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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.
77 lines
2.5 KiB
Text
77 lines
2.5 KiB
Text
created: 20260507101500000
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modified: 20260507101500000
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tags: ResearchStack Erdos Gyarfas GraphTheory DetectorAnomaly FourPrimitives
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title: Erdos Gyarfas Detector Anomaly
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type: text/vnd.tiddlywiki
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! Erdos Gyarfas Detector Anomaly
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The local Erdős-Gyárfás run must be treated as a detector/generator anomaly, not as a mathematical counterexample.
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!! What The Local Run Claimed
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`4-Infrastructure/shim/test_erdos_gyarfas_4primitive_results.json` reports:
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* `total_min_degree_3`: 9
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* `has_power_of_two_cycle`: 0
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* `conjecture_holds`: false
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* tested `n_values`: 10, 15, 20
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* samples per `n`: 3
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That is suspect because the conjecture is still open, and known searches rule out very small counterexamples. Any claimed failure at `n = 10` or `n = 15` should be classified as a detector anomaly until independently certified.
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!! Likely Failure Modes
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* Cycle detector misses simple cycles.
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* Detector accidentally searches only induced or chordless cycles.
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* Detector does not check every power-of-two length `4, 8, 16, ... <= n`.
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* Generated graph does not actually satisfy simple-graph assumptions.
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* Graph has invalid artifacts such as multiedges, self-loops, disconnected fragments, or bad adjacency.
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* Result variable is inverted or over-promoted from a diagnostic boolean.
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!! Required Γ_fail Packet
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Any claimed counterexample must emit:
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```
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Gamma_fail = {
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graph_id,
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n,
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edge_list,
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adjacency_list,
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degree_sequence,
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min_degree,
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checked_lengths: [4, 8, 16, ... <= n],
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cycles_found_by_length,
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independent_verifier_result,
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sha256(edge_list)
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}
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```
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!! Promotion Rule
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`Conjecture holds: false` is allowed only if:
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* `min_degree >= 3`
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* the graph is a valid simple graph
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* no simple cycle has length `2^k`
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* the checked lengths cover all `2^k <= n`
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* an independent verifier confirms the cycle absence
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* the edge list and verifier output have receipt hashes
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Until then, the correct status is:
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> Detector anomaly: claimed counterexample requires verification.
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!! Four-Primitive Interpretation
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* `ρ(x)` = edge and minimum-degree density.
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* `C = UΛU^T` = adjacency spectral structure.
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* `G = A^T A` = graph rigidity / degree deformation.
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* `Γ_i` = cycle packet plus missing-cycle residual.
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Keeper conclusion: the four-primitives model found the right obstruction shape, but the Gyárfás packet must carry a verifiable cycle certificate before the result can be trusted.
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!! External Status
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The Erdős-Gyárfás conjecture remains open. Reference status checked against the public Erdős-Gyárfás summaries and graph-theory references on 2026-05-07.
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