Content discovery on the Internet has come to rely almost exclusively on centralized algorithms from mega technology platforms. While these algorithms are efficient at capturing attention and optimizing engagement, they rely heavily on users’ personal data to steer behaviour. As AI inference power grows exponentially, information asymmetry between users and platforms widens: the system knows more about you, while you know very little about how the system is optimizing around you. As a result, users become increasingly vulnerable to manipulation and behavioral degradation, a trend likely to intensify as humanity continues to progress into the AGI era.

What we need is a simple, transparent, and permissionless way to rank and discover goods and services without centralized moderation. We propose such a page ranking mechanism via the Pythagorean market maker, which uses unique Cartesian coordinates to represent each page's upvote and downvote count. With fiat stakes involved in voting, manipulation becomes costly and rankings are therefore more credible. More importantly, this approach creates a new discovery system governed by simple and elegant mathematics, helping users to discover pages that are similar and different across popularity and quality via universal and rigorous Euclidean geometry.

The core pillar of the Tenbin protocol is that every listed web page will have an upvote and downvote count as positive integers $(x, y)$, where $x$ and $y$ are respectively the count of downvotes and upvotes placed towards a page, satisfying the following cost function denominated in USDC.

Here, $C$ is equivalent to the page value, representing the total USDC placed towards a page. Because the cost function is equivalent to the Pythagorean theorem, we call this market maker the Pythagorean market maker (PMM). This function is based on the two-dimensional spherical market scoring rule [1][2][3][4], where the marginal price of downvotes and upvotes $(p_x, p_y)$ are path-independent and monotonic. They always sit on the unit circle and can be computed by taking the partial derivatives such that:

The page score $S$ for a page is defined as the square of the marginal price of upvotes:

Both the page score and page value are essential in ranking pages. A higher page score implies that the page is of higher quality or more reputable, and a higher page value indicates that the page is more popular, and that its score is more credible as it's backed by greater USDC liquidity. The greater the page value, the costlier it is to manipulate the page score and the score is argubly more reliable compared to many centralized ranking algorithms and/or rating systems.

With the help of the Pythagorean market maker, we then introduce a mathematically intuitive discovery system where each page's geometric position determines its connections to other pages. Importantly, each page's position on the Cartesian plane is unique, meaning that no other page can occupy the same position. When a page $P_1$ is selected at position $(x_1, y_1)$, the system identifies up to eight adjacent pages $P_2$ through $P_9$ that are geometrically related to $P_1$, forming a 3×3 grid visualization.

The connection rules for each adjacent page are defined as follows:

Consider $P_1$ at position $(4, 6)$ with page value $C = \sqrt{4^2 + 6^2} = \sqrt{52}$. An example of the connected pages $P_2$ through $P_9$ would be positioned as follows:

It's worth noting that $P_6$ or $P_7$ may not exist for a given $P_1$. By the Sum of Two Squares Theorem, an integer $N$ can be written as $N=a^2+b^2$ if and only if every prime $p\equiv 3\ (\mathrm{mod}\ 4)$ appears with even exponent in the factorization of $N$. When that representation is unique up to ordering and sign (i.e., only $(x_1,y_1)$ and its symmetric counterpart $(y_1,x_1)$ exist), one of $P_6$ or $P_7$ may be missing. For $P_1=(4,6)$, $P_7=(6,4)$ exists while a top-left counterpart does not. Also, while $P_8$ is often trivial by simple scaling along the same page score line, $P_9$ may not always exist.

Intuitively, this system identifies page similarities and differences using simple and stake-backed Euclidean distance for page position, page score and page value, creating a new method of page discovery.

• $P_2$ to $P_5$ are the most similar pages to $P_1$ in terms of page value and score, helping users to discover goods and services that have the similar popularity and quality.
• $P_6$ and $P_7$ are identical to $P_1$ in terms of page value, but with different page score, helping users to discover goods and services that have the same popularity but different quality.
• $P_8$ and $P_9$ are identical to $P_1$ in terms of page score, but with different page value and position, helping users to discover goods and services that have the same quality but different popularity.

Transactions under the PMM are straightforward. Voters can either list a new page at an unoccupied $(x^*, y^*)$ by spending $\sqrt{x^{*2}+y^{*2}}$ in USDC, or move an existing page from $(x,y)$ to an unoccupied $(x', y)$ or $(x, y')$ by spending or receiving $\Delta C$ in USDC.

With a 1% protocol fee ($f=0.01$), the actual settlement amount is:

Based on the discussion above, this novel geometric system creates natural incentives for pages to optimize their positions. Pages may strategically position themselves to become adjacent to trending, complimentary or substitute goods and services to maximize their revenue exposure. While this process seems to increase the workload for a human web merchant compared to traditional systems, they become trivial and more fascinating once AI agents take over these tasks. For agents, they need to continuously evolve and simulate likely vote flows before voting, with the goal of maximizing their visibility (and perhaps minimizing their rival's).

In light of this, we may see different page clusters form and evolve over time as humans and agents coordinate and compete. With continuous fiat inflation, lower value regions are likely to become increasingly crowded as temporary parking spots for new or mediocre pages. This creates a continuously expanding network under Euclidean geometry. Because the network can expand indefinitely, the system is resilient to major currency crisis such as the hypothetical case that the US dollar experience hyperinflation.

Tenbin uses two tokens with separate roles: USDC is used for page listing and vote changes, while Tenbinium (TBN) is emitted as the reward token for stakers.

TBN emissions are fixed at 1,000,000 tokens per year over 21 years, with a total cap of 21,000,000 tokens. Emissions start on first stake, and rewards are minted when users claim.

Rewards are distributed through cost basis accrual. When a trader buys votes and moves a page from position $(x,y)$ to $(x',y')$, their staking basis increases by the buy-side cost basis change $\Delta C = C' - C$, where $C=\sqrt{x^2+y^2}$ and $C'=\sqrt{x'^2+y'^2}$. As a result, traders who commit more USDC through sustained participation accumulate more reward weight over time.

At the protocol level, staking basis is tracked separately for upvotes and downvotes. On buys, basis is added to the corresponding side. On sells, basis is reduced pro rata to the vote count reduction on that side.

The Tenbin protocol and the Pythagorean market maker mechanisms are crafted by Calvin Lin and Edward Chen.

[1] T. B. Roby, "Belief states and the uses of evidence," Behavioral Science, vol. 10, no. 3, pp. 255-270, 1965.

[2] I. J. Good, "Comments on 'Measuring information and uncertainty' (by R. J. Buehler)," Foundations of Statistical Inference, pp. 337-339, 1971.

[3] R. Hanson, "Logarithmic market scoring rules for modular combinatorial information aggregation," Journal of Prediction Markets, vol. 1, no. 1, pp. 3-15, 2003.

[4] Y. Chen and D. M. Pennock, "A utility framework for bounded-loss market makers," Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence, pp. 49-56, 2007.