Ordinary Leech Peirce Multiplicities
Abstract
The actual ordinary Golay Leech axis has the full fixed energy-two Peirce multiplicities.
Use the actual admissible Golay integer construction, with real pairing equal to the raw dot product divided by eight. Its coordinate equivalence pulls the raw vector (4,4,0^22) back to a norm-four charge a in an even unimodular positive definite lattice D of rank twenty-four. Let S be the intersection of the actual theta fixedSpace with the eigenvalue-two space of the canonical inverse-Gram Sugawara zero mode. The operator is the actual zero mode L=NormFourVirasoroVector.mode(D,a,1,0).
Theorem 1.1 (The actual invariant restriction, trace and complete decomposition).
Lean statement: D5/S3/VertexAlgebra/Lattice/LeechPeirceMultiplicities.actual_leech_peirce_multiplicities
Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/LeechPeirceMultiplicities.actual_leech_peirce_multiplicities (✓ 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.
The mode preserves all of S. Its restricted full-basis trace is 8215/2. The literal ambient intersections of S with its eigenspaces at 0, 1/16, 1/2 and 2 form an independent family whose sum is S, and have dimensions 49152, 47104, 2323 and 1 respectively. Rank, unimodularity, positive definiteness, the full real pairing, rootlessness and the exact raw-axis identity remain in the conclusion. The numbers, trace and decomposition are conclusions of the constructed example.
The complete ordinary basis consists of symmetric oscillator pairs and sign-orbits of all norm-four charges. The actual quadratic and charged mode identities prove invariance on every basis vector. On a neutral pair (i,j), the diagonal coefficient is (a_i g_i+a_j g_j)/8, including repeated colours. Summing unordered pairs counts each colour rank plus one times; the charge contraction is four. Thus the neutral trace is 25/2. On a charged orbit the diagonal coefficient is B(a,b)^2/16. The reflection partners at pairing plus or minus two lie in distinct sign-orbits. The entire unbounded Golay shell distribution, transported by the coordinate-compatible shell equivalence, gives charged trace 1+4600/4+47104/16=4095.
The actual quartic L(L-1/16)(L-1/2)(L-2)=0 gives four cubic Lagrange projections. Multiplication by L minus the corresponding eigenvalue annihilates each projection; each projection fixes that literal eigenspace. Their sum is the identity and their eigenvalue-weighted sum is L. Projection traces therefore equal eigenspace dimensions. The complete one-sixteenth dimension is 47104, and the complete two-eigenline is the span of the nonzero actual norm-four vector. With total dimension 98580 and trace 8215/2, the remaining dimensions are 2323 and 49152.
The lattice vertex construction follows Bakalov-Kac and Frenkel-Lepowsky-Meurman. The norm-four Ising formula is due to Dong-Mason-Zhu, Theorem 6.3, as recalled by Shimakura, arXiv:1202.6107, formula (1.3). The result concerns the ordinary untwisted fixed energy-two space. It asserts no orbifold extension, Monster representation, fusion completion or string-theoretic realization.