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The Ordinary Leech One-Sixteenth Multiplicity

Abstract

The complete ordinary fixed energy-two one-sixteenth eigenspace is counted by the actual positive pairing-one shell fiber, giving 47104 for an explicit Golay Leech axis.

Let D be positive definite even integral charge data with no norm-two charge, and let B(a,a)=4. Write S for the literal intersection of fixedSpace(D) with eigenvalue two of sugawaraMode(D,inverse gramComplex(D),0), and L for the actual mode(D,a,1,0). The eigenspace considered here is the intersection of S with the ambient eigenvalue-one-sixteenth space of L. All oscillator and charge sectors of S are included.

Theorem 1.1 (A whole-space dimension formula).

Lean statement: D5/S3/VertexAlgebra/Lattice/LeechPeirceDimension.actual_one_sixteenth_finrank

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/LeechPeirceDimension.actual_one_sixteenth_finrank (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Acknowledgement. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

For every such D and a, the complex finrank of S intersect eigenspace(L,1/16) equals Nat.card of the entire norm-four charge fiber B(a,b)=1. No count or finite-model hypothesis occurs in this formula.

The complete actualFixedEnergyTwoBasis consists of neutral unordered colour pairs and norm-four sign orbits, whose charged vector is E(b)=single(b,1)+single(-b,1). The character chi fixes neutral basis vectors and multiplies E(b) by the parity sign of B(a,b). For odd pairing, b cannot equal a or -a. The rootless norm-four bound then forces B(a,b)=1 or -1, and the actual middle action gives L E(b)=E(b)/16. The odd-orbit vectors are linearly independent as a subfamily of the full basis.

For completeness, set P=(I-chi)/2 on S, with values in the actual carrier. Its values on the full basis are zero on even vectors and the original vector on odd ones; therefore its entire image lies in the odd-orbit span. The existing whole-space spectral-sign identity chi=I-(8192/217)L(L-I/2)(L-2I) evaluates to chi(v)=-v whenever L(v)=v/16. Such a v equals P(v), so every vector of the full one-sixteenth eigenspace lies in that span. Conversely the middle action places every odd basis vector in the eigenspace. Each odd sign orbit has exactly one representative with positive pairing one, proving the dimension formula.

Theorem 1.2 (The explicit actual Golay Leech axis has multiplicity 47104).

Lean statement: D5/S3/VertexAlgebra/Lattice/LeechPeirceDimension.actual_leech_one_sixteenth_dimension

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/LeechPeirceDimension.actual_leech_one_sixteenth_dimension (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Acknowledgement. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

There exist actual positive even unimodular rank-twenty-four data D, an integral linear equivalence e to the real Golay Leech lattice preserving every pairing, and a norm-four charge a whose raw real coordinates are (4,4,0,…,0), such that the literal intersection fixedSpace(D) intersect eigenspace(sugawaraMode(D,inverse gramComplex(D),0),2) intersect eigenspace(mode(D,a,1,0),1/16) has complex dimension 47104. The theorem also retains rootlessness and the exact real coordinate identity.

The complete coordinate-compatible shell equivalence from LeechLowSpectrum is used to pull back the sphere point supplied by LeechMinimalCount.sphereEquiv: support {0,1}, with both signs zero. Its literal raw vector is single(0,4)+single(1,4). The real lattice stores these raw cast coordinates and pairs by dot product divided by eight; equivalently its standard Euclidean coordinates are divided by square root of eight. Full pairing compatibility identifies B(a,b)=1 with dot(raw(a),raw(b))=8 on every admissible square-sum-thirty-two integer vector. LeechAxisDistribution.axis_distribution counts this entire unbounded fiber as 47104. No transitivity, assumed incidence, selected finite shell or inference from total shell size is used.

The lattice Fock construction follows Bakalov-Kac and Frenkel-Lepowsky-Meurman. The norm-four Virasoro formula retains the Dong-Mason-Zhu attribution described in the imported NormFourVirasoroVector source. The statement concerns the ordinary theta-fixed space and its actual zero mode. It asserts no twisted-sector completion, reflection orbifold, Monster action or string-theoretic identification.

References