The paper presents a detailed analysis of Anthropic Claude's subscription model, specifically comparing its cost-effectiveness against the API usage, and dissecting the true underlying limits of its 'Max 5x' and 'Max 20x' plans. The central finding is that for "agent-like" usage patterns, the subscription model offers significantly higher cost efficiency, up to 36 times better value than API usage.

The core methodology employed is a sophisticated reverse-engineering technique to deduce the internal credit and token structures of Claude's web UI. The motivation stemmed from the observation of "suspiciously precise" floating-point numbers displayed for usage rates (e.g., 0.16327272727272726). The hypothesis was that these precise decimals were artifacts of an internal, integer-based fractional credit system. To recover the underlying integer ratios, the author utilized the Stern–Brocot tree algorithm. The Stern–Brocot tree is a binary search tree that generates all positive rational numbers exactly once. By finding the fraction $\frac{p}{q}$ that best approximates a given floating-point number, where $p$ and $q$ are relatively prime integers and $q$ is within a reasonable bound, the method allowed the reconstruction of the original fractional values representing usage and limits. This allowed for the precise recovery of actual session limits, weekly limits, and credit units, which were found to be more generous than indicated by the marketing names.

A key differentiator identified is the cost structure for context reusability. In the Claude web UI subscription, there is zero cost for re-referencing cached context within the same conversation session. Conversely, the API requires the entire context to be re-transmitted with each request. Even if the context is cached on Anthropic's side, a 10% input token cost is still incurred for repeated transmission. This difference leads to a drastic cost divergence for agent-like coding or other tasks that involve lengthy, iterative interactions relying on persistent context.

The analysis provides a granular breakdown of the subscription tiers:

• 'Max 5x' Plan: This plan is identified as the optimal choice, offering a session limit that is 6 times that of the Pro plan and a weekly cumulative usage limit that is 8.3 times higher than Pro. The actual capacities are significantly more generous than the '5x' moniker suggests, making it highly cost-effective for continuous, long-duration tasks, akin to a "marathon" usage pattern. Users frequently report rarely hitting limits even with heavy daily use.
• 'Max 20x' Plan: The '20x' multiplier misleadingly applies only to a 5-hour session usage limit. The crucial weekly cumulative usage limit is merely ~2 times that of the 'Max 5x' plan. While beneficial for short bursts of intense activity (a "sprint" usage pattern), it is significantly less efficient for sustained, long-term operations compared to 'Max 5x'.

The paper concludes that for tasks like long coding sessions or agent-based development (e.g., using the Claude Code extension), the subscription model is overwhelmingly superior. A practical tip highlighted is that the Claude Code VS Code extension can be used by logging in with a claude.ai subscription account, bypassing API key usage and leveraging subscription limits without incurring API credits.

Community feedback largely validated the numerical analysis, concurring that 'Max 5x' offers the best value for heavy users. However, a significant caveat raised is Anthropic's lack of transparency; the derived limits are based on reverse-engineering and can change without notice, posing a risk for long-term business decisions. Additionally, privacy and terms of service differences between the web UI and API are noted, with the API generally perceived as safer for sensitive or proprietary code. Despite these concerns, the prevailing sentiment is that "Max 5x is the optimal choice for now," balanced with the understanding that "this structure can change at any time."