It is a bit sad to work on a project that no one cares about. You're not sure if you're crazy or a visionary. And it is kind of lonely.

I sometimes wonder if I would be happy doing this for the rest of my life if I could. And if it would have any impact at all no matter for how long I do it. My feelings in that area go from slightly depressed to slightly excited about the potential a few times every week.

As we all know, living and making life choices means sacrificing other things that could have been. When I was in France in 2015, I started a masters course in AI/robotics with the idea of doing a PhD and AGI research later on but quit half way because I felt university was such a waste of time.

But come now the AI boom, and although I still believe Education is broken, I might have been much better off financially/reputationally if I had withstood the bullshit followed that path. Instead I sacrificed that for nerding about low level programming and open educational content.

It is hard to get such ideas off one's mind. But the fact is, for better or worse, I've started walking the path of educational reform and sacrificed others along the way, and this is the path that I'm further ahead than other people, and perhaps I should pursue it further to a possible conclusion. Also this path has the advantage that it is not fully exclusive from other academic endeavors as we will always need content about the new flashy things that keep coming up.

So yeah, it's hard, but here I am, and I'll go as far as I can without going into Charles Bukowski levels of personal sacrifice.

Announcements:

OK, I need to do content. I know :-) At the university I'm at, the only department that is open is the mathematics one. Both:All other courses extremely closed, notably Physics, which is the other course I'd consider. There are upsides and downsides for going for Mathematics:If I were free to choose, I might go for Physics instead. But maths isn't hard, and I think I'll just go with the hand I'm dealt this time to start with.

Tech wise, the big things are the following ones to which I have given different levels of architectural consideration (i.e. read: I'm afraid they'll be fucking hard and that I'll spend a month on yet another useless feature that won't help get a single user). I don't think I'll do those before at least a little bit of content, we'll see:

Smaller cute tech that I might do before content "real quick" include:

Another thing I really want to do before time is up is to create a video summarizing my philosophy of education. I want it to be as fun and funny and sad as possible, with silly moving animated images and slides, not just me talking to the camera. Although all of the points I intend to talk about have undoubtedly been covered by others, it is something that I feel so strongly about that I would like to tell others about it more personally. If I start it it will likely take a few days to get done, and I'm not sure wha the final quality would be. It is a bit sad to not do "project work", but I think I'll end up doing it regardless. Class it under "fundraising" if you will, as it may help to find other like minded but rich people.

In any case, the outcome of that is that the tech has improved. And I have done a relatively good job of clearly publishing any "more user visible" improvements to docs.ourbigbook.com/news and social media such as though it is important to note that there have been more than one "fix a hard bug" weeks that were not published because they would just bore readers.

During this period the main focus has been on improving OurBigBook Web, i.e. the dynamic website that powers OurBigBook.com. There are two reasons for that:As a result, Web is now way less buggy and much more usable.

If you look through the list of Web updates, there is nothing specifically mind blowing. The core ideas have largely crystallized, and we are just trying to making them click. I have a few more punches up my sleeve, but the core is decided.

Web process has been somewhat slower than what I'd like. Of course, it is the case of any project that things are easily said than done. But there are two other main structural factors that have played into it:

Aside from Web, there was only one thing that received a significant improvement, and that was the OurBigBook VS Code extension. The extension is not perfect, and it is not the "final UI", which has to be some WYSIWYG implementation, and there are some fundamental limitations that cannot be overcome without patching VS Code itself. However, the extension is already extremely usable, and I'm writing this on it right now. Basics like syntax highlighting, jump to definition and autocomplete are very useful and usable.

At first I had intended to create a lot more content for the world class university located where I lived, but I ended up not doing that and just improving the project tech instead.

There are a few reasons for this, good or bad:

The downsides of going for tech first are severe:There are however counterpoints to these as for anything else:

Long story short, the project is so far a complete failure on the most important metric: number of regular users, which current sits at exactly one: myself.

There were notable users who found the project online and who actually tried to use the website for some content and provided extremely valuable feedback:Unfortunately after the period of a few weeks they stopped using it to follow their other priorities instead. Which is of course totally fine, however sad.

I still believe that the OurBigBook Web feature is a significant tech innovation that could make the website go big.

I also believe that the project gets many fundamentals of braindumping right, notably the infinitely deep table of contents without forced scoping, e.g.:does not make Calculus have an ID orr URL of mathematics/calculus, rather it's just calculus.

But there is a fundamental difficulty in reaching critical mass to that self-sustaining point, as people don't seem to be convinced by these logical "my system is better" argument alone, as opposed to having them Google into stuff they need now and then understand that the project is awesome.

A closely related critical mass issue is that existing big multiuser knowledge base websites such as Stack Overflow and Wikipedia have a tremendous advantage on PageRank. No matter how useless a Wikipedia article about something is, it will always be on top of Google within a week of creation for title hits. And since the main goal of publishing your stuff is to get it seen, it makes much more sense for writers to publish on such existing websites whenever possible, because anywhere else it is way way less likely to be seen by anybody.

Even I end up writing way more on Stack Overflow than on OurBigBook as a programmer. But I still believe that there is a value to OurBigBook, for the usual reasons of:

Perhaps what saddens me the most is that even on GitHub stars/Twitter/Hacker news terms there is almost no interest in the project despite the fact that I consider that it has innovations, while many other note taking apps as well in the thousands of stars. Maybe I'm just delusional and all the tech that I'm doing is completely useless?

Part of the issue is probably linked to the fact that most other note taking apps focus on "help me organize my ideas so I can make more money" and often completely ignore "I want to publish my knowledge", and stuff that helps you make money is always easier to sell and promote.

OurBigBook on the other hand a huge focus on "I want to publish me knowledge". It aims almost single mindedly in being the best tool ever for that. However this doesn't make money for people, and therefore there are going to be way less potential users.

I do believe strongly that all it takes is a few users for the project to snowball. For some people, once you start braindumping, it is very addictive, and you never want to stop basically. So with only a few of those we can open large parts of undergrad knowledge to the world. But these people are few, and so far I haven't been able to find even a single one like me, and on top of that convince them that I have created the ultimate system for their knowledge publishing desires.

Another general lesson is that I should perhaps aimed for greater compatibility with existing systems such as Obsidian. Taking something that many people already know and use can have a huge impact on acceptance. E.g. anything that touches Obsidian can reach thousands of stars: github.com/KosmosisDire/obsidian-webpage-export. Note taking apps that aim for "markdown" compatibility also tend to fare better, even if in the end you inevitably have to extend the Markdown for some of your features. And WYSIWYG, which I want but don't have, is perhaps the ultimate familiarity.

Another issue compared to other platforms is that OurBigBook just came out late. Obsidian launched in 2020. Roam Research and Trillium Notes also came earlier. And it is hard to fight the advantage already gained by those on the "I'm going to take some personal notes" area. I do believe however that there a strong separation between "these are my personal notes" and "I want to publish these". Once you decide to publish your knowledge, you immediately start to write in a different way, and it is very hard to convert pre-existing "private" notes into ones suitable for public consumption.

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The opposite of the AGI blues. In 2025 Ciro Santilli fell well in this camp.

A hero.

1. Introduction

As System-on-Chip (SoC) architectures incorporate billions of transistors, the ability to accurately predict design properties has become paramount ${}^5$. Early-stage architectural design and physical synthesis rely heavily on robust models that quantify the relationship between logic complexity and the communication requirements between disparate system blocks.

The foundational model in this domain is Rent's Rule ${}^1$. Discovered empirically by E. F. Rent at IBM and later formalized by Landman and Russo ${}^2$, the rule establishes a power-law relationship between the number of external signal connections (terminals) to a logic block and the number of internal components (gates or standard cells) it contains:Where:
• $T$: Number of external terminals (pins) of the block.
• $g$: Number of internal logic components (gates/cells).
• $K$: Rent's empirical constant (average pins per block).
• $p$: Rent exponent ($0<p<1$).

While Rent's Rule is an indispensable tool for wire length ${}^3{,}^4$ and placement optimization, its empirical origins lead to inherent limitations—especially when applied to modern, heterogeneous architectures. This paper discusses New Law ${}^5$ and a new generalization, which addresses these shortcomings by incorporating explicit structural constraints, extending its utility to the next generation of complex computing systems.
________________________________________
2. Overview of Rent's Rule and Current Drawbacks
2.1. Applications and Interpretation

Rent's Rule describes a statistical self-similarity in digital systems. The Rent exponent ($p$) provides insight into a design's topological complexity:
• $p\approx 0.4$: Highly regular structures.
• $p\approx 0.5$: Structured designs with high locality (e.g., SRAM).
• $p\approx 0.75$: "Random logic" or complex, unstructured designs.

2.2. Limitations

The power-law form suffers from two primary drawbacks ${}^6{,}^8$:
1. Terminal Constraint Deviation (Region II) ${}^7$: The power law breaks down as partitions approach the total system size ($>25\%$ of the chip). Physical I/O pins are finite; thus, the log-log plot flattens as $g$ approaches $N$.
2. Undefined Constants: There is an absence of methodology relating design metrics to the empirical constants $K$ and $p$.

________________________________________
3. The New Rule: Generalization for Autonomic Systems

We utilized a graph-mathematical model to generalize Rent’s Rule, specifically addressing its limitations when applied to autonomic systems. We demonstrated that the classical power-law form of Rent’s Rule is valid only under the restrictive conditions where the system contains a large number of blocks, and the number $g$ of internal components in a block is much smaller than the total number of components ($N$) in the entire system ${}^9$.

The generalized formulation, referred to as the New Graph-based Rule, extends the applicability of the scaling law across the entire range of partition sizes, including the problematic Rent's Region II. The New Rule is expressed as ${}^9{,}^{10}{,}^{11}$:Where:
• $T$ is the number of external terminals for the block partition.
• $N$ is the total number of components in the system.
• $g$ is the number of components in the block partition.
• $t$ represents the average number of pins of a component in the system.
• $P_g$ is the generalized Rent exponent, derived by the described graph-partitioning method.

The rule was derived by modeling the system as a graph, where each component is represented as a vertex, and each net is represented as a tree connecting its components.

Figure 1. "All Net Components Are in the Block" illustrates the case when a net connects three components ($A$, $B$, and $C$) and is represented as a net tree. In this example, all net components are in the same block; thus, there is no need for a block external terminal—none of the net edges exit the block.

Figure 2. "An external terminal" illustrates the same case, but only components $A$ and $B$ are in the same block, while component $C$ is located in another block. In this scenario, an edge exits the block to connect to component $C$, necessitating a block external terminal for the net under consideration.

3.1. Derivation Logic

Initially, we assumed that each block has randomly assigned components. Under this assumption, the probability $Q^{\prime }$ that a given pin of a given component has an edge to another component outside of the block is:If the net has only two components to connect (the net tree is a single edge), the above formula is straightforward. In this case, the edge goes outside the block, creating one block terminal. If the net has $m>2$ pins to connect, we still have only one outside terminal—all components of the net within the block are connected by an internal net-tree, requiring only one tree edge to exit the block.

Because the component under consideration has $t$ pins on average, the probability $Q$ that the component will have $t$ edges (block terminals) to components in other blocks is:With $g$ components in the block, the number of expected block terminals is:The drawback of formula $[2]$ is the assumption of random component assignment. In reality, blocks are not designed randomly; highly connected components are partitioned into the same block to minimize communication overhead. Therefore, formula $[2]$ produces conservative results. To account for the effect of optimized partitioning that minimizes terminals, we introduce a correction constant $P_g<1$ (analogous to the Rent exponent), which reduces the estimated number of terminals:

By substituting the variables from $[2]$ into $[3]$, we arrive at the generalized New Rule $[1]$.

3.2. Behavioral Cases

• Case 1 ($g=1$): Simplifies to $T=t$, matching classical expectations.
• Case 2 ($g=N/2$): Yields the maximum terminal count, reflecting the peak communication requirement when a system is halved.
• Case 3 ($g=N$): $T=0$. This accurately models Region II, as a closed system has no external signals.

________________________________________
4. A Hypergraph Model

Above, we utilized a graph-mathematical model to generalize Rent’s Rule. We will show that if we use a hypergraph model of the system, we can further improve the accuracy of the generalized Rent’s Rule by taking into account an additional and known design property: the average number of components, $m$, that a net connects.

Let’s represent a net that connects $m$ pins as a hyperedge, instead of a tree as used in the previous graph-based model. Note that $m$ is a known design property and is the average value that can be obtained for any real design.

Figure 3. "All three components and the hyperedge are within the Block" illustrates the case when a net connects three components ($A$, $B$, and $C$) and is represented as a hyperedge (an oval encompassing all components). In this example, all net components are in the same block, and there is no need for a block external terminal—the hyperedge does not cross the block boundary.

4.1. Hypergraph Derivation

Again, let’s initially assume that each block has randomly assigned components. Then, the probability $V^{\prime \prime }$ that a given pin of a given component within the block is connected to another component within that same block is:The probability $V^{\prime }$ that the remaining $m-1$ vertices (components) within the hyperedge are all located in the block (resulting in no block terminal for this net) is:This implies that the probability $Q^{\prime }$ that the hyperedge will exit the block (necessitating a block terminal) is:Because the component under consideration has $t$ pins on average, the probability $Q$ that the component will have $t$ hyperedges (block terminals) connecting to components in other blocks is:The above formula reflects the physical reality that the more components of a net are located within the block, the lower the probability that the net will exit the block. If all $m$ components of a net are in the block, the net requires no block terminal.
With $g$ components in the block, the number of expected block terminals is:Again, the drawback of formula $[3]$ is the assumption of random component assignment. In reality, highly connected components are partitioned together to minimize external terminals. Thus, formula $[3]$ produces conservative results. To account for optimized partitioning, we introduce a correction constant $P_h<1$ (similar to $P_g$) to reduce the estimated number of terminals:After substituting the variables into $[4]$, we arrive at the New Hypergraph-based Rule:It is easy to see that if each net connects only two components ($m=2$), the New Hypergraph-based Rule becomes equivalent to the New Graph-based Rule.

Our comparative study of graph-based versus hypergraph-based rules showed that the hypergraph model is approximately 1.2% more accurate.

Figure 4. "Comparison of the Rules" illustrates a comparison of Rent’s Rule against the new generalizations.

Properties used: $N=10,000$; $K=4$; $P_{rent}=0.6$; $t=4$; $P_g=0.6$; $m=3.2$; $P_h=0.23$.

4.2 Justification Summary

The following final points support the justification of the new rules:
• Experimental Alignment: They provide a superior match to experimental data across all regions.
• Convergence: Terminal counts are close to Rent’s predictions when $g$ is small.
• Structural Commonality: There is a fundamental commonality in the rule structures; they can be effectively approximated by Rent’s Rule for very small $g$.

________________________________________
5. Conclusion

The proposed New Rules resolve long-standing issues in VLSI modeling by explicitly incorporating $N$ (system size), $t$ (average pins), and $m$ (net fan-out). By naturally constraining terminal counts at $g=N$, these rules provide a mathematically sound bridge across both Region I and Region II of Rent's curve.
________________________________________
References

1. Rent, E.F. (1960): Original discovery (often an internal IBM memorandum).

2. Landman, L.A. and Russo, R.L. (1971): "On Pin Versus Block Relationship for Partitions of Logic Graphs," IEEE Transactions on Computers, vol. C-20, no. 12, pp. 1469-1479.

3. Donath, W.E. (1981): "Wire Length Distribution for Computer Logic," IBM Technical Disclosure Bulletin, vol. 23, no. 11, pp. 5865-5868.

4. Heller, W.R., Hsi, C. and Mikhail, W.F. (1978): "Chip-Level Physical Design: An Overview," IEEE Transactions on Electron Devices, vol. 25, no. 2, pp. 163-176.

5. Natalie Heroux (2025): “Tetelbaum’s Law of Computer Systems, ourbigbook.com/nheroux/tetelbaum%E2%80%99s-law-of-computer-systems

6. Sutherland, I.E. and Oosterhout, W.J. (2001): "The Futures of Design: Interconnections," ACM/IEEE Design Automation Conference (DAC), pp. 15-20.

7. Davis, J. A. and Meindl, J. D. (2000): "A Hierarchical Interconnect Model for Deep Submicron Integrated Circuits," IEEE Transactions on Electron Devices, vol. 47, no. 11, pp. 2068-2073.

8. Stroobandt, D. A. and Van Campenhout, J. (2000): "The Geometry of VLSI Interconnect," Proceedings of the IEEE, vol. 88, no. 4, pp. 535-546.

9. TETELBAUM, A. (1995). "Generalizations of Rent's Rule", in Proc. of 27th IEEE Southeastern Symposium on System Theory, Starkville, Mississippi, USA, March 1995, pp. 011-016.

10. TETELBAUM, A. (1995). "Estimations of Layout Parameters of Hierarchical Systems", in Proc. of 27th IEEE Southeastern Symposium on System Theory, Starkville, Mississippi, USA, March 1995, pp. 123-128.

11. TETELBAUM, A. (1995). "Estimation of the Graph Partitioning for a Hierarchical System", in Proc. of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, San Francisco, California, USA, February 1995, pp. 500-502.
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