The Speed of Life Theories by Alexander Tetelbaum 2 2026-08-07 2026-08-07
Alexander Y. Tetelbaum
Independent Researcher.
alex.tetelbaum@gmail.com
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Abstract
This paper introduces the concept of the Speed of Lifea 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.
1. Introduction
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.
2. Definition of the Speed of Life
Let denote normalized age, where represents birth and represents death. The normalization maps any individual lifespan, regardless of its actual duration, onto the unit interval.
Let denote the Speed of Life at normalized age , where represents complete temporal stillness and represents maximum life speed.
The Speed of Life is defined as the normalized density of meaningful activities, obligations, experiences, and demands per unit of chronological time, relative to the maximum such density observed across the individual lifespan.
Formally:
Where 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.
3. Theory 1: The Monotonically Increasing Speed of Life
3.1 Theoretical Justification
A newborn enters the world with zero obligations, zero commitments, and zero responsibilities. By the definition above, and therefore .
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, is monotonically increasing, approaching its maximum at the moment of death.
Figure 1.
Speed of Life: Theory 1A
.
For :
Properties:
-
-
- for all ✓ (monotonically increasing)
- for (decelerating rate of increase)
Figure 2.
Speed of Life: Theory 1B
.
4. Theory 2: The Monotonically Decreasing Speed of Life
4.1 Theoretical Justification
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 .
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, .
Properties:
-
-
- for all ✓ (monotonically decreasing)
- for (decelerating rate of decrease)
Figure 3.
Speed of Life: Theory 2A
.
Figure 4.
Speed of Life: Theory 2B
.
5. Theory 3: The Inverted U — Maximum Speed at Midlife
5.1 Theoretical Justification
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. 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. .
The descending phase: from peak midlife through old age and death, obligations dissolve, novelty depletes, and activity density declines. falls back toward zero.
Properties:
-
-
- ✓ (peak at midlife)
- , zero at
- (concave everywhere)
For asymmetric peak placement at :
Where and normalizes the maximum to 1.
Figure 5.
Speed of Life: Theory 3
.
6. The Epistemological Point
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.
7. Conclusion
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.
Updates / Post OurBigBook job search round 2025 by Ciro Santilli 40 Created 2025-05-07 Updated 2025-07-16 2025-07-16
I shouldn't be doing this on funded OurBigBook time which is until the end of May, but I was getting too nervous and decided to start a casual job search to test the waters.
In particular I want to see if I can get past the HR lady step without toning down my online profiles. If nothing works out for the next round I'll be hiding anything too spicy like:
Another interesting point is to see if French companies are more likely to reply given that Ciro Santilli studied at École Polytechnique which the French worship.
Figure 1. . Ciro's profiles may be a bit too much for the HR ladies who reject his job applications on the spot. To be fair, perhaps not enough years of experience for certain applications and job hopping may have something to do with it too. But since they don't ever tell you anything not to get sued, we'll never know.
I'm looking in particular either for:
I spent the last two weeks doing that:
I'll make sure to add two weeks of OurBigBook work after May to make up for this.
Updates / 60 new CIA website screenshots discovered on CQ Counter by Ciro Santilli 40 Created 2025-04-15 Updated 2025-07-16 2025-07-16
While procrastinating I suddenly remembered that cqcounter.com/siteinfo/ has screenshots of many many old websites, and I decided to look at possible hits in known IP ranges for which the Wayback Machine archive was broken.
Luckily I had already maintained a clear list of known domains in IP ranges which had no or broken wayback machine archive, so I just went over those.
This led to finding 60 novel screenshots of previously examined domains that are in common CIA-style, thus confirming them as hits beyond reasonable doubt in my mind. This also publicly revealed for the first time how a few new websites looked like, and what was their content, and in particular the target language, which could sometimes not be easily determined from the domain name alone.
This novel CQ Counter screenshot interpretation, plus a few new random discoveries and a slight relaxation of fingerprint requisites described described below moves us to 473 hits up from the previous 397!
The newly found websites were all just soulless bulk or mildly cute like the vast majority of them, but I did find found a few new screenshots of CIA websites that targeted other democracies:
I've also decided to now classify garanziadellasicurezza.com (Italy) as a hit due to various forms of supporting evidence being present. The archive is very broken however unfortunately.
Figure 3.
2011 cqcounter archive of ordenpolicial.com targeting Spain
. Source.
Figure 4.
2011 cqcounter archive of vejaaeuropa.com targeting Brazil
. Source.
Figure 5.
2011 cqcounter archive of european-footballer.com targeting Croatia
. Source.
The fingerprint of "having a visually similar CQ Counter screenshot" is definitely weaker than a Wayback Machine archive as we only have a screenshot and can't inspect the HTML to find the communication mechanism. But when the screenshot is perfectly in CIA style and in a known IP range, the evidence is too strong and we'll consider it as a hit moving forward.
I'm also going to reclassify a few previously known domains in confirmed IP ranges as hits as hits either when:
This is a slight moving of goalposts, but those cases just feel overwhelmingly probably.
I love how this project has led me to use whatever random sources come in hand! CQ Counter is the ONLY website that I know of besides the Wayback Machine that has historical screenshots of a huge number of domains. Their database is VERY complete. But they are so obscure!
They even have the old IP of the domain. But because they don't have reverse IP to domain reverse search, and are heavily CAPTCHAed preventing search engines from properly indexing them, we can't use them to fill in existing IP ranges... So the search for the most complete DNS database that doesn't cost 15k USD like DomainTools continues www.reddit.com/r/OSINT/comments/1j8uasm/does_domaintools_offer_historical_reverse_ip_ie/
Interestingly a large number of the websites with broken Wayback Machine are from regions outside of the USA, presumably being slower to load from Wayback Machine US-based servers makes he archives more likely to break.
Updates / 44 new CIA websites by Ciro Santilli 40 Created 2025-03-13 Updated 2025-07-16 2025-07-16
I found 44 new covert websites made by the CIA around 2010 bringing the total to 397!
Most websites were boring as usual, but one was slightly cooler: webofcheer.com is a comedy fansite featuring Johnny Carson, Charles Chaplin, Rowan Atkins (of Mr. Bean fame), The Three Stooges and some other Americans no one knows about anymore. There must have been a massive Johnny Carson amongst the contractors at that time, given that we previously also knew about alljohnny.com, a site dedicated fully to him! Both of these sites also serve as some of the earliest examples we've got so far, dating back to 2004 and 2005.
Figure 2.
2011 Wayback Machine archive of webofcheer.com scrolled to show Johnny Carson
. Source.
Figure 3. . Source. This one was a previously known website featuring Johnny Carson.
Another cool discovery is that I found the Getty Images source of the Jedi boy on their Star Wars themed site starwarsweb.net: web.archive.org/web/20101230033220/http://starwarsweb.net/ The photo can still be licensed today as of 2025: www.gettyimages.co.uk/detail/photo/little-jedi-royalty-free-image/172984439. I found it by searching for "jedi boy" on gettyimages.co.uk. The photo is credited to username madisonwi, presumably an alias of a photographer from Madison, Wisconsin. Inspired by this I reverse image searched and found the source of many other stock images from other websites, and I pinged their authors whenever I could locate them e.g. x.com/cirosantilli/status/1899750172260806711.
Figure 4.
Stock photo of a Jedi boy from Getty Images used on starwarsweb.net
. Source.
There were two small advances that led to the discovery of new domains:
With those two new developments, I then kicked off my pre-existing search pipelines searching for domain names with the word news on them, an amazingly efficient heuristic because many of the websites were disguised as news aggregators, and after a few hours theses new hits emerged. A few of those also led to the discovery of new IPs which then led to new domains.
One entirely new IP range was found around fastnews-online.com from 208.93.112.105 to 208.93.112.125. There were many domain names with very promising names in the range, but unfortunately for some reason most didn't have Wayback Machine Archives so I didn't count them as hits as per my guidelines.
Also the newly found todaysengineering.com at 208.254.38.39 appears to form an IP range with the previously known nejadnews.com at 208.254.38.56, but I couldn't find any other domains in the region with our current data sources.
All other domains either slot into previously known IP ranges, or more commonly don't currently have a known IP, though they would likely just slot in existing ranges if we had better data.
Thanks to Jack Rhysider from the Darknet Diaries podcast for pointing me to the existing of the 2022 Reuters article that kickstarted my research on the subject!
One outcome of this update is that I've increased my jq level to better automate the maintenance of the hits.json file were I store all the known websites in JSON format. I love that tool so much, I managed to merge two JSONs with it removing duplicates and then sort the JSON as desired. Beauty.
Test data 2 by cirosantilli2 0 Created 2025-03-09 Updated 2025-04-18 2025-04-18
Caroline Ulbricht by Ciro Santilli 40 Created 2025-02-11 Updated 2025-09-11 2025-09-11
Caroline Ulbricht is the wife of Ross Ulbricht, founder of the Silk Road website which allowed users to buy and sell illegal drugs online.
Caroline's maiden name is Caroline Debrion. She is French, and they met soon after Ross was put into jail. The couple married in 2024, one year before Ross' pardon in 2025. Ordinary this girl is not.
The spectacular comeback tour of Ross Ulbricht explains:
Shortly after Mr. Ulbricht was sentenced, he received a letter from a French tech worker in Los Angeles, Caroline Debrion. She told him that she had followed the trial and thought he could use support. They exchanged letters almost every day.
Other interesting quotes from the article related to Caroline:
“I dated so many jerks in L.A., and Ross was just very real and just super kind,” she said in 2021, according to a recording of a conversation with an associate obtained by The Times. “Ross is very, very romantic.”
Their relationship caused friction in the family. Ms. Ulbricht and Ms. Debrion sometimes butted heads, two people familiar with the women said. Against the wishes of Ms. Ulbricht, who had shielded her son from the media, Ms. Debrion helped persuade Mr. Ulbricht to participate in a Silk Road documentary, a last-ditch effort to sway public opinion.
Ciro Santilli had deduced caroline's maiden name "Caroline Debrion" before the New York Times article by stalking of her online accounts below. Also, her middle name might start with an L giving "Caroline L. Debrion", but this is less certain.
Likely accounts, all of which were live after Ross left prison, but were since taken down tested as of July 2025, including any Wayback Machine archives which are now marked "This URL has been excluded from the Wayback Machine":
Googling "Caroline Debrion" previously found a: cryptonews.com/fr/exclusives/saga-ross-ulbricht-et-silk-road-scandales-derriere-la-peine-3271/ (archive from January 2025) specificlly describing her in French as "voluntary and projct manager for the website FreeRoss", so that is almost certainly be her maiden name. As of July 2025, her family name "Debrion" had been removed from the article, which is marked "Last updated: June 18, 2025", so they presumably found that article after either reading this article or Googling her and requested the change. She is also quoted in French in that article, but before The spectacular comeback tour of Ross Ulbricht it was unclear if she actually spoke in French or if this was just translated from English. Translation of one of her quotes:
Caroline, a volunteer and project manager for the FreeRoss website, clarified the case: "Ross has never been prosecuted for injury or death, or for endangering another person."
As of 2025 most Google hits currently give "Rebecca Caroline Ulbricht Ferreira" however, an unrelated Brazilian molecular biology researcher from UNICAMP. So a good Google search is "caroline ulbricht" -Rebecca to remove the overlap. This leads to www.linkedin.com/in/carolineld/ hits.
x.com/pete_rizzo_/status/1882489065167356024 has a photo of them hugging at the jail parking lot just after he was released in 2025.
Figure 1.
Ross Ulbricht with his wife soon after he was released from jail in 2025 in front of a mural
. Source. He still seems to be in jail clothes?
Figure 2.
Ross Ulbricht with his wife soon after he was released from jail in 2025 on the road
. Source.
Figure 4.
Ross Ulbricht's wife's wife credly.com profile picture according to facecheck.id
.
Figure 5.
Ross Ulbricht's wife's wife meetup.com profile picture according to facecheck.id
.
Test data by cirosantilli2 0 Created 2022-11-29 Updated 2025-04-18 2025-04-18

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
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    • a Wikipedia where each user can have their own version of each article
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    Screenshot of the "Derivative" topic page
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    Visual Studio Code extension tree navigation
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    Figure 5.
    Web editor
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    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
    Video 4.
    OurBigBook Visual Studio Code extension editing and navigation demo
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