Inserting P0 after 000106 | EACH charged factor kills it since P0 v=0. A bare opposite-root vertex 000107 | mode can include the given cocycle phase; the displayed positive1 is 000108 | explicitly an adjoint norm, not an arbitrary bare-cocycle convention. 000109 | This vacuum is a normalization control, NOT a replacement of the original 000110 | E0/quarter/conditional or coupled state. The sector identities hold as 000111 | operator/field statements independently of that control state. 000112 | 000113 | There are also balanced three-charged root words. For the existing 000114 | lambdaA, beta=e2-e6 and gamma=-e2-e3 all have triality1, sum0 and 000115 | lambdaA+beta is a root. The original bracket amplitude has magnitude1 000116 | because the root string has length1. Its PHASE remains the original 000117 | cocycle phase. This is a coefficient control, not a new physically 000118 | selected sequence or a new E6 cubic derivation. 000119 | 000120 | 5. Actual local observable, without global Fourier-mode localization 000121 | Work in the GIVEN unitary E8_1 local current net. For a smooth test f 000122 | supported inside a proper interval I, take the closure 000123 | A=closure(J_lambdaA(f)). 000124 | Affine current energy bounds on the finite-energy core give this closed, 000125 | densely defined operator affiliated with the original A_E8(I). Its center 000126 | character is zeta, with its entire domain transported by Zc. Therefore 000127 | B=A* A and R=(I+B)^-1 000128 | are center-neutral; R is bounded, selfadjoint and belongs to A_E8(I)^Z3. 000129 | R and its adjoint preserve H0=closure(V0). Its restriction is consequently 000130 | a genuine local bounded observable of the CENTER-retained source branch. 000131 | No particular f is selected as a physical intervention by this existence 000132 | statement; the result holds for the whole specified supported test class. 000133 | It neither executes R as an instrument nor chooses a new source state. 000134 | A global Fourier coefficient J_-1 is not localized in I. 000135 | 000136 | For the strongly local affine source, CKLW Example8.7 supplies its current 000137 | net and Proposition7.6 identifies the fixed-point net with the VOA fixed 000138 | point. Thus, on its own invariant vacuum sector, this retained local net 000139 | is the existing E6_1 tensor SU3_1 net. The construction preserves actual 000140 | interval support. This is a conditional local-source realization, not an 000141 | OS reconstruction of physical3+1 space or identification of all Padm factors. 000142 | 000143 | 6. Test the first attempted fermion identification before extending it 000144 | The intersector root CURRENT fields above are bosonic E8 fields. Their 000145 | triality is not fermion parity. This cannot be fixed by merely calling 000146 | V1,V2 odd. On L0* / L0=Z3 x Z3 the inherited conformal quadratic form is 000147 | q(k,l)=2k^2/3+l^2/3 mod1. 000148 | All9 sectors have q0 five times,q1/3 twice,q2/3 twice. NONE has q1/2. 000149 | The actual original E8 glue is an isotropic diagonal order3 subgroup; 000150 | its two nontrivial modules have total lowest weight1 as already shown. 000151 | 000152 | In the precise class of integral lattice local superextensions containing 000153 | L0 with its SAME conformal vector, no odd lattice extension is possible. 000154 | Indeed det L0=9. If L' is an integral finite-index overlattice, its index 000155 | d has d^2 dividing9, hence d1 or3. If L' were odd, its even-norm kernel 000156 | L'_even would have index2 and contain L0, forcing [L':L0] even, a 000157 | contradiction. Equivalently none of the discriminant sectors provides a 000158 | half-integer conformal weight for such an odd local lattice field. 000159 | This is a whole-class exclusion, not a search over a few trial fields. 000160 | 000161 | It does NOT prohibit physical3+1 fermions, primitive CAR, nonlocal field 000162 | intertwiners, another originally justified graded source/cover, or a 000163 | non-point-local physical dictionary. It does not select an extra free 000164 | fermion sector. The original graded source, domains, actual quarterfold 000165 | and physical localization/statistics must supply the missing map. A 000166 | chiral conformal weight is not an internal Spin10 index or physical 000167 | Lorentz spin. The prior NS-vector/local-composite exclusion remains in 000168 | its own scope; no new full-source/TFPT no-go is inferred. 000169 | 000170 | 7. Full theory, clocks and the next actual arrow 000171 | The old all-E6/A2 native-J obstruction is retained: J^2=-I cannot preserve 000172 | these root lattices because its action on the Z3 discriminant would 000173 | square to identity while -I acts by nontrivial inversion. Thus this is 000174 | a FIXED COLOR retained branch, not a native-J closed clock dynamics. 000175 | Transport the original mark/selector and ALL source fields if the actual 000176 | clock requires that action. NativeJ, actual charged source quarterfold, 000177 | Cartan H8 and physical time are distinct existing objects. The known 000178 | [H8,Zc]=0 survives, with no new physical time claim. 000179 | 000180 | The original color selector has now been executed on the full stipulated 000181 | source: local invariant algebra, charged field transitions, weak mediator 000182 | and complete balanced words are explicit. The next actual comparison is 000183 | the original GRADED/word/field map to the existing96 matter consumer and 000184 | C_Sigma/P_F or Z_adm, with support, adjoints, phase, representation, 000185 | state/time/domain/ref and H before/after integration retained. Use neutral 000186 | balanced two/four/higher words as the comparison, not a charged-state 000187 | embedding or a bosonic-current=CAR declaration. The actual original 000188 | fermionic/statistics map is decisive before any new coefficient search. 000189 | Carry R067/R068/R054/R063 jointly when that map supplies their fields. 000190 | No new Y/H/scale/geometry/projector scan; no discarding charged paths. 000191 | Physical localization, common source measure,3+1 and spin2 remain F1. 000192 | No new Lean build, ledger promotion, physical validation or final closure. 000193 | 000194 | Primary local-net method: Carpi,Kawahigashi,Longo,Weiner, arXiv1503.01260v4, 000195 | Example8.7 and Proposition7.6, https://arxiv.org/pdf/1503.01260 . The actual 000196 | marked lattice/kernel calculation and elementary no-odd-overlattice proof 000197 | are explicit above. Fixed-point-free lattice-isometry orbifold theorems 000198 | are NOT applied to this different inner-character automorphism.