The Actual Level-Six Vacuum Null Vector
Abstract
The production norm-four lattice vector satisfies the level-six vacuum null relation.
Let D have an integral symmetric Gram matrix with even diagonal, let B be its bilinear form, and let a be a charge with B(a,a)=4. The oscillator algebra is the complex multivariate polynomial ring on Fin(rank(D)) times N; the carrier is its finite-support charge direct sum. For s in C with s squared equal to one, use the production vector v=(1/16)single(0,P(0) squared)+(s/4)(single(a,1)+single(-a,1)), where P(r)=sum_i a_i X(i,r). Write L_m=Y(v)_(m+1) and vacuum=single(0,1).
Theorem 1.1 (Level-six vacuum relation in the production carrier).
Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVacuumNull.level_six_vacuum_null
Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourVacuumNull.level_six_vacuum_null (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every such D,a,s, the actual vector 64 L_(-2)(L_(-2)(L_(-2)(vacuum))) +93 L_(-3)(L_(-3)(vacuum)) -264 L_(-4)(L_(-2)(vacuum)) -108 L_(-6)(vacuum) equals zero. No positivity, rootlessness, unimodularity, nondegeneracy, rank restriction or determinant premise is used.
The calculation uses the private production calculus in NormFourVacuumNullCalculus for the actual current modes, charged-field transport and creation series. The creation recurrence determines coefficients through degree six after multiplication by 720. Current annihilation bounds give finite support for both unbounded normal-ordering sums. Transport of the eight required oscillator monomials gives finite charged coefficients, including vanishing same-sign charge channels. Nineteen atom actions and seven descendant actions reduce the final relation to polynomial cancellation separately in charges 0,a,-a. Integral scaling is cancelled only after these actual carrier equalities.
In the finite coefficient notation, 12288 L_(-2) cubed vacuum=3 V222, 2880 L_(-3) squared vacuum=V33, 11520 L_(-4)L_(-2)vacuum=V42, and 96 L_(-6)vacuum=V6. The coefficient identity is 15 V222+31 V33-22 V42-1080 V6=0. Multiplying the asserted relation by 960 reduces it to this identity. Virasoro commutators alone do not supply the zero vector.
The norm-four vector and its classical attribution are those of NormFourVirasoroVector: Shimakura, arXiv:1202.6107, equation (1.3), attributing the Ising-vector assertion to Dong-Mason-Zhu, Theorem 6.3. The imported lattice-field construction follows Bakalov-Kac, arXiv:math/0402315, section 4.1; imported normal-product and Sugawara sources retain their Carnahan and Kytola Apache-2.0 attribution.
This relation establishes a concrete annihilated vacuum vector. Ising simplicity, fusion rules, Monster moonshine, string theory and AdS/CFT remain outside its conclusion.