Monster Fano Reconstruction

Abstract

Seven-point unique pair incidence forces symmetric-difference complement closure.

Theorem 1.1 (Distinct blocks share at most one point).

Lean statement: D5/S3/VertexAlgebra/MonsterFanoReconstruction.block_intersection_le_one

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/MonsterFanoReconstruction.block_intersection_le_one (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Tathagata Basak (2017). The octonions as a twisted group algebra. URL: https://arxiv.org/abs/1702.05705.

Commentary.

Unique pair incidence alone bounds the intersection of two distinct blocks by one point. Two distinct common points would make both blocks witnesses to the same unique block, a contradiction. No block cardinality, additive closure, labeling, or enumeration is used.

This is the first formalized incidence premise in the Fano closure argument for the Monster short-support theory. It does not construct VOA modules, fusion products, conformal weights, or a CFT.

A block system consists of three-element subsets of seven points. Every two distinct points lie in exactly one block. No labeling, additive presentation, intersection rule, or block count is assumed.

Theorem 1.2 (Complement of a symmetric difference is a block).

Lean statement: D5/S3/VertexAlgebra/MonsterFanoReconstruction.complement_symmDiff_mem

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/MonsterFanoReconstruction.complement_symmDiff_mem (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Tathagata Basak (2017). The octonions as a twisted group algebra. URL: https://arxiv.org/abs/1702.05705.

Commentary.

Two different blocks cannot share a pair. Disjoint blocks would leave a single outside point, forcing incompatible blocks through cross-pairs. Thus different blocks meet at exactly one point.

The unique block through the two points outside their union must also contain their common point. It is exactly the complement of their symmetric difference. This is a finite incidence theorem; it does not construct VOA modules, fusion, or conformal spectra.

References

• Truth anchor: D5/S3/VertexAlgebra/MonsterFanoReconstruction.block_intersection_le_one
• Truth anchor: D5/S3/VertexAlgebra/MonsterFanoReconstruction.complement_symmDiff_mem