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Created refined investigation script for Erdős–Gyárfás conjecture where previous test found no power-of-two cycles (conjecture holds: False). Refinements: - Regular graph construction (all vertices same degree) - Exhaustive DFS cycle detection - More samples per n (5 instead of 3) - Extended n values [8, 10, 12, 14, 16] Goal: Determine if previous negative result was due to random graph construction or if regular graphs with exhaustive cycle detection find power-of-two cycles. Script created: investigate_erdos_gyarfas_refined.py Execution canceled by user - awaiting further instructions.
61 lines
1.8 KiB
Text
61 lines
1.8 KiB
Text
created: 20260507101500000
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modified: 20260507101500000
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tags: ResearchStack Erdos Selfridge CoveringSystems SmokeTest FourPrimitives
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title: Erdos Selfridge Finite Smoke Test
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type: text/vnd.tiddlywiki
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! Erdos Selfridge Finite Smoke Test
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The local Erdős-Selfridge run is a valid finite smoke test, not a proof of the full odd covering-system conjecture.
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!! What The Local Run Claimed
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`4-Infrastructure/shim/test_erdos_selfridge_4primitive_results.json` reports:
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* `total_tests`: 12
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* `conjecture_violations`: 0
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* `conjecture_holds`: true
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* `n_moduli_values`: 3, 4, 5, 6
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* `max_modulus`: 100
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* samples per `n_moduli`: 3
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This should be classified as:
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> PASS: finite generated samples produced no odd-moduli counterexample.
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and not as:
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> PROOF: the Erdős-Selfridge conjecture is solved.
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!! Four-Primitive Interpretation
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* `ρ(x)` = residue/modulus coverage density.
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* `C = UΛU^T` = covering overlap spectrum.
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* `G = A^T A` = even/odd modulus balance.
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* `Γ_i` = covering-system packet plus violation witness.
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!! Required Violation Packet
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A stronger future smoke test should emit a packet for every candidate covering:
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```
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Gamma_cover = {
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candidate_id,
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moduli,
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residues,
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all_moduli_odd,
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moduli_distinct,
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coverage_window,
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uncovered_residues,
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independent_verifier_result,
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sha256(candidate)
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}
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```
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!! Status
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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.
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!! Keeper Conclusion
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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.
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