TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt Original file SHA-256: 7c0cd92dfc5eca88260baae0af9edad66f958a27ce0d15d7965414b3971f8da0 Inclusive original lines: 17-171 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. 000017 | We retain the explicitly declared eightfold charged-CAR fold, its Cartan 000018 | frame, source charge q=(1,1,1,1,1,1,1,1), and source time 000019 | 000020 | K(p,N)=||p||^2/2 + delta.p + N, 000021 | delta_k=(k+alpha)/8 - 1/2, k=0,...,7, N>=0. 000022 | 000023 | N in this equation is oscillator level, not the source U(1) charge. 000024 | Physical quarter cases are alpha=1/4,3/4. Our time convention is the previous 000025 | ordered-plane convention; reversing the charge/time dictionary consistently 000026 | does not change ground dimensions or lattice divisibility. 000027 | 000028 | The original marked packet supplied an exact, but conditional, dictionary 000029 | 000030 | q -> q'=2(e3+e6), 000031 | 000032 | under the previously checked rational E8 Weyl isometry M. The target half 000033 | q'/2 is a genuine E8 root. This dictionary is an assumption to preserve, 000034 | not a derivation of the raw source map Xi. Source originals and ledger retain 000035 | their separate scopes. 000036 | 000037 | 2. Complete lattice classification in this comparison class 000038 | 000039 | Define D8={n in Z^8 : sum n_i even}, s=(1/2)^8 and 000040 | c=(-1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2). The two positive even unimodular 000041 | overlattices containing these original D8 roots are 000042 | 000043 | Ls=D8 union(s+D8), Lc=D8 union(c+D8). 000044 | 000045 | D8 has determinant4; a unimodular overlattice must have index2. Its three 000046 | nonzero discriminant classes are vector v (norm1) and the two spinor 000047 | classes s,c (norm2). The vector class cannot generate an even extension; 000048 | the two spinor classes are the two possibilities. Each is an E8 lattice. 000049 | On the full half-integer cosets it is the parity of sum p_i that defines 000050 | chirality. Counting minus signs is valid only on the norm2 root shell. 000051 | 000052 | The classification does not cover an arbitrarily changed source algebra, 000053 | non-lattice extensions or different stress/time laws. 000054 | 000055 | 3. The full-lattice state calculation, not merely a finite scan 000056 | 000057 | For 0|delta_k|. Other coordinates contribute nonnegative increments. 000075 | This gives the full, unbounded-lattice result 000076 | 000077 | min_(p in c+D8) K(p,0)=min_k |delta_k| 000078 | =min(alpha,1-alpha)/8 >0. 000079 | 000080 | The minimum is unique except at alpha=1/2, where two excited spinor states 000081 | tie. The global Lc vacuum is still unique. In either quarter case the 000082 | lowest spinor energy is 1/32, or 1/4 in the source normalization H=8K. 000083 | This matches a possible gap scale; matching a scale does not identify fields. 000084 | 000085 | 4. Charge divisibility forbids the proposed simultaneous repair 000086 | 000087 | q/2=s is a norm2 vector in Ls but is not a vector in Lc. If a lattice 000088 | isometry U:Lc -> canonical E8 satisfied Uq=q'=2r for a target root r, 000089 | linearity and bijectivity would imply q/2=U^{-1}r in Lc. Contradiction. 000090 | 000091 | Therefore the tested twice-root Ward marking selects Ls, whereas preserving 000092 | a unique original vacuum under the declared K requires Lc. No extension 000093 | in this comparison class satisfies both. This obstruction is invariant 000094 | under a change of frame; it does not concern just the old matrix M. 000095 | 000096 | Independent full root charge histograms give another usable fingerprint: 000097 | 000098 | Ls: Q=-4,-2,0,2,4 with multiplicities 1,56,126,56,1; 000099 | Lc: Q=-3,-2,-1,0,1,2,3 with multiplicities 8,28,56,56,56,28,8. 000100 | 000101 | All Ls lattice charges q.p are even; Lc spinor-coset charges are odd. 000102 | If exp(i*pi Q) remains the physical CAR statistics grading, an h=1 000103 | bosonic Lc current has incompatible odd grading. This additional statement 000104 | requires physical statistics preservation. An internal Z2 automorphism 000105 | may act oddly on a bosonic current and is not itself contradictory. 000106 | 000107 | For a different triality-related frame Lambda containing transported actual 000108 | D8 bilinear currents, pulling Lambda back along their isometric operator 000109 | dictionary produces an overlattice of the original D8. The same two-case 000110 | classification applies. Merely rotating a lattice while discarding actual 000111 | bilinear currents is a changed source assumption, not a rescue by coordinates. 000112 | 000113 | 5. The necessary neutral memory algebra within the stipulated extension 000114 | 000115 | In the unitary level1 lattice-current theory of Ls, fix root phases so that 000116 | J_p(z) J_-p(w) has double pole 1 and simple pole p.J(w) for ||p||^2=2. 000117 | The affine modes obey 000118 | 000119 | [J_p,m, J_-p,n]=H_p,m+n + m delta_(m+n,0) I. 000120 | 000121 | Set p=p*, A=J_p,-1, B=J_-p,1=A^dagger, Z=H_p,0-I. Then 000122 | 000123 | [K,A]=[K,B]=0, [A,B]=Z, [Z,A]=2A, [Z,B]=-2B. 000124 | 000125 | On the two ground vectors |0>,|p*> the lattice OPE gives 000126 | 000127 | A|0>=|p*>, A|p*>=0, B|p*>=|0>, B|0>=0, 000128 | Z|0>=-|0>, Z|p*>=|p*>. 000129 | 000130 | These are the spectrally-flowed horizontal modes of affine SU(2) at level1, 000131 | not its ordinary J_0 horizontal modes. Their zero-mode algebra acts as 000132 | the canonical spin1/2 doublet. 000133 | It supplies a concrete necessary two-state information carrier if this 000134 | marked Ls extension is to be the source. This algebra is derived inside 000135 | the stipulated lattice-current extension; none of A,B is thereby supplied 000136 | as an actual operator of the unchanged microscopic CAR source. 000137 | 000138 | q.p*=0 and mu.p*=-1, mu_k=(k+alpha)/8. Hence source U(1), the internal 000139 | fold clock g=exp(2*pi*i mu.H), its square and deck g^4 act trivially on 000140 | both ground vectors and commute with A,B there. Projecting to invariants 000141 | of these particular groups retains both. Projecting this particular SU2 000142 | doublet to singlets would retain neither; no general statement about all 000143 | gauging prescriptions follows. 000144 | 000145 | For every uniform delta -> delta+eta q, |0> and |p*> still have energy0. 000146 | If the original vacuum remains a ground state, the extra degeneracy remains. 000147 | For unrestricted eta some other states can have negative energy; in that 000148 | case the original vacuum has been lost, not rendered uniquely minimal. 000149 | K restricts to zero on the doublet, so its unitary time evolution alone 000150 | does not supply a nontrivial event law on this necessary memory. 000151 | 000152 | There is a positive distinction between selecting an initial density and 000153 | requiring a one-dimensional Hamiltonian ground eigenspace. In this specified 000154 | multiplicity-one representation, any normal state supported on the whole 000155 | ground doublet has density rho=[[a,c],[conjugate(c),b]], a,b>=0, a+b=1, 000156 | |c|^2<=ab. The source-normal-ordered Cartan current H_p*,0 restricts to 000157 | diag(0,2). Preserving the full source values =0 gives 000158 | 000159 | Tr(rho H_p*,0)=2b=0. 000160 | 000161 | Hence b=0; positivity forces c=0, and a=1. The original full Cartan Ward 000162 | data therefore select the initial vacuum density |0><0| uniquely within 000163 | this normal ground-state class. Source total charge, clock and deck alone 000164 | cannot do so, since they act trivially on both ground vectors. 000165 | 000166 | This fixes the conditional initial state without a further arbitrary 000167 | ground-doublet mixture. It does not remove the second zero eigenvector, 000168 | restore the original spectral gap, derive the new source operators or 000169 | generate events. It is not a uniqueness theorem for states on arbitrary 000170 | local/type-III extensions or other representations. Normal ordering, the 000171 | specified representation and ground support are retained premises.