Erdős spiral

Rank every open problem on erdosproblems.com by how attackable it is for exact computation and finite certificates, then spiral multi-agents outward from the easiest. Grades, not vibes.

SkyLab’s honesty HUD, transposed to maths: every attempt is PROOF / DISPROOF / PARTIAL / HEURISTIC / NOTHING. Computation stays heuristic. Literature checks and independent referees gate anything that looks like a claim.

The spiral

Pinned catalogue in, ranked queue out, agents outward from the easiest attack surface, every landing graded and kill-switched.

Pipeline

flowchart LR
  A["teorth/erdosproblems
pinned YAML"] --> B["rank.py
tractability"]
  B --> C["queue
easy → hard"]
  C --> D["multi-agents
FOSS sandboxes"]
  D --> E["REPORT.md
+ artifacts"]
  E --> F{"kill switches"}
  F -->|pass| G["grade:
PROOF / PARTIAL
HEURISTIC / NOTHING"]
  F -->|fail| H["downgrade
or NOTHING"]
  G --> I["board.json
+ /erdos page"]
  H --> I

What landed

#506 - minimum circles

PARTIAL independently verified

Minimum number of circles determined by n points in the plane (not all collinear, not all concyclic). Elliott’s asymptotic formula fails at small n.

min(4)=3 · min(5)=5 · min(6)=8 · min(7)=11

Exact integer certificates; dual-Fano lower bounds; second-agent referees for n=6 and n=7. Certified upper bound min(8)≤18; Round 3 closed many profiles at t≥18 (including disjoint 4-lines) but equality min(8)=18 stays conjectural. Likewise min(9)≤25.

#506 small-n ladder

flowchart LR
  n4["n=4
min=3"] --> n5["n=5
min=5"]
  n5 --> n6["n=6
min=8"]
  n6 --> n7["n=7
min=11"]
  n7 --> n8["n=8
≤18"]
  n8 --> n9["n=9
≤25"]
  n4:::v
  n5:::v
  n6:::v
  n7:::v
  n8:::u
  n9:::u
  classDef v fill:#143528,stroke:#6ee7b7,color:#e9edf4
  classDef u fill:#2a2414,stroke:#e8b64c,color:#e9edf4,stroke-dasharray: 5 4

erdosproblems.com/506 · attempt · referee n≤6 · referee n=7

#287 - Egyptian gaps

PARTIAL reproduction confirmed

No gap-{1,2} Egyptian fraction of 1 with first denominator ≤ 12. Finite exhaustive search, independently checked. Already known on the forum; recorded here as reproduction, not a new result.

erdosproblems.com/287 · attempt

Also probed (heuristic / null)

HEURISTIC NOTHING

Erdős-Straus (#242) OEIS extension work held behind a publish brake. Murty-Simon (#742) SAT curve through small n. Ramsey / extremal wave (#547, #551, #556, #23, …). Superseded catalogue hits (#848 Sawhney, #67 Tao discrepancy, #90 unit distance) graded NOTHING. Full family coverage in COVERAGE.md.

Grading law

flowchart TD S["attempt lands"] --> L{"literature still open?"} L -->|no| N1["NOTHING
superseded"] L -->|yes| U{"upstream sunk?"} U -->|yes| N1 U -->|no| C{"computation only?"} C -->|yes| H["HEURISTIC
no laundering"] C -->|no| V{"independent verify?"} V -->|no| D["downgrade / silence"] V -->|yes| G{"full close?"} G -->|yes| P["PROOF / DISPROOF"] G -->|no| R["PARTIAL"]

Kevin Weil literature check · upstream sink · verification-or-silence · no laundering computation into proof.

How the spiral ranks

Pinned snapshot of teorth/erdosproblems → local YAML → rank.py scores decidable / falsifiable / finitely verifiable first. Agents work the queue top-down. Tooling: sympy, networkx, PySAT, nauty/geng, PARI; exact tracks for verification. See TOOLING.md.

flowchart TB R["rank.py score"] --> T1["1. decidable
finite certificates"] R --> T2["2. falsifiable
search for counterexample"] R --> T3["3. finitely verifiable
bounded check"] R --> T4["4. heuristic-only
open-ended"] T1 --> Q["queue head"] T2 --> Q T3 --> Q T4 --> Q2["queue tail"]

Erdős spiral

The spiral

What landed

#506 - minimum circles

#287 - Egyptian gaps

Also probed (heuristic / null)

Grading law

PROOF / DISPROOF

PARTIAL

HEURISTIC

NOTHING

How the spiral ranks

Famous-family coverage

Attempt board

Reproduce