created: 20260507101500000
modified: 20260507101500000
tags: ResearchStack Erdos Gyarfas GraphTheory DetectorAnomaly FourPrimitives
title: Erdos Gyarfas Detector Anomaly
type: text/vnd.tiddlywiki

! Erdos Gyarfas Detector Anomaly

The local Erdős-Gyárfás run must be treated as a detector/generator anomaly, not as a mathematical counterexample.

!! What The Local Run Claimed

`4-Infrastructure/shim/test_erdos_gyarfas_4primitive_results.json` reports:

* `total_min_degree_3`: 9
* `has_power_of_two_cycle`: 0
* `conjecture_holds`: false
* tested `n_values`: 10, 15, 20
* samples per `n`: 3

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.

!! Likely Failure Modes

* Cycle detector misses simple cycles.
* Detector accidentally searches only induced or chordless cycles.
* Detector does not check every power-of-two length `4, 8, 16, ... <= n`.
* Generated graph does not actually satisfy simple-graph assumptions.
* Graph has invalid artifacts such as multiedges, self-loops, disconnected fragments, or bad adjacency.
* Result variable is inverted or over-promoted from a diagnostic boolean.

!! Required Γ_fail Packet

Any claimed counterexample must emit:

```
Gamma_fail = {
  graph_id,
  n,
  edge_list,
  adjacency_list,
  degree_sequence,
  min_degree,
  checked_lengths: [4, 8, 16, ... <= n],
  cycles_found_by_length,
  independent_verifier_result,
  sha256(edge_list)
}
```

!! Promotion Rule

`Conjecture holds: false` is allowed only if:

* `min_degree >= 3`
* the graph is a valid simple graph
* no simple cycle has length `2^k`
* the checked lengths cover all `2^k <= n`
* an independent verifier confirms the cycle absence
* the edge list and verifier output have receipt hashes

Until then, the correct status is:

> Detector anomaly: claimed counterexample requires verification.

!! Four-Primitive Interpretation

* `ρ(x)` = edge and minimum-degree density.
* `C = UΛU^T` = adjacency spectral structure.
* `G = A^T A` = graph rigidity / degree deformation.
* `Γ_i` = cycle packet plus missing-cycle residual.

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.

!! External Status

The Erdős-Gyárfás conjecture remains open. Reference status checked against the public Erdős-Gyárfás summaries and graph-theory references on 2026-05-07.