GPT-5.6 Cracks 50-Year Math Problem, Claude Opus 5 Revealed Shortly After
N.R. Finch
OpenAI's GPT-5.6 proved the cycle double cover conjecture — unsolved for 50 years — in under an hour, while Anthropic's Claude Opus 5 surfaced almost simultaneously, extending the AI race from raw capability into pricing strategy.
A 50-year-old math problem — how did AI crack it?
The cycle double cover conjecture — a graph-theory problem asking whether every bridgeless graph has a set of cycles covering each edge exactly twice — was posed independently by Tutte, Szekeres, and Seymour decades ago. Only partial progress had been made in half a century.
GPT-5.6 Sol Ultra deployed 64 parallel agents simultaneously. The system forced the first round onto divergent paths — algebra, structural induction, flow fields, embedding — and barred most agents from knowing which direction looked strongest.
Dedicated "checker" agents attacked every candidate proof; only those surviving scrutiny advanced. This means → the system wasn't guessing — it used institutional adversarial filtering to weed out errors.
The winning proof path: recast the graph-theory problem as a system of linear-algebra equations, invoked Tutte's group-flow theorem, and completed the derivation by constructing a dual vector space.
The key variable — why one hour instead of one day?
OpenAI research scientist Noam Brown credited the breakthrough to parallel test-time compute (TTC): scaling up parallelism compressed a potentially day-long task to under one hour.
In plain terms = the model itself didn't leap a full tier in intelligence — sending more "workers" down separate paths at once is what drove the speed gain.
Brown also stressed: the model that produced this proof was the same version publicly available to all users that day, not an internal build.
Where did Opus 5 come from — and how does it relate to Fable 5?
A model codenamed "Claude Honeycomb" briefly appeared in Cursor's model list before being pulled. Its specs: 1-million-token context, ultra-high compute tier, per-item safety controls — architecture closely resembling the flagship Fable 5, with a fallback chain pointing to Opus 4.8.
Claude Opus 5 then surfaced on Google's Vertex platform. Blogger leo reported Anthropic is in a final sprint, with a launch expected this week or next at the latest.
This means → Opus 5 is not meant to replace Fable 5 — it fills the gap between Sonnet 5 and Fable 5, offering near-flagship capability at a lower price tier.
The pricing battle — can Opus 5's value play win?
Fable 5 is currently priced at $10 per million input tokens and $50 per million output tokens — double Opus 4.8. Its subscription-included free quota has been extended three times in five weeks.
In plain terms = the flagship is too expensive and the quota keeps running short — the market wants a "near-flagship but affordable" option.
If Opus 5 launches with Fable-5-level capability at Opus-level pricing, it will compete head-to-head with GPT-5.6 on value for money.
This reflects a broader shift: AI model competition has moved beyond raw capability into pricing strategy — whoever strikes the right balance between performance and cost captures the larger share.
Content is for reference only, not financial advice.