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Abstract

The divergence between executive compensation and median employee wages has reached historic levels, yet current methods for determining "fair" pay often rely on peer benchmarking and market heuristics rather than structural logic. This paper proposes a new mathematical framework for determining the CEO-to-Employee Pay Ratio ($R_{ceo}$) based on the internal architecture of the corporation. By integrating the Pareto Principle with organizational hierarchy theory, we derive a scalable model that calculates executive impact as a function of the company's size, span of control, and number of management levels.

Our results demonstrate that a scientifically grounded approach can justify executive compensation across a wide range of organization sizes—from startups to multinational firms—while providing a defensible upper bound that aligns with organizational productivity. Comparison with empirical data from the Bureau of Labor Statistics (BLS) suggests that this model provides a robust baseline for boards of directors and regulatory bodies seeking transparent and equitable compensation standards.

________________________________________
1. Introduction

The compensation of Chief Executive Officers (CEOs) has evolved from a matter of private contract into a significant issue of public policy and corporate ethics. Over the past four decades, the ratio of CEO-to-typical-worker pay has swelled from approximately 20-to-1 in 1965 to over 300-to-1 in recent years ${}^1$.

Developing a "fair" compensation model is not merely a question of capping wealth, but of aligning the interests of the executive with those of the shareholders, employees, and the broader society. As management legend Peter Drucker famously noted:
"I have over the years come to the conclusion that (a ratio) of 20-to-1 is the limit beyond which it is very difficult to maintain employee morale and a sense of common purpose." ${}^2$

________________________________________
2. Overview of Existing Works and Theories

The academic literature on CEO compensation generally falls into three primary schools of thought: Agency Theory ${}^3$, Managerial Power Hypothesis ${}^4$, and Social Comparison Theory ${}^5$. While these provide qualitative insights, they often lack a predictive mathematical engine that accounts for the physical size and complexity of the firm.

________________________________________
3. Principles and Assumptions

We propose a framework for estimating the CEO-to-Employee Pay Ratio ($R_{ceo}$) based on five realistic and verifiable assumptions:

Assumption 1: The Pareto Principle. We utilize the 80/20 rule, assuming that the top 20% of a leadership hierarchy is responsible for 80% of strategic results ${}^6$.

Assumption 2: Span of Control. The model incorporates the total number of employees ($N$), hierarchical levels ($K$), and the average number of direct reports ($D$), benchmarked at $D=10$ ${}^7$.

Assumption 3: Productivity Benchmarking. The average worker's productivity ($P$) is set to 1 to establish a baseline for relative scaling.

Assumption 4: Hierarchical Scaling. Strategic impact increases as one moves up the organizational levels, but at a decaying rate of intensity ($H$).

Assumption 5: Occam’s Razor. We prioritize the simplest mathematical explanation that fits the observed wage data ${}^8$.

________________________________________
4. The CEO-to-Employee Pay Ratio ($R_{ceo}$)

The fair compensation of a CEO ($S_{ceo}$) is expressed as:Where $S$ is the average worker's salary. For an organization with $K$ hierarchical levels, we calculate the number of levels as:The total CEO productivity ratio is then modeled as a geometric progression of impact:

________________________________________
5. Model Discussion

To validate the model, we compared our theoretical $R_{ceo}$ against Bureau of Labor Statistics (BLS) data groups ${}^9$.

The current statistics: Ranges for employee salaries (S1, S2), CEO Compensation (CEO1, CEO2), and CEO-to-Employee Pay Ratios (R:1) (R1, R2) are presented in the table below.

Using values of $D=10$ and $H=0.7$, the model tracks the reported ratios of mid-to-large cap companies with high accuracy.

Notes: This table compares empirical (reported) CEO-to-employee pay ratios from large public firms against modeled estimates (Model Rceo), which adjust for factors like company size, industry, and equity components. Data is illustrative based on 2024–2025 benchmarks; actual ratios vary widely.

Figure 1. "CEO Payment Model vs Data" illustrates the comparison.

Special cases like Tesla (2024) demonstrate that while traditional hierarchy explains baseline pay, performance-based stock options can create extreme outliers reaching ratios of 40,000:1 ${}^{10}$.

________________________________________
6. Conclusion

This paper has introduced a consistent and scientifically grounded framework for determining CEO compensation. By shifting the focus from "market guessing" to hierarchical productivity scaling, we provide a transparent justification for executive pay. As an additional feature, the upper bounds of managerial remuneration at all hierarchical levels can be identified across corporations of any size.

The strength of this model is its mathematical consistency across all scales of enterprise. While determining the exact hierarchical decay constant ($H$) remains an area for further empirical refinement, the framework itself provides a logical and defensible constraint on executive compensation, ensuring alignment between leadership rewards and structural organizational impact.

________________________________________
7. References

1. Mishel, L. and Kandra, J. (2021). "CEO pay has skyrocketed 1,322% since 1978," EPI.
2. Drucker, P. F. (1984). "The Changed World Economy," Foreign Affairs.
3. Jensen, M. C. and Meckling, W. H. (1976). "Theory of the firm," J. Finan. Econ.
4. Bebchuk, L. A. and Fried, J. M. (2004). Pay Without Performance. Harvard University Press.
5. Adams, J. S. (1963). "Towards an understanding of inequity," J. Abnorm. Soc. Psych.
6. Koch, R. (1998). The 80/20 Principle. Currency.
7. Gurbuz, S. (2021). "Span of Control," Palgrave Encyclopedia.
8. Baker, A. (2007). "Occam's Razor," Stanford Encyclopedia.
9. BLS (2024). "Occupational Employment and Wage Statistics," U.S. Dept of Labor.
10. Hull, B. (2024). "Tesla’s Musk pay package analysis," Reuters.
________________________________________

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.
________________________________________

Alexander Y. Tetelbaum
Independent Researcher.
alex.tetelbaum@gmail.com
----------------------------------

This paper introduces the concept of the Speed of Life — a normalized, fuzzy measure of experiential life velocity across the human lifespan — and demonstrates that three mutually contradictory theories of its behavior can each be rigorously justified, mathematically formalized, and rendered internally consistent using the same set of observations, considerations, and starting assumptions. Theory 1 holds that the Speed of Life increases monotonically from birth to death. Theory 2 holds the exact opposite: that it decreases monotonically from birth to death. Theory 3 proposes an inverted U-shaped trajectory, accelerating from birth to a peak in middle age and decelerating thereafter. We argue that the coexistence of three contradictory yet defensible theories is not a failure of analysis but a feature of fuzzy, subjectively defined variables — and that this methodological observation has broad implications for how we evaluate theories built on loosely defined constructs in psychology, social science, and philosophy.

Keywords: Speed of Life, time perception, subjective time, fuzzy logic, temporal experience, lifespan psychology, life velocity, mathematical modeling

Time is the one resource distributed with perfect equality. Every human being receives exactly twenty-four hours per day, from birth to death, without exception. And yet the experience of time — its subjective velocity, its felt passage, its qualitative texture — varies enormously across individuals, circumstances, and stages of life.

Children waiting for a birthday report that the hours crawl. Adults reflecting on a passed decade report that the years flew. The elderly frequently observe that life accelerates with age. But careful observation also reveals the opposite: that the very young live with maximum experiential intensity, that retirement slows the daily pace to a crawl, and that the frantic middle years represent a peak of temporal compression from which life descends into gradual deceleration.

Which of these observations is correct? All of them. And that is precisely the problem.

This paper introduces a formal — if deliberately fuzzy — definition of the Speed of Life and uses it to construct three mutually contradictory mathematical theories of how life speed evolves across the human lifespan. Each theory is internally consistent. Each is supported by plausible observations and reasoning. Each contradicts the other two. And all three can be derived from essentially the same set of starting considerations, depending on which factors are weighted and how the core construct is defined.

The goal is not to determine which theory is correct. The goal is to demonstrate that when a variable is loosely defined — when it is fuzzy in the technical sense — contradictory conclusions can be rigorously derived from the same premises. This observation has implications well beyond temporal perception, touching on the epistemological foundations of any science that relies on subjectively defined constructs.

Let $t\in [0,1]$ denote normalized age, where $t=0$ represents birth and $t=1$ represents death. The normalization maps any individual lifespan, regardless of its actual duration, onto the unit interval.

Let $S(t)\in [0,1]$ denote the Speed of Life at normalized age $t$, where $S=0$ represents complete temporal stillness and $S=1$ represents maximum life speed.

Formally:

Where $A(t)$ is the activity density function

This definition is deliberately and necessarily fuzzy. "Meaningful activities," "obligations," and "experiences" resist precise operationalization. It is precisely this fuzziness that enables the contradictions that follow.

A newborn enters the world with zero obligations, zero commitments, and zero responsibilities. By the definition above, $A(0)\approx 0$ and therefore $S(0)\approx 0$.

As the child grows, activity density increases. School imposes schedule. Friendships require cultivation. Adolescence brings examinations, social obligations, and the first experiences of time pressure. Young adulthood adds career, partnership, and financial responsibility. Each decade compounds the accumulation.

Crucially, the subjective experience of time passing accelerates with age. William James observed in 1890 that "the same space of time seems shorter as we grow older." The ratio of any given year to total life experience decreases monotonically: a year at age five represents 20% of all experienced life; a year at age fifty represents approximately 2%. As this ratio decreases, subjective time accelerates.

By this reasoning, $S(t)$ is monotonically increasing, approaching its maximum at the moment of death.

For $\beta =2$: $S_1(t)=2t-t^2$

- $S_1(0)=0$ ✓
- $S_1(1)=1$ ✓
- $S_1^{\prime }(t)>0$ for all $t\in [0,1)$ ✓ (monotonically increasing)
- $S_1^{\prime \prime }(t)<0$ for $\beta >1$ (decelerating rate of increase)

The same newborn, viewed through a different lens, is not idle — they are maximally engaged. Every sensation is novel. Every experience requires full conscious processing. The infant brain forms approximately one million new neural connections per second. By any neurological measure, the infant is living at maximum Speed of Life, so $S(0)=1$.

As routines develop, novelty depletes. With each passing year, a larger fraction of daily life is managed by automaticity rather than conscious engagement — reducing experienced life density even as clock-time obligations increase.

By retirement, obligations have dissolved. Time stretches. The elderly person who reports that "the days are long but the years are short" is describing a Speed of Life that has declined to near its starting asymptote. At death, $S(1)\approx 0$.

- $S_2(0)=1$ ✓
- $S_2(1)=0$ ✓
- $S_2^{\prime }(t)<0$ for all $t\in (0,1)$ ✓ (monotonically decreasing)
- $S_2^{\prime \prime }(t)>0$ for $\gamma <1$ (decelerating rate of decrease)

Theory 3 reconciles Theories 1 and 2 by observing that both are partially correct — each describing a different phase of the same inverted U-shaped trajectory.

The ascending phase: from birth through young adulthood, activity density increases as the individual acquires obligations, relationships, and responsibilities. $S(t)$ rises.

The peak: in middle age — roughly when career demands, parenting obligations, household responsibilities, and care for aging parents simultaneously converge — activity density reaches its maximum. $S(t)=1$.

The descending phase: from peak midlife through old age and death, obligations dissolve, novelty depletes, and activity density declines. $S(t)$ falls back toward zero.

- $S_3(0)=0$ ✓
- $S_3(1)=0$ ✓
- $S_3(0.5)=1$ ✓ (peak at midlife)
- $S_3^{\prime }(t)=4(1-2t)$, zero at $t=0.5$ ✓
- $S_3^{\prime \prime }(t)=-8<0$ (concave everywhere)

For asymmetric peak placement at $t^{\ast }\ne 0.5$:

Where $t^{\ast }=a/(a+b)$ and $C$ normalizes the maximum to 1.

The coexistence of three contradictory yet internally consistent theories is not an accident of careless analysis. It is a direct consequence of the fuzziness of the central construct.

The Speed of Life is not a single thing. It is a family of related but distinct phenomena:

- Subjective time perception (Theory 1's domain)
- Cognitive engagement intensity (Theory 2's domain)
- Obligation and activity density (Theory 3's domain)

Each is real. Each is measurable in principle. And each yields a different theory when treated as the definition of Speed of Life.

This observation generalizes. Wherever a construct in psychology, social science, or philosophy admits multiple reasonable operationalizations, contradictory theories can be built on the same empirical foundation. The contradiction is not between the theories — it is between the hidden definitional choices embedded in each.

We have introduced the Speed of Life as a normalized fuzzy measure of experiential life velocity and derived three mutually contradictory mathematical theories of its behavior across the human lifespan. Each theory is internally consistent, empirically motivated, and mathematically well-formed. Each contradicts the other two. And all three emerge from essentially the same starting observations.

This is not a paradox. It is a demonstration. When a variable is sufficiently loosely defined, the conclusions that follow are determined less by the evidence than by the definitional choices made before the analysis begins.

The Speed of Life accelerates from birth to death. It decelerates from birth to death. It peaks at midlife and decelerates thereafter. All of these are true, in the sense that each can be rigorously derived. None of them is true in the sense that none is uniquely forced by the evidence.

It would be remiss not to acknowledge that this paper is, in part, a deliberate provocation. The Speed of Life is not a variable anyone has measured, and the three theories presented here are not competing empirical claims. They are demonstrations. What they demonstrate is that mathematical rigor is a necessary but insufficient condition for truth — that a proof is only as sound as the definition on which it rests. When that definition is fuzzy, almost anything can be proved. This paper has proved three contradictory things. The reader is invited to draw their own conclusions.

One final admission: this is, at heart, a playful paper. But its playfulness carries a serious point. Given a sufficiently loose definition, a mathematician can prove almost anything — rigorously, elegantly, and wrongly. The Speed of Life is the proof of that proof.

Acknowledgments: The author thanks life itself for providing the dataset.

Conflicts of interest: None declared.

References
- James, W. (1890). The Principles of Psychology. Henry Holt and Company.
- Eagleman, D. (2011). Incognito: The Secret Lives of the Brain. Pantheon Books.
- Friedman, W. J., & Janssen, S. M. J. (2010). Aging and the speed of time. Acta Psychologica, 134 (2), 130–141.
- Wittmann, M., & Lehnhoff, S. (2005). Age effects in perception of time. Psychological Reports, 97 (3), 921–935.
- Langer, E. J. (1989). Mindfulness. Addison-Wesley.
- Modigliani, F. (1966). The life cycle hypothesis of saving. Social Research, 33 (2), 160–217.
- Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8 (3), 338–353.
- Ornstein, R. E. (1969). On the Experience of Time. Penguin Books.

Literally: East of the Mountain.

Literally: West of the Mountain.

Fucking greedy bastards. They take down every YouTube video showing his talks from time to time. Then more people come and reupload, but the links keep churning. Goes entirely against the spirit of the thing. If only Alan knew more about copyright and had more clearly put his stuff on the public domain. Shame!

Maybe Ciro Santilli should do something useful and remarkable so that someone might actually want to read his biography in the first place. But hey, procrastination.

Ciro Santilli was born in Brazil in the small/medium city of Rio Claro, São Paulo (~200k people in 2020) in the State of São Paulo in 1989 AD.

When he was 3, the family moved within Rio Claro to Av. 44 - Parque Universitario, Rio Claro - SP, 13504-360.

In 1995 the family moved to Jundiaí (Rua José Milani, CEP 13207-691) for his father's work. He played a lot on the streets. Ciro learned to cycle there on a slight downhill. He was mildly bullied by some kids on the street. He played NES games that were far too hard for him which made him made and required his parents to call the teenage neighbors to beat. The neighbors had some DOS games which felt like mystery and magic.

In 1997 family finally moved to Santos, São Paulo, Brazil again for his father's work.

At the age of 10, Ciro Santilli spent 10 months in Coventry, United Kingdom, where he greatly improved his English.

After Coventry, Ciro's family went back to Santos, São Paulo, Brazil, which made a deep impression on Ciro, until he Ciro Santilli's undergrad studies at the University of São Paulo in 2007.

In 2010, as mentioned at Section "Ciro Santilli's formal education", Ciro as admitted in a double degree program at the École Polytechnique, France, where he stayed until 2013. Going to France was a mind blowing, life changing event.

Welcome to my home page!

Welcome to my home page!

www.tritoncycles.co.uk/urban-bikes-c4/hybrid-c30/trek-fx-3-equipped-urban-bike-2021-p39860/s102315 XL color Dnister Black. £675.00 plus £35.00 delivery. Disc breaks, 2x9 gears. Bought 2023 on a bit of a sale.

The 2020 one on official website: www.trekbikes.com/gb/en_GB/bikes/hybrid-bikes/fitness-bikes/fx/fx-3-disc/p/28474/ archive.ph/wip/jMDdw TODO find 2021.

Specs upon buying:

Weight after bought measured differentially on bathroom scale: 13.5 kg

Gears: 11-36T on back, 46-30T on front. So min speed is 30/36 = 0.83 and max is 46/11 = 4.18, so range is 4.18 / 0.83 ~= 5. Compared to Kross which was 11-32 on back vs 48-28 on front, so min 28/32 = 0.875 and max 48/11 = 4.36. So the ranges are very comparable! This confirms Ciro's suspicion that 3 speeds on cranckset is just useless.

I was sized just at the lower limit of XL, but decided to go for it. The seatpost goes all the way into the tube, which is a good thing, might help with noises?

Frame serial under bottom bracket: WTU173C0374S. Registered at: www.bikeregister.com/

Initial impressions:

I'm unable to place a U-lock carrier on this bike in a way that doesn't block one of the two bottles inside frame, or hits leg on if on side. Tried both Kryptonite and Abus ones I have lying around. Shame. Also the Topeak MTS Trunkbag does not fit well on the pannier rack, it is not wide enough to strap: 11cm outer width, 9cm inner width, same geometry as that of the Liv bike.

Log:

2026-08-08: failed to unclip at slow speed, fell over, and broke derailleur hanger... I'm unable to unbolt the frame bot, hex is rounded off even though I never touched it I think. I might to need help for this... Buying:2026-08-08: failed to unclip at slow speed, fell over, and broke derailleur hanger... I'm unable to unbolt the frame bot, hex is rounded off even though I never touched it I think. I might to need help for this... Buying:The repair process was a shitshow:

2026-04-15: was hearing back wheel hit something periodically. First I separated the disk breaks with the tool but didn't solve. Then I noticed broken spoke, 9th clockwise from tube valve. I replaced it, but unfortunately spreading the disk break made me lose back breaks so I had to go into break fluid territory. Some conclusions: you can't do break fluid without the break pads and disk break in the break. This opens it up for contamination. The correct thing to do is to put a plastic protection on top of them. I spilled fluid on break at one point, it was horrible, I ended up changing the break pad which improved things a bit. Overall break felt slightly worse than before but passable.

2026-02-13: chain reched .7, changed it to new KMC X9. x9 9 speed chain 114 links. www.amazon.co.uk/dp/B07J38GX7L Started slipping like crazy on cassette when I pedaled hard, changed it to www.amazon.co.uk/dp/B073H3DST7 SHIMANO CS-HG400 9-speed cassette (11-36 teeth) which internet said was compatible and it seems to be. Didn't manage to remove it and needed help from bike shop as usual. After that it stopped slipping.

2025-09-30: kickstand broken while taking it on car. Proceeding to buy Bontrager Rear Mount Adjustable Kickstand www.trekbikes.com/gb/en_GB/equipment/bike-accessories/bike-kickstands/bontrager-rear-mount-adjustable-kickstand/p/36015/ 20 pounds + 4 delivery. The new kickstand is slightly different from the previous one, it is less slick looking, but perhaps a bit more sturdy, notably the little arm is shorter which might help prevent the previous failure mode. It came with 3 types of screws, but the two of correct diameter were a bit shorter or longer than the old ones so I just kept the old ones.

2025-09-13: chain reached 0.75, rebuying KMC X9. x9 9 speed chain 114 links. www.amazon.co.uk/dp/B07J38GX7L

2025-08: rear mudgard broke in the exact same way as last time, so I'm saying fuck this bullshit and I just fixed it up with some wire which will likely be more resistant than the shitty metal thing they had.

2025-06-26: I suddenly lost the ability to shift the front gear, cable seemed fine, upon inspection I noticed that the pedal could move sideways. Maybe linked to screeching sound that has been coming from that area for a while. As per www.youtube.com/watch?v=cPQyQnNdews leading to www.youtube.com/watch?v=QqBtB8Kyl2U this is a different system from the Kross one, two piece compression slotted rather than 3 piece. The side to side issue is mentioned precisely at: youtu.be/QqBtB8Kyl2U?t=357 Buying the standalone removal tool: www.amazon.co.uk/Professional-Extracto-Compatible-Hollowtech-Maintenance/dp/B0DP6QF11X Fixing it fixed the shift issue, but I noticed that the little plastic piece was broken. Also a few days later it became loose again during a ride, but thankfully I had the tool and put it back mid rid. TODO wanted to rebuy it but it's really hard to find... Going to try this one: www.bikeparts.co.uk/products/shimano-altus-fc-mt210-chainset-46-30-9-speed-black-170-mm-w-o-chainguard 175 mm and see if it comes with both arms. Bought it. has both arms. But I can't fit it, appears to be 1 or 2 mm short compared to the other cranck, the new arm does go on the old crank. There is a small rubber seal that is perhaps not too well inserted which might be to blame? Fuck my life! On the old one, it seems to be fully inserted into a tight hole. I didn't manage to insert the new one enough to make it fit, but if I remove it then it fits... just removing it for now then, how important could that be? Famous last words. Web says 12-14 nM torque on holding screws.

2025-06: rebuying 2x Shimano B05S brake pads, breaking is not too great and I don't want to repeat last year's episode of on-road failure. Will also be bleeding for the second time, God help me. The old pads weren't too bad, but I replaced them anyways. The bleeding was a failure and the brakes were too soft to be useful at the end, fuck. Went to two bike shops, both said bleeding not too bad, one said likely problem was contamination/friction issues. Tried cleaning rotors further, improved. Buying sandpaper and new rotors as well 2x SHIMANO SM-RT26 6-Bolt Disc Rotor and also got two more brake pads, totalling 3 new before I start putting them in with new rotor.

2025-04: rear shift cable snapped near shifter when going over a cattle grid, was already a bit off in the previous days after last adjust I then noticed. I'll buy a SHIMANO Gear Inner Wire Y60098070 on Amazon and might pass by bike shop as well I'll see.

2025-03: rear middle metal connection broke off read mudguard. Rebuying £21.04 from: www.tritoncycles.co.uk/frames-forks-c6/frame-fork-spares-c152/eurofender-snello-700c-fenders-46mm-rear-p33330/s93941 Quite disappointed with that, didn't come with any screws, though perhaps these are frame dependant. And the drilling was a bit different, this model seems to be intended to have some wiring passing inside via some rubber tubes that I removed, and unfortunately the hole on front was too high up and didn't match up with the screw hole on frame. I just removed the broken metal piece from it and put it on the old mudguard.

2024-09: buying CHUMXINY Brake Bleed Kit: www.amazon.co.uk/dp/B0CXDQ2S1S I hope I don't regret this. But I can't find an original one on Amazon either so... £12.74. Quality is a bit crap but seems to work. The 7mm wrench however is so bad that it can't unscrew a 7mm nut.

2024-09: chain reached 0.75, rebuying KMC X9. x9 9 speed chain 114 links: www.amazon.co.uk/dp/B07J38GX7L

2024-09: when going down a hill in intense breaking, noise started coming from back disk brake. Tried to adjust, but made it worse, had to bail out on nearby bike shop. Readjusted, told me to replace back pads and likely bleed. Bought two Shimano B05S brake pads to replace the B01S that came with the bike, Internet says compatible. Brake maintenance finally caught up with me, it was a great summer and I kept putting it off. Changed just the back one for now, front still has life it seems.

2024-06: noticed a sideways lump on back tyre between spoke 23 and 24 when washing. The only sign of damage was internal near grip part, I could see some clear shearing... God those tyres are crap, or at least crap to what I've subjected them. Inner tube and outside seem perfect, so weird. If only the Schawlbe had worked! But OK now I'm forced to get something new and better, thinking:They had a Schwalbe Marathon Plus 700x35c (non-Tour, as that one didn't fit as per previous experiment) on the shop and I went for it. They said 35c was the maximum recommended for that mudgard which is a bit of a shame, but good to know. Also getting one from Amazon to match on front as shop only had one.

2024-04: micropuncture on front tube? Could not identify during ride, refilled and was OK to get home. Almost no pressure in middle of ride. Passed through some deep water which may have made things worse? Had a small pice of glass-like material stuck in tyre, but didn't seem to penetrate? One day and a half later, was full empty again after full fill without riding, so confirming micropuncture theory. Buying 2x Conti Tube Cross 28 (700C) presta inner tubes.

Tried to put a spare Schwalbe Men's Marathon Plus Tour 700x40C on the bike to get better stability. It seemed to fit the wheel quite well, but unfortunately failed to go well in the fork because it was too tall and hit the mudgard. Without mudgard it might work. Measured height was 6cm for Schwalbe vs 5.3cm for the stock one.

2024-02: another broken spoke... what crappy weak wheels!!! Bought Bavel 290 mm spokes 36pcs 14G J Bend Silver. Broken position was 22nd clockwise from inner tube valve hole.

2023-12: chain reached 0.75, rebuying KMC X9. x9 9 speed chain 114 links.

2023-12: broken spoke! Also had to replace nipple because broke inside nipple. At least learned how to change tire at last.

2023-05: buying another pair of Shimano 324 combination pedals for it: www.wiggle.com/p/shimano-m324-combination-pedals

2023-05: rebuying another Topeak Aero Wedge saddle bag for this new bike

2023-05: 2x Elite custom race cage water bottle frame cages, felt great, much better than those on the Kross! www.elite-it.com/en/products/bottle-cages/fibre-reinforced-material-frp/custom-race

2023-05: ParkTool TR1 tyre levers + GP-2 super patch kit

2023-05: Kask Rapido Medium Black/Yellow fluo helmet, got a more racy helmet too! 69 pounds, way more any previous helmet I've owned: www.kask.com/en/cycling/helmets/rapido_24_2.htm

2023-05: Lezyne hand pump, I think Sport Drive HP? ride.lezyne.com/collections/hand-pumps/products/sport-drive-hp

2023-05: Lezyne V-Pro 10 multi-tool ride.lezyne.com/products/v-pro-tool No spoke wrench unfortunately.

Initial components:

Annotated copypasta:Weight limit: This bike has a maximum total weight limit (combined weight of bicycle, rider and cargo) of 136 kg (300 lb)

OMG, Ciro Santilli only learned about this in 2021 after: twitter.com/ryancdotorg/status/1375484757916672000

Check out: OurBigBook.com, the best way to publish your scientific knowledge. It's an open source note taking system that can publish from lightweight markup files in your computer both to a multi-user mind melding dynamic website, or as a static website. It's like Wikipedia + GitHub + Stack Overflow + Obsidian mashed up. Source code: github.com/ourbigbook/ourbigbook.

Sponsor me to work on this project. For 1M USD I will quit my job and work on OurBigBook full time for three more years to try and kickstart The Higher Education Revolution. Status: ~44k / 400k USD. At 4M USD I retire/tenure and work on open STEM forever. How to donate: Section "Sponsor Ciro Santilli's work on OurBigBook.com".

I first quit my job 1st June 2024 to work on the project for 1 year after I reached my initial 100k goal mostly via a 1000 Monero donation.

Mission: to live in a world where you can learn university-level mathematics, physics, chemistry, biology and engineering from perfect free open source books that anyone can write to get famous. More rationale: Section "OurBigBook.com"

Explaining things is my superpower, e.g. I was top user #39 on Stack Overflow in 2023^[ref][ref] and I have a few 1k+ star educational GitHub repositories^{[ref][ref][ref][ref]}. Now I want to bring that level of awesomeness to masters level Mathematics and Physics. But I can't do it alone! So I created OurBigBook.com to allow everyone to work together towards the perfect book of everything.

My life's goal is to bring hardcore university-level STEM open educational content to all ages. Sponsor me at github.com/sponsors/cirosantilli starting from 1$/month so I can work full time on it. Further information: Section "Sponsor Ciro Santilli's work on OurBigBook.com". Achieving what I call "free gifted education" is my Nirvana.

This website is written in OurBigBook Markup, and it is published on both cirosantilli.com (static website), outbigbook.om/cirosantilli (multi-user OurBigBook Web instance) and github.com/cirosantilli/cirosantilli.github.io-md (censorship resistant Markdown render directly on github.com). Its source code is located at: github.com/cirosantilli/cirosantilli.github.io and also at cirosantilli.com/_dir and it is licensed under CC BY-SA 4.0 unless otherwise noted.

To contact Ciro, see: Section "How to contact Ciro Santilli". He likes to talk with random people of the Internet.

GitHub | Stack Overflow | LinkedIn | YouTube | Twitter | Wikipedia | Zhihu 知乎 | Weibo 微博 | Other accounts

Besides that, I'm also a freedom of speech slacktivist and recreational cyclist. I like Chinese traditional music and classic Brazilian pop. Opinions are my own, but they could be yours too. Tax the rich.

Let's create an educational system with:

I offer:

My approach is to:For minors, parents are welcome to join video calls, and all interactions with the student will be recorded and made available to parents.

I have a proven track of explaining complex concepts in an interesting and useful way. I work for the learner. Teaching statement at: Section "How to teach". Pricing to be discussed. Contact details at: Section "How to contact Ciro Santilli".

I am particularly excited about pointing people to the potential next big things, my top picks these days are:I am also generally interested in:

 -------------------------------------
|  Force of Will               3 U U  |
|  ---------------------------------  |
| |                  ////////////   | |
| |                ////() ()\////\  | |
| |               ///_\ (--) \///\  | |
| |        )      ////  \_____///\\ | |
| |       ) \      /   /   /    /   | |
| |    ) /   \     |   |  /   _/    | |
| |   ) \  (  (   /   / /   / \     | |
| |  / ) ( )  / (    )/(    )  \    | |
| |  \(_)/(_)/  /UUUU \  \\\/   |   | |
| .---------------------------------. |
| Interrupt                           |
| ,---------------------------------, |
| | You may pay 1 life and remove a | |
| | blue card in your hand from the | |
| | game instead of paying Force of | |
| | Will's casting cost.  Effects   | |
| | that prevent or redirect damage | |
| | cannot be used to counter this  | |
| | loss of life.                   | |
| | Counter target spell.           | |
| `---------------------------------` |
|                                     l
| Illus.  Terese Nelsen               |
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A quick 2D continuous AI game prototype for reinforcement learning written in Matter.js, you can view it on a separate page at cirosantilli.com/_raw/js/matterjs/examples.html#top-down-asdw-fixed-viewport. This is a for-fun-only prototype for Ciro's 2D reinforcement learning games, C++ or maybe Python (for the deep learning ecosystem) seems inevitable for a serious version of such a project. But it is cute how much you can do with a few lines of Matter.js!

<iframe src="_raw/js/matterjs/examples.html#top-down-asdw-fixed-viewport" width="1000" height="850"></iframe>