# Singer's Theorem — Lean 4 Formalization

A complete, zero-sorry Lean 4 formalization of Singer's theorem on cyclic
difference sets and Sidon set infrastructure for additive combinatorics.

## Overview

This repository contains a verified Lean 4 proof of Singer's theorem
(1938).  It formalizes the prime-field construction and its prime-power
extension: for every prime power `q = p^k`, the projective-line construction
over a degree-three extension of `GaloisField p k` gives a Sidon set modulo
`q² + q + 1` of size `q + 1`.

The formalization builds reusable infrastructure for Sidon set theory,
including:

- **`IsSidon`** / **`IsSidonMod`** / **`IsIntervalSidon`** — predicates
  for Sidon sets in various contexts
- **Singer construction** — the full algebraic chain from finite field
  trace through kernel dimension to the combinatorial Sidon property,
  including the prime-power family theorem
- **Lindström's bounds** — the classical √(2N) counting bound and the
  improved √N + N^(1/4) + O(1) Johnson/Lindström upper bound
- **Unconditional two-sided Θ(√N) bounds** — matching upper and lower
  bounds for the extremal Sidon function
- **Structural Sidon identities** — exact cardinalities for sumsets and
  difference sets, additive energy, and a sumset-cardinality
  characterization of finite Sidon sets
- **Representation functions** — ordered pair-sum bounds and exact L¹/L²
  identities for integer Sidon sets
- **Conditional reduction** — a verified reduction from prime gap
  hypotheses to asymptotic Sidon set bounds (Erdős Problem B₃₀)

## Prime-power export provenance

The prime-power Singer modules were exported from
`d0d1/erdos/work/erdos-30/formal/Erdos30/` at commit
`ca225a0bb39ce131d31d8a949133b3c17f60acfa`.  The public-export edits are
limited to neutralized comments/docstrings plus a local `PrimeGapExponent`
definition in `PrimePowerLowerBound.lean`, needed because this public artifact
predates that interface in its older `ConditionalLowerBound.lean`.

## Mathematical Content

The core algebraic chain establishes:

1. `finrank_ker_trace`: The kernel of the finite field trace has
   dimension 2.
2. `no_proper_invariant_subspace`: No proper subspace is invariant under
   the Frobenius action.
3. `finrank_inf_of_distinct_twodim`: Distinct Frobenius orbits yield
   distinct two-dimensional subspaces.
4. `singer_sidon_mod`: The prime-field Singer set is Sidon modulo
   p² + p + 1.
5. `singer_theorem`: Prime-field statement — existence of a Sidon set of size
   p + 1 in ℤ/(p² + p + 1)ℤ.
6. `SingerPrimePower.singer_primePower_sidon_set_KL`: Prime-power statement for
   `q = p^k`, giving a Sidon set of size q + 1 in ℤ/(q² + q + 1)ℤ.
7. `singerPrimePowerFamilyHypothesis_holds`: Discharge of the prime-power
   Singer family hypothesis used by the prime-power lower-bound reduction.

The structural Sidon modules also prove:

1. `IsSidon.card_add`: exact sumset cardinality
   `|A + A| = |A|(|A|+1)/2`.
2. `isSidon_iff_card_add`: the corresponding characterization by maximum
   sumset cardinality.
3. `IsSidon.card_sub`: exact difference-set cardinality
   `|A - A| = |A|² - |A| + 1` for nonempty Sidon sets.
4. `IsSidon.addEnergy_eq`: exact additive energy `2|A|² - |A|`.
5. `IsSidon.card_sub_ge_card_add`: difference-dominance
   `|A - A| ≥ |A + A|` for nonempty Sidon sets.

The extremal-bound and representation-function modules additionally prove:

1. `sidonMaximum_le_lindstrom`: the improved upper bound
   `h(N) ≤ ⌊√N⌋ + ⌊√⌊√N⌋⌋ + 2` for `N ≥ 16`.
2. `sidonUpperBound_quarter`: the upper-bound hypothesis holds
   unconditionally for every exponent `ε ≥ 1/4`.
3. `erdos30_of_gap_and_upper`: after discharging the Singer-family hypothesis,
   subpolynomial prime gaps plus the Sidon upper-bound hypothesis imply
   `Erdos30Statement`.
4. `IsSidon.repr_le_two`: for an integer Sidon set, every ordered
   pair-sum representation count is at most `2`.
5. `IsSidon.sum_addConvolution_sq`: exact L² identity
   `∑ r_A(n)^2 = 2|A|² - |A|`.
6. `IsSidon.sum_deficiency`: exact deficiency identity
   `∑ (2 - r_A(n)) = |A|`.

## Project Structure

```
├── Erdos30.lean              # Root module (imports all)
├── Erdos30/
│   ├── Sidon.lean            # Sidon set predicates
│   ├── Interval.lean         # Interval Sidon infrastructure
│   ├── Singer.lean           # Core algebraic construction
│   ├── SingerBridge.lean     # Algebra → combinatorics bridge
│   ├── SingerSidon.lean      # Modular Sidon property
│   ├── SingerTheorem.lean    # Full Singer theorem
│   ├── SingerPrimePower*.lean # Prime-power Singer family theorem
│   ├── Lindstrom.lean        # Classical upper bound
│   ├── LindstromImproved.lean # Johnson/Lindström improved upper bound
│   ├── RepresentationFunction.lean # Pair-sum representation bound
│   ├── RepresentationL2.lean # L¹/L² representation identities
│   ├── AdditiveEnergy.lean   # Exact additive energy for Sidon sets
│   ├── DifferenceSet.lean    # Exact difference-set cardinality
│   ├── SumsetCard.lean       # Exact sumset cardinality
│   ├── SidonCharacterization.lean # Sumset-cardinality characterization
│   ├── SumDiffComparison.lean     # Sumset/difference-set comparison
│   ├── FormalStatement.lean  # Extremal function definition
│   ├── UnconditionalBounds.lean  # Two-sided Θ(√N)
│   ├── PrimePowerLowerBound.lean # Prime-power lower-bound reduction
│   ├── ConditionalLowerBound.lean # Conditional reduction
│   ├── ConditionalErdos30.lean    # Discharged conditional Erdős 30
│   ├── FinalTheoremSurface.lean   # Final theorem assembly
│   ├── Boundary*.lean        # Boundary analysis infrastructure
│   └── ...                   # Supporting modules
├── lakefile.toml             # Lake project configuration
├── lean-toolchain            # Lean version (v4.29.0)
├── lake-manifest.json        # Dependency lock file
└── LICENSE                   # GPL-3.0-only
```

## Building

### Prerequisites

- [Lean 4](https://lean-lang.org/) (v4.29.0, specified in `lean-toolchain`)
- [Lake](https://github.com/leanprover/lean4/tree/master/src/lake)
  (bundled with Lean)

### Steps

```bash
lake exe cache get  # Fetch precompiled Mathlib artifacts for the pinned manifest
lake build          # Build the full project (requires ~32 GB RAM for Mathlib)
```

The cited theorem targets build successfully and contain no `sorry` proof
placeholders.  Some pre-existing modules currently emit linter warnings; the
new Lindström and representation-function targets build without warnings.

### Verification

To confirm zero sorry usage:

```bash
lake env lean --run Erdos30.lean 2>&1 | grep -i sorry
# Expected: no output
```

Or inspect the `.olean` files:

```bash
grep -r "sorry" .lake/build/lib/ 2>/dev/null
# Expected: no output
```

## Dependencies

- **Mathlib** v4.29.0 (`leanprover-community/mathlib4`)

## License

This project is licensed under the **GNU General Public License v3.0
only** — see the [LICENSE](LICENSE) file for details.

## Citation

If you use this formalization in your work, please cite:

```bibtex
@software{hulak2026singer,
  author = {Hulak, David B. and Ramos, Arthur F. and de Queiroz, Ruy J. G. B.},
  title  = {Singer's Theorem --- Lean 4 Formalization},
  year   = {2026},
  url    = {https://github.com/d0d1/singer-theorem-lean}
}
```