created: 20260507101500000
modified: 20260507101500000
tags: ResearchStack Erdos Selfridge CoveringSystems SmokeTest FourPrimitives
title: Erdos Selfridge Finite Smoke Test
type: text/vnd.tiddlywiki

! Erdos Selfridge Finite Smoke Test

The local Erdős-Selfridge run is a valid finite smoke test, not a proof of the full odd covering-system conjecture.

!! What The Local Run Claimed

`4-Infrastructure/shim/test_erdos_selfridge_4primitive_results.json` reports:

* `total_tests`: 12
* `conjecture_violations`: 0
* `conjecture_holds`: true
* `n_moduli_values`: 3, 4, 5, 6
* `max_modulus`: 100
* samples per `n_moduli`: 3

This should be classified as:

> PASS: finite generated samples produced no odd-moduli counterexample.

and not as:

> PROOF: the Erdős-Selfridge conjecture is solved.

!! Four-Primitive Interpretation

* `ρ(x)` = residue/modulus coverage density.
* `C = UΛU^T` = covering overlap spectrum.
* `G = A^T A` = even/odd modulus balance.
* `Γ_i` = covering-system packet plus violation witness.

!! Required Violation Packet

A stronger future smoke test should emit a packet for every candidate covering:

```
Gamma_cover = {
  candidate_id,
  moduli,
  residues,
  all_moduli_odd,
  moduli_distinct,
  coverage_window,
  uncovered_residues,
  independent_verifier_result,
  sha256(candidate)
}
```

!! Status

The odd covering-system problem remains open in general. Known restricted results rule out important cases, including square-free moduli under strong conditions, but that does not promote this finite sample to a proof.

!! Keeper Conclusion

The run is useful because it makes the packet shape explicit: covering density, overlap spectrum, parity shear, and witness receipts. Keep it as a smoke test and require Γ packets before promoting any result claim.