In John Carreyrou’s account of the Theranos fraud, there is a passage that allows you to understand how they were actually able to fool so many people for such a long time. A couple of employees were tasked with the job of retesting blood samples over and over to measure how much their results varied, and calculating each blood test’s coefficient of variation (CV). A test is generally considered precise if its CV is less than 10 percent. Theranos’s devices were proving unreliable, and these employees discovered that data runs failing to meet the required CV thresholds were simply discarded. The experiments were repeated until a satisfactory result appeared, and that result alone was recorded, reported, and eventually presented to investors, regulators, and patients. Carreyrou frames this as scientific misconduct, which it was — but he also translates it into mathematical terms that expose something deeper: a systematic exploitation of how probability actually works, dressed up as laboratory science.

The structure of what Theranos did can be stated precisely. If you flip a fair coin ten times, the probability that all ten land heads is exactly 1 in 1,024 — roughly 0.1 percent.¹ That outcome is rare, but rare is not the same as impossible, and crucially, rare does not mean unrepeatable. If you run that ten-flip experiment enough times, the all-heads result will eventually appear. In 1,000 attempts, the probability of seeing at least one perfect run of heads is approximately 62 percent — better than even odds.² What Theranos did was run the coin-flip experiment hundreds of times, wait for the inevitable all-heads sequence, and then present that sequence as proof that their coin always lands heads. The other 999 runs were not mentioned because they were never recorded. The denominator — the number of attempts it took to produce the result — was the one piece of information that would have made the whole picture legible, and blown the lid off, yet it was the one piece that was never disclosed.

This is not only a story about fraud. It is a story about how probability is used selectively — in science, in finance, and in the spaces between — to present results that cannot be evaluated without information that is almost never disclosed. The Theranos mechanism operates in legitimate finance with enough frequency that it deserves a structural framing. The coin flip makes that structure visible.

Two Ways to Lie With a Coin Flip

The Theranos mechanism contains two distinct mathematical sins, and it is worth separating them because each one operates independently and each one appears, in a different form, throughout financial markets.

The first sin is the hidden denominator. Theranos did not get lucky on the first attempt and mistake luck for truth. They ran the experiment many times — the passage makes this explicit — and kept running it until the favorable result appeared. That is precisely how probability works: given enough attempts, a 1-in-1,024 event will eventually occur, and in 1,000 attempts the probability of seeing it at least once is already 62 percent. The all-heads run was not a miracle. It was an inevitable consequence of sampling a distribution until its tail appeared. The fraud was in then presenting that tail event without its context — stripping away the hundreds of failed runs and offering the single success as if it had emerged from a single, unbiased trial. The denominator, the number of attempts required to produce the result, was the one piece of information that would have converted the result from compelling to meaningless. It was never disclosed because disclosing it would have ended the story immediately.

The second sin is more subtle and in some ways more damaging, because it is the error that persists even when people understand the first one. Theranos did not merely claim that their technology had worked once. They claimed it always worked — that the all-heads result revealed a permanent property of the system. This is a claim that no finite sequence of coin flips can ever support, regardless of how that sequence was generated. A run of ten heads, even if it were genuinely the first and only attempt with no selection involved, tells you nothing about what the eleventh flip will produce. The coin has no memory. A single realised outcome, however dramatic, cannot establish the distribution that generated it.

What Theranos presented as a stable, repeatable diagnostic capability was a single draw from a process they had deliberately sampled until a favorable draw appeared — and then they generalised from that one draw to a universal law about their technology. The probability of the observed result, given their methodology, was close to certain. The probability of the result they claimed — that the system reliably performed at that level — was precisely what had never been measured.

These two errors compound. The hidden denominator makes the result look like rare evidence. The false generalisation then treats that apparent evidence as a settled conclusion. Together they construct a narrative of validated technology from what is, mathematically, a single selected observation with no inferential content whatsoever.

How Markets Produce Winners Without Producing Skill

The Theranos structure — one actor, repeated attempts, hidden denominator — requires at least some degree of intention. Someone has to decide to discard the failed runs. The more pervasive version of the same mathematical problem requires no intention at all, because the selection happens automatically across a large population rather than deliberately within a single organisation.

The mechanism is straightforward. At any given moment there are thousands of fund managers each running their own ten-flip experiment — each operating a strategy with its own risk profile, leverage, and market exposure. In a population of 1,024 managers all running genuinely fair coins, the mathematics guarantees that roughly one of them will produce all-heads in any given ten-flip period. That manager did not cheat. They did not repeat their experiment until a favorable result appeared. They simply drew the outcome that a population of that size will always contain. But the investor observing only that manager’s record, without visibility into the other 1,023 records, cannot distinguish a lucky draw from genuine skill. Only the denominator — the full population that generated it — would tell them which one they are looking at, and that is precisely what is never disclosed.

This is structurally different from the Theranos case in one critical respect. Theranos required active concealment — someone had to decide not to record the failed runs. Survivorship bias in fund management requires no such decision. The failed funds simply close, their records become inaccessible, and the population that remains visible is automatically selected for strong performance. The denominator disappears not through fraud but through the ordinary mechanics of capital allocation and fund closure. The mathematical consequence is identical. An investor evaluating the visible population of funds with strong three-year records is in the same position as an investor shown only Theranos’s successful diagnostic runs — they are looking at a sample that has been selected from a much larger distribution, and the selection process has been designed, by structure rather than by intent, to show them only the tail.

The numbers make this concrete, and they are worth tracing across time because the story is more instructive than a single snapshot. There are approximately 3,000 hedge funds reporting returns at any given time, but that population is already the survivor — funds that closed are no longer in it. Over the decade from 2011 to 2020, the average hedge fund returned 5% annually against 14.4% for the S&P 500, a gap so large that $100,000 invested in the average fund grew to $160,000 while the same amount in an index fund grew to $364,000. The industry’s defenders argued that the environment changed after 2022 — rising interest rates and higher volatility were precisely the conditions their strategies needed. In 2023, the average hedge fund returned 4.4%, below even the decade average that had already been condemned as inadequate, while the S&P 500 returned 26.3%. In 2024 returns improved to 11.9% — and the S&P 500 returned 25%.³ Across the full period from 2011 to 2024, the average hedge fund has not beaten a passive index fund in a single year on a net-of-fee basis, through bull markets, bear markets, and rate cycles alike. The funds that produced the all-heads records within that period raised the capital. The funds that did not closed quietly. The investor who allocated to the visible winners was not observing skill. They were observing the tail of a distribution that the structure of the industry was always going to produce, and paying substantially for the privilege.

The within-firm version of this problem — backtesting — sits between the two cases in terms of intention. A quantitative fund testing hundreds of parameter combinations across historical data and then presenting the best-performing set is running the Theranos experiment deliberately, but without necessarily understanding it as selection. The strategy that survived the testing process looks, in the presentation, like the strategy that was identified through rigorous research. What it actually is, mathematically, is the all-heads run drawn from a large sample of attempts — the one that the process was always going to produce, because any process that searches a distribution until a favorable result appears will find one. The number of parameter combinations that failed to produce a strong backtest is not in the pitch deck because it was never written down, and because writing it down would immediately raise the question the denominator always raises: given how many attempts this required, what does the result actually tell us?

There is a more precise way to state how uninformative a short track record actually is, and the result is more extreme than you would expect. To reach conventional statistical confidence that a fund manager’s outperformance is real rather than lucky, assuming a realistic level of skill and typical return volatility, you need decades of continuous data.⁴ At three years you have noise. At ten years you have slightly less noise. The industry sells three-year records because that is what the capital allocation process rewards. The mathematics says a three-year record is not evidence of skill. It is evidence that the manager has been flipping long enough for a run to appear.

Proof of Returns Can Be Indistinguishable From a Coin Flip

There is a single question that, if applied consistently, would make a sophisticated observer dramatically harder to deceive across laboratory science, financial markets, and business narratives alike. That question is not “what did the data show?” — it is “how many times did you flip before showing me this?”

Markets run the same procedure with less intention and more systemic force. There is no single decision-maker at the center of survivorship bias who chooses which funds to hide. The mechanism operates through a distributed process of capital allocation and fund closure that, in aggregate, produces a population of visible managers whose records look exactly like what you would expect if you had selected them specifically for strong performance. The population that generates this effect is not conspiring. It is doing what all populations of coin-flippers do — producing some runs of heads, some runs of tails, and a small number of long consecutive streaks that, once they appear and are presented without their context, look like something other than what they are.

The denominator is not a technical footnote. It is the number that converts a result from evidence into noise, or from noise into evidence, depending on its size. Any performance record, any clinical result, any backtest, and any diagnostic threshold presented without its denominator is uninterpretable — a result whose meaning cannot be established until you know how many attempts it took to produce it. The practice of omitting it is so widespread, and so rewarded, that demanding it consistently marks you immediately as someone who understands what the data actually contains. In most rooms, that remains a minority position.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

Notes

¹ Each flip of a fair coin has a 1/2 probability of landing heads. Ten independent flips multiply ten times: 1/2 x 1/2 x…x 1/2 = (1/2)¹⁰ = 1/1,024 ≈ 0.098%.

² The probability of never seeing an all-heads result across 1,000 independent attempts is (1,023/1,024)1,000 ≈ e-0.977 ≈ 0.376, so the probability of seeing it at least once is 1 − 0.376 ≈ 62.4%.

³ Return data for hedge funds draws on the Barclay Hedge Fund Index and HFRI Fund Weighted Composite Index, as compiled by the American Enterprise Institute (2011–2020 figures), HFR (2023 figures), and Hedge Fund Alpha (2024 figures). S&P 500 annual return figures are total returns including reinvested dividends, sourced from standard index data for each calendar year. The $160,000 and $364,000 terminal values assume $100,000 invested at the start of 2011 compounded at the respective average annual rates through end of 2020.

⁴ To test whether a fund manager generates genuine alpha rather than luck, we use a t-statistic of the form t = (mean annual excess return ÷ volatility) × √years. Assuming an alpha of 2% annually and return volatility of 15%, the calculation to reach the conventional significance threshold of t = 2 is: (2% ÷ 15%) × √years = 2, giving √years = 15, and therefore years = 225. At three years the t-statistic is 0.23. At ten years it is 0.73. Neither approaches significance. If assumed alpha is raised to 5% — already a generous assumption — the required period falls to 36 years. To reach 19 years, which is the figure sometimes cited in academic literature, requires assuming alpha of roughly 6.5%, which most serious researchers would consider optimistic. The direction of the conclusion is robust across all realistic assumptions: the data horizon required to distinguish skill from luck in fund management is measured in decades, not years.

The Coin Always Lands Heads was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

On any given Tuesday in 2026, someone opens a laptop and uses an AI the way a previous generation used a search engine — except the interaction is longer, the output is richer, and it happens more often. In the morning, they ask it to summarize a report before a meeting. At lunch, they draft a difficult email to a client. In the afternoon, they ask it to explain a clause in a contract, check whether a connection leaves enough time between flights, and help structure a presentation they have been putting off for a week. By the time they close the laptop, they have had roughly twenty exchanges with the system across three sessions.

Those twenty exchanges generated approximately 16,000 tokens.¹ And behind that consumption — behind those three sessions and the feeling of frictionless intelligence — sits the most capital intensive industrial buildout in the history of the private economy.

The numbers involved dwarf every major industrial and infrastructure program in American history — individually and combined.² This column is not an analysis of artificial intelligence as a technology. It is not an assessment of what these systems can do, how they reason, or how close they are to human cognition. Those are important questions and other people are better positioned to answer them. This column is an attempt to do something more uncomfortable: to apply the same mathematical discipline that governs any capital-intensive industrial system to the one being built right now at a speed and scale that has no historical precedent — and to ask whether the economics actually close. Whether the physical infrastructure required to power what is being promised can be built on the timeline the capital assumes. Whether what is being described as the most transformative technology in human history is being funded with the efficiency it demands.

The Chain

A token is a unit of computation — a compressed fragment of language, roughly three-quarters of a word in English, that a neural network processes as a discrete mathematical object.³ What happens between the moment a user submits a prompt and the moment a response appears is a sequence of matrix multiplications executed across thousands of processing cores. For a model with seventy billion parameters — the scale of the frontier inference models most widely deployed as of 2026 — each token generated requires on the order of 140 billion floating-point operations (FLOPs).⁴ The speed feels instantaneous. The arithmetic is not reduced by that speed. Every floating-point operation still occurs. Every one consumes energy.

An NVIDIA H100 GPU draws approximately 700 watts under realistic inference workloads and processes roughly 24,000 tokens per second on a seventy-billion-parameter model at measured utilization.⁵ The energy cost at the chip is therefore approximately 0.10 joules per token. After accounting for the cooling systems, power conversion equipment, and facility overhead that every data center requires — measured by the industry as Power Usage Effectiveness, or PUE, which runs between 1.2 and 1.5 at hyperscaler facilities⁶ — the actual draw from the grid is approximately 2,160 joules for one user’s Tuesday. Taken alone that number is unremarkable. Multiplied across the estimated 300 million people using AI assistants daily across all major platforms — ChatGPT, Gemini, Copilot, Claude, Grok, and others — it becomes approximately 180,000 megawatt-hours of electricity consumed in a single day from conversational AI interactions alone.⁷ That is enough electricity to power approximately 6 million American homes for a full day.

That number is not what makes this interesting. What makes it interesting is that it is being multiplied simultaneously by two compounding growth rates that do not cancel each other and have no natural ceiling in sight.

The first is users. ChatGPT alone reported 900 million weekly active users as of early 2026, more than double the figure twelve months earlier.⁸ The second is tokens per user per day. In 2023, the typical AI interaction was occasional and exploratory. In 2026, it is continuous and functional. AI agents — systems that execute tasks autonomously on a user’s behalf, without requiring a prompt or a keypress — are beginning to multiply this figure again, because an agent running in the background consumes tokens whether or not its user is at a keyboard.

The mechanism that governs the relationship between these two growth rates and total energy demand was identified in 1865 by the economist William Stanley Jevons, studying coal consumption as steam engine efficiency improved. He observed that more efficient engines did not reduce total coal consumption — they reduced the cost of coal-powered work, which expanded the range of applications viable for steam power, which increased total consumption. Efficiency did not shrink the market. It grew it. The same mechanism operates in AI token consumption with unusual force: as inference costs fall — the population of economically viable use cases expands faster than the per-unit cost declines, and total energy demand rises even as energy per token falls.

US data centers consumed 183 terawatt-hours of electricity in 2024 — roughly equivalent to the annual electricity demand of Pakistan.⁹ The IEA projects that figure will double by 2030 in its base case, and triple for AI-focused data centers specifically. Goldman Sachs projects a 165% increase in data center power demand by 2030 versus 2023.¹⁰ Both projections were made before the full adoption of agentic AI systems, which Morgan Stanley estimates require twelve times more processing overhead per session than conversational chatbots.¹¹ The inference cost deflation driving this expansion is itself without precedent: a 280-fold reduction in the cost of equivalent AI capability between late 2022 and late 2024, continuing through 2025 at rates Epoch AI estimates at nine to 900 times per year.¹²

The chain, stated completely, runs as follows. A token requires floating-point operations. Floating-point operations require chips. Chips require data centers. Data centers require electricity. Electricity requires generation. Generation requires transmission. Transmission requires permits. Permits require time. And time, unlike capital, cannot be deployed faster by deciding to spend more of it.

Chart 1: What AI would consume if the only limit were demand. Three scenarios — conservative IEA base case (15%/yr), agentic AI (30%/yr), and AGI-level adoption (50%/yr) — show global data center electricity demand in terawatt-hours from 2024 to 2032, uncapped by any physical supply constraint. Even the most conservative scenario doubles current consumption by 2030. The chart is not a forecast. It is the demand side of the equation before physics enters the room.

Chart 2: The same demand lines, now measured against what the physical supply of each constraint can actually deliver. Panel A shows chip supply in millions of H100-equivalents — growing fast, partially self-correcting as new TSMC fabs come online in 2027–2028. Panel B shows grid supply in terawatt-hours — growing slowly, bounded by interconnection queues and transmission timelines that cannot be shortened by capital alone. The chip constraint is the immediate bottleneck. The grid constraint is the structural one. The key observation: by the time chip supply catches up with demand around 2027–2028, the grid becomes the binding constraint — and it does not self-correct.

The Three Clocks

There are three distinct physical layers in the AI infrastructure stack, and each operates on a different construction timeline. Understanding the difference between those timelines is the prerequisite for understanding where the capital currently being committed will — and will not — arrive on schedule.

The first layer is chips. An NVIDIA H100 GPU costs approximately $30,000 to $40,000. Its successor generations deliver progressively more compute per watt, which means each generation requires fewer chips and less electricity per unit of intelligence produced. TSMC, which manufactures the overwhelming majority of the world’s advanced AI chips, is building new fabs in Arizona, Germany, and Japan with volume production expected in 2027 and 2028, and has committed $165 billion to its US expansion alone.¹³ The chip constraint is real and binding today — H100 rental prices rose roughly 30% between November 2024 and early 2026 as customers unable to access newer generations reverted to older hardware¹⁴ — but it is self-correcting. Better chips do more per watt. New fabs come online. The constraint eases as technology advances.

The second layer is data centers. A hyperscaler data center takes eighteen to thirty-six months to design, permit, and construct in favorable jurisdictions. Google, Amazon, Microsoft, Meta, and Oracle — the five largest hyperscalers — committed a combined $690 billion in capital expenditure for 2026 alone, nearly double 2025 levels, with the majority directed at data center construction. Google committed between $175 billion and $185 billion for 2026, including $40 billion for three new Texas data centers through 2027. This layer is moving at the speed of capital and construction. It is the fastest layer to scale, although lately it has started to experience some political opposition.

The third layer is the grid. Generation, transmission, and distribution infrastructure operates on a timeline that neither capital nor engineering can accelerate past a physical and regulatory minimum. A new high-voltage transmission line in the United States takes seven to ten years from planning to energization. A new gas-fired combined-cycle plant takes three to five years. A nuclear plant takes ten to fifteen. The median time from an interconnection request to commercial operation for any new power project in the United States is currently four to five years, more than double from under two years for projects completed in the early 2000s. As of the end of 2024, approximately 2,300 gigawatts of generation and storage capacity were actively waiting in US interconnection queues — more than twice the country’s current total installed generating capacity.¹⁵ That pipeline documents the scale of ambition. It does not shorten the timelines. The physical constraints described above apply to every project in it regardless of queue position, and of those that entered the queue between 2000 and 2019, only 13% had reached commercial operation by end of 2024.

The handoff between these three clocks produces the system’s central paradox. Chips are the binding constraint today — but they are self-correcting. As new TSMC fabs come online in 2027 and 2028 and efficiency gains reduce chips required per token, the chip constraint eases. At precisely that moment, the grid constraint becomes dominant — and unlike the chip constraint, it does not self-correct with efficiency gains. Efficiency makes tokens cheaper, which expands demand through the Jevons mechanism, which increases total electricity consumption even as consumption per token falls. The capital is flowing to the layers that move fastest. The layer that moves slowest — the one governed by permits and physics — is receiving the least.

This is where the AGI scenario sharpens from ambition into arithmetic. Epoch AI estimates that reaching the lower bound of the compute threshold required for a single AGI-level training run — approximately 20 million H100-equivalents — will occur around 2028 on the current supply trajectory, as available compute grows at roughly 2.25 times per year from its 2024 base of 8.5 million H100-equivalents. But powering a cluster of 20 million H100-equivalents simultaneously requires approximately 34 gigawatts of continuous electricity — roughly equivalent to the entire electricity consumption of Norway — when accounting for full system power including supporting hardware, cooling, and data center overhead. By 2028 — when available compute crosses that threshold — the maximum power deliverable to a single data center site on optimistic projections is approximately 0.9 gigawatts. The grid delivers roughly 3% of what the compute demands.

The paradox is precise: the compute and the power requirement arrive on different clocks. The compute timeline is governed by chip manufacturing, which is accelerating. The power timeline is governed by grid construction, which is not. By 2028, the compute may exist. The 34 gigawatts required to turn it on will still be years away from delivery. The constraint is not intellectual. It was set in motion, in permitting offices and interconnection queues, years before the training run was scheduled.

Chart 3: The AGI paradox in gigawatts. The blue line shows the power that available AI compute would require if running simultaneously — growing at 2.25 times per year from 1.4 GW in 2024 to approximately 950 GW by 2032, using full system power of 1,700 watts per H100-equivalent as estimated by Epoch AI (chip plus supporting hardware, cooling, and data center overhead). The green dashed line shows the maximum power actually deliverable to a single AI training cluster, growing at an optimistic 15% per year from 0.5 GW today. The two amber horizontal lines mark the AGI power thresholds using full system power: 34 GW for the lower bound (20 million H100-equivalents) and 680 GW for the upper bound (400 million H100-equivalents). The blue line crosses the lower threshold in 2028 — at which point the grid delivers approximately 0.9 GW to a single site, or roughly 3% of what is needed. It crosses the upper threshold around 2031–2032 — at which point the grid delivers approximately 1.4 GW, or 0.2% of what is needed. The compute arrives. The power does not.

The Capital Stack

The mental experiment this column proposes is straightforward in structure. Take three physical layers — chips, data centers, and grid. For each layer, calculate the investment required under three scenarios (two from projections from authoritative sources and the third one derived), all on the same 2026–2030 five-year horizon: the Goldman Sachs baseline, the McKinsey baseline, and an AGI scenario derived from first principles. Then compare the required investment against what is actually being committed — using the most optimistic reasonable assumptions for committed capital. The exercise produces two findings, which is why it requires two charts.

The two authoritative published baselines are as follows. McKinsey’s April 2025 analysis projects $6.7 trillion of total AI infrastructure investment required through 2030, decomposed as $3.1 trillion for chips and silicon, $1.3 trillion for grid and energy infrastructure, and $800 billion for physical data center construction, with the remaining $1.5 trillion representing traditional IT capex outside the three AI-specific layers.¹⁶ Goldman Sachs’ April 2026 baseline projects $7.6 trillion cumulatively through 2031, with annual capex growing from $765 billion in 2026 to $1.6 trillion in 2031.¹⁷ Subtracting the 2031 year alone yields approximately $6.0 trillion for the comparable 2026–2030 period — slightly below McKinsey’s figure, reflecting different model anchors and assumptions.

Neither baseline models AGI. That scenario requires a separate derivation. Epoch AI estimates the lower bound of the compute threshold for a single AGI-level training run at approximately 20 million H100-equivalents. At a current cost of approximately $35,000 per GPU, that is $700 billion in silicon alone for a single training run — nearly equal to Goldman Sachs’ entire projected annual capex for 2026. Applying Goldman Sachs’ own data center unit cost of $15 million per megawatt to the 34 gigawatts required to power that cluster yields $510 billion in dedicated data center infrastructure. Power infrastructure at Goldman Sachs’ assumed $2,500 per kilowatt adds a further $85 billion. A single lower-bound AGI training run therefore requires approximately $1.3 trillion in purpose-built infrastructure. Projected over the 2026–2030 horizon assuming one AGI-level training run per year with supporting inference infrastructure, total capital requirement reaches approximately $12 to $14 trillion.¹⁸

The committed investment figures are drawn from public filings, earnings calls, and announced partnerships — taken at their most optimistic, on the same 2026–2030 horizon. Hyperscaler capex at 25% annual growth totals approximately $4.0 to $4.5 trillion. Global semiconductor industry capex at 20% annual growth from its $200 billion 2026 base totals approximately $1.5 trillion. Grid investment actually committed totals approximately $400 to $500 billion. The aggregate across all three layers on the most optimistic assumptions is approximately $6.0 to $6.5 trillion — with $6.5 trillion used as the optimistic ceiling.

The first finding is reassuring at the headline level and dangerous beneath it. At the aggregate, committed capital of $6.5 trillion exceeds Goldman Sachs’ 2026–2030 figure of $6.0 trillion and McKinsey’s AI-specific requirement of $5.2 trillion. The base case appears funded at the aggregate level. The AGI scenario is not — the gap between $6.5 trillion committed and $12 to $14 trillion required is $5.5 to $7.5 trillion. That is the total picture and to cover that gap it would require the largest public-private partnership commitment the world has ever seen.

The second finding is the one that matters for the base case. The distribution of committed capital across the three layers is structurally misaligned with the requirement — and the misalignment is concentrated in the layer that cannot be corrected after the fact. Data centers are receiving approximately 69% of committed capital while representing approximately 15% of the McKinsey AI-specific layer requirement. The grid is receiving approximately 8% of committed capital while representing approximately 25% of the McKinsey AI-specific layer requirement. The money is there. It is going to the fastest layer to build and away from the slowest. By the time the misallocation becomes operationally visible — when built data centers cannot run at capacity because the transmission infrastructure is not ready — no capital decision made at that point can close the gap. The grid’s clock does not respond to urgency. It responds to permits. And permits were needed years earlier.

Google has seen this math. The evidence is in its capital allocation. The December 2025 acquisition of Intersect Power for $4.75 billion — bringing multiple gigawatts of solar and storage projects directly onto Google’s balance sheet — is not a sustainability commitment. It is the response of a company that has concluded the grid layer in the capital stack cannot be filled by waiting for utilities to act. The August 2025 collaboration with Kairos Power and the Tennessee Valley Authority to deploy the first Generation IV advanced nuclear reactor connected to the US grid, under a 500-megawatt nuclear capacity initiative, is the same conclusion applied to firm baseload power. The $40 billion commitment to three Texas data centers through 2027, structured with co-located power generation rather than grid connection, is the third expression of the same insight: the interconnection queue is not a temporary inconvenience. It is a structural feature of the US grid that no company building at this scale can afford to treat as someone else’s problem.

The IEA reported in April 2026 that conditional offtake agreements between data center operators and small modular reactor projects grew from 25 gigawatts at the end of 2024 to 45 gigawatts in a single year. That is not a policy development. It is a capital allocation signal from the companies that have done this calculation and arrived at the same answer: the grid gap is real, it is structural, and it cannot be closed through the conventional interconnection process on any timeline that the data center construction schedule requires.

Chart 4A: Four bars on the same 2026–2030 horizon, all on an AI-specific basis: committed capital ($6.5T), Goldman Sachs baseline ($6.0T), McKinsey AI-specific baseline ($5.2T, excluding $1.5T of traditional IT capex), and AGI scenario derived from first principles ($13T midpoint). The amber dashed line marks committed capital at $6.5T. At the aggregate level, committed capital not only covers both published base cases — it exceeds Goldman Sachs by $0.5T and McKinsey’s AI-specific requirement by $1.3T. The AGI scenario is a different story: the gap is $6.5T. But aggregate adequacy is the wrong conclusion to draw. The money is there for the base case. The question is where it is going. That is in Chart 4B. Sources: McKinsey (2025), Goldman Sachs (2026), Epoch AI (2025), company filings. AGI scenario is a first-principles derivation — not a published figure.

Chart 4B: The distribution. This is the finding. Colored bars show what McKinsey’s AI-specific layer requirement looks like as a percentage of the $5.2T AI-specific total. Gray bars show how committed capital is actually allocated as a percentage of the $6.5T committed total. Data centers receive 69% of committed capital but represent only 15% of the requirement — overfunded by $3.7T. Chips receive 23% of committed capital but represent 60% of the requirement — underfunded by $1.6T. The grid receives 8% of committed capital but represents 25% of the requirement — underfunded by $0.8T. The grid deficit is the most consequential: it is the only layer whose shortfall cannot be corrected after the fact. Sources: McKinsey (2025), company filings.

The Price of a Tuesday

Return to the person at the laptop. The 16,000 tokens. The three sessions that felt like nothing.

That Tuesday is the demand signal of the most capital intensive industrial buildout in the history of the private economy. The total capital being committed — approximately $6.5 trillion through 2030 on optimistic assumptions — is roughly adequate for the base case. It is not adequate for AGI, where the gap reaches $6.5 to $7.5 trillion. But the more important finding is not the total. It is the distribution. The capital is flowing to the layer that builds fastest and away from the layer that builds slowest. Data centers get built in thirty-six months. Transmission lines take seven to ten years. And the grid, receiving 8% of committed capital against 25% of the requirement, is the only layer where a funding gap today cannot be corrected by a decision made tomorrow.

You can add a data center in thirty-six months. You can add chip capacity in twenty-four to thirty-six months as new fabs come online. You cannot add a transmission line in under seven years. You cannot add a nuclear plant in under ten. The grid is the only layer in the capital stack where a funding gap today produces a physical gap in 2029 that no decision made in 2028 can close.

The public conversation about AGI is almost entirely about intelligence — about whether the models are capable enough, safe enough, aligned enough. Those are important questions. But they are not the binding question on the timeline the capital currently assumes. The binding question is physical. It is measurable. It is visible in interconnection queue data, in PJM (Pennsylvania-New Jersey-Maryland Interconnection) capacity auction prices that rose eleven-fold in two years, in the CEO of TSMC saying there are no shortcuts and a new fab takes two to three years, in 45 gigawatts of nuclear offtake agreements that exist on paper and not yet in the ground.

Ask not if we can build AGI models. The models are being built. Ask if we can power them — on the timeline the capital stack assumes, at the scale the demand curve requires, through the physical infrastructure whose clock speed has never, in the entire history of American industrial development, been governed by the urgency of the private sector alone.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

Notes

¹ OpenAI’s May 2025 study of 1.5 million conversations found that the average daily active user conducts approximately twenty interactions per day across multiple sessions, with average session length of twelve to fourteen minutes. A typical exchange runs between 500 and 1,500 tokens depending on task complexity. 16,000 tokens per day represents a conservative midpoint estimate for a user conducting a mix of question-answering (49%), task completion (40%), and exploratory interactions (11%). Source: OpenAI, How People Are Using ChatGPT, May 2025.

² The Interstate Highway System cost approximately $634 billion in 2024 dollars, and took thirty-five years to build. Source: Wikipedia, Interstate Highway System, citing Federal Highway Administration data. The Apollo program cost approximately $288 billion in inflation-adjusted dollars. Source: The Planetary Society, Reconstructing the Cost of the One Giant Leap. The Manhattan Project cost approximately $30 billion in 2024 dollars. Source: Congressional Research Service. Combined: approximately $952 billion in 2024 dollars across all three programs. In 2025, US technology capital expenditure as a share of GDP reached approximately 1.9% — comparable in scale to all three of those programs combined, in a single year. Source: KobeissiLetter analysis of BEA data, 2025; Empower Investment Insights, The AI Revolution Rolls On, 2026. The five-year AI infrastructure requirement of $6.7 to $12 trillion (McKinsey and Goldman Sachs, see notes ¹⁶ and ¹⁷) exceeds this by an order of magnitude.

³ One token corresponds to approximately four characters of English text, or roughly three-quarters of a word. OpenAI’s Tiktoken library confirms this ratio for GPT-family models.

⁴ The standard approximation for inference compute in a transformer model is 2N floating-point operations per token per forward pass, where N is the number of model parameters. For a 70-billion-parameter model: 2 × 7 × 1⁰¹⁰ = 1.4 × 1⁰¹¹ FLOPs per token. This figure is consistent with the MLPerf Inference v4.1 benchmarks for the Llama-2 70B model. Larger frontier models — including mixture-of-experts architectures that activate a subset of parameters per token — may require significantly more or fewer effective FLOPs depending on architecture. The 70B figure is used here as the empirically grounded benchmark for which measured performance data exists.

⁵ MLPerf Inference v4.1 benchmarks for Llama-2 70B in offline configuration report approximately 24,525 tokens per second on a single H100-SXM GPU at approximately 700 watts, corresponding to 0.029 milliwatt-hours per token (0.10 joules per token). Source: ScienceDirect, Green AI Techniques for Reducing Energy Consumption in AI Systems, December 2025.

⁶ PUE (Power Usage Effectiveness) is the ratio of total data center facility energy to IT equipment energy. A PUE of 1.0 is the theoretical minimum. Hyperscaler facilities report PUEs of 1.2–1.4. Google has reported a trailing twelve-month average PUE of 1.10 for its global fleet. A midpoint of 1.35 is used here as a conservative estimate between the hyperscaler best case and the broader industry average of approximately 1.5–1.6.

⁷ Total AI assistant daily active users estimated at approximately 300 million across all major platforms. Methodology: DataReportal’s Digital 2026 Global Overview Report estimates more than 1 billion people use AI monthly across platforms including ChatGPT, Gemini, Copilot, Claude, Grok, DeepSeek, and others. Applying a daily-to-monthly active user ratio of approximately 30% — consistent with the 193 million daily active users reported for ChatGPT against its approximately 900 million weekly active user base — yields approximately 300 million daily active users industry-wide. Platform-specific anchors: ChatGPT 193 million daily active users (DemandSage, citing OpenAI, February 2026); Google Gemini approximately 750 million monthly active users (DemandSage, 2026), implying approximately 35 million daily active users at a 5% daily ratio consistent with Gemini’s reported engagement patterns; Microsoft Copilot, Grok, Claude, and others collectively account for the remainder. Aggregate daily grid draw calculation: 300,000,000 users × 2,160 joules per user = 648,000,000,000 joules ÷ 3,600,000 joules per MWh = approximately 180,000 MWh = 180 GWh per day. Average US household daily electricity consumption: approximately 30 kWh per day (EIA, Residential Energy Consumption Survey). 180,000,000 kWh ÷ 30 kWh per home = approximately 6 million homes. This figure covers conversational inference only and excludes training runs, API calls, enterprise batch workloads, and embedded AI in productivity software — all of which represent substantial additional consumption not captured here.

⁸ ChatGPT weekly active users: 900 million as of February 2026, versus 400 million in February 2025. ChatGPT processes approximately 2.5 billion queries per day as of July 2025. Source: DemandSage, ChatGPT Statistics, March 2026.

⁹ IEA, Energy and AI, April 2025. US data center electricity consumption of 182.61 TWh in 2024. Pakistan total electricity consumption approximately 180–190 TWh in the same period. IEA Key Questions on Energy and AI, April 2026: data center electricity demand soared 17% in 2025.

¹⁰ Goldman Sachs Research, AI to Drive 165% Increase in Data Center Power Demand by 2030, February 2025. Goldman Sachs base case: global electricity generation for data centers rises from 460 TWh in 2024 to more than 1,000 TWh in 2030.

¹¹ Morgan Stanley estimates agentic AI systems require approximately one CPU for every GPU, compared with one to twelve for chatbot systems — implying a twelve-fold increase in processing overhead per session as agent usage proliferates. Source: The Economist, Silicon Ceiling, May 2026.

¹² The cost of querying an AI model performing at GPT-3.5 level on the MMLU benchmark fell from $20 per million tokens in November 2022 to $0.07 per million tokens by October 2024 — a 280-fold reduction in approximately eighteen months. Source: Stanford HAI, AI Index Report 2025, Chapter 1: Research and Development. The deflation has continued through 2025: Epoch AI estimates LLM inference costs are falling between nine and 900 times per year depending on the task and capability level. The Jevons paradox (The Coal Question, 1865) holds that efficiency gains in resource use tend to increase rather than decrease total consumption by expanding the economic viability of the resource. IEA, Key Questions on Energy and AI, April 2026, confirms: “power consumption per AI task is declining rapidly… however, more people are using AI, and energy-intensive uses — such as AI agents — are on the rise.”

¹³ TSMC announced in March 2025 an expansion of its US investment to $165 billion, including three new fabs, two advanced packaging facilities, and an R&D center in Arizona. 2026 capex guidance: $52–56 billion. High-volume manufacturing at Arizona Fab 2 (3nm): expected second half 2027. Source: TSMC SEC Form 6-K, March 2025.

¹⁴ H100 GPU rental price increase of approximately 30% since November 2024. All three major HBM producers — SK Hynix, Samsung, and Micron — report most of their 2026 supply is sold out. Source: The Economist, Silicon Ceiling, May 2026, citing SemiAnalysis.

¹⁵ Lawrence Berkeley National Laboratory, Queued Up: 2025 Edition. Of the 2,300 GW in active US interconnection queues as of end of 2024, only 13% of projects that entered the queue between 2000 and 2019 had reached commercial operation by end of 2024. The queue documents ambition, not delivery. Physical construction timelines — four-to-five-year median interconnection wait, seven to ten years for transmission, three to five years for gas generation, ten to fifteen for nuclear — apply to every project in it regardless of queue position.

¹⁶ McKinsey, The Cost of Compute: A $7 Trillion Race to Scale Data Centers, April 2025. Total AI infrastructure investment required through 2030: $6.7 trillion. Decomposition: technology developers and silicon suppliers $3.1 trillion (60%); energisers — utilities and power providers — $1.3 trillion (25%); builders — physical construction — $0.8 trillion (15%).

¹⁷ Goldman Sachs, Tracking Trillions: The Assumptions Shaping the Scale of the AI Build-Out, April 2026. Baseline cumulative capex 2026–2031: approximately $7.6 trillion. Annual capex: $765 billion in 2026 growing to $1.6 trillion in 2031. Key unit cost assumptions used in this column’s AGI derivation: $15 million per MW for data centers; $2,500 per kW for new power infrastructure; PUE of 1.2. To standardize to the 2026–2030 horizon used throughout this column: $7.6T cumulative minus $1.6T for 2031 = approximately $6.0T for 2026–2030. Goldman Sachs notes this is a scenario-based framework, not a forecast.

¹⁸ AGI scenario capital requirement derived from first principles using Goldman Sachs’ own unit cost assumptions (Goldman Sachs, Tracking Trillions, April 2026) and Epoch AI’s published compute threshold. All figures on 2026–2030 horizon. Inputs per training run: (1) Silicon: 20M H100-equivalents × $35,000 = $700B. (2) Data centers: 34,000 MW × $15M per MW = $510B. (3) Power infrastructure: 34,000,000 kW × $2,500 per kW = $85B. Single training run total: approximately $1.3T. Projected over 2026–2030 assuming one AGI-level training run per year plus supporting inference infrastructure at comparable scale: approximately $12–14T total, with $13T used as midpoint. This is the column’s own derivation, not a published figure, and is labeled as such throughout. Committed capital (2026–2030, optimistic): hyperscaler capex at 25% annual growth = $4.0–4.5T; semiconductor industry at 20%/yr from $200B base = $1.5T; grid committed = $0.4–0.5T. Total: $6.0–6.5T, with $6.5T used as optimistic ceiling. Committed capital vs McKinsey AI-specific layers ($5.2T): committed exceeds by $1.3T. Committed capital vs Goldman Sachs 2026–2030 ($6.0T, AI-specific): committed exceeds by $0.5T. In both base cases aggregate committed capital is sufficient — the deficit is not in the total but in the distribution across layers. Gap vs AGI scenario: approximately $6.5T ($13T required minus $6.5T committed).

Ask Not if We Can Build AGI Models, Ask if We Can Power Them was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

In the summer of 1998, Long-Term Capital Management was running models built by two Nobel laureates and a team of some of the most sophisticated quantitative minds on Wall Street. The models were calibrated to decades of market data, stress-tested across historical scenarios, and internally consistent in ways that most institutional risk systems could not match. The process they described was well-defined. The mathematics was correct. And in September of that year, the fund lost more than $4 billion in equity in under five weeks, requiring a Federal Reserve-organized bailout to prevent broader market contagion.

The standard explanation is that LTCM took too much risk, that their models underweighted tail events, that they were overconfident in relationships that broke down under stress. That explanation is not wrong, but it is incomplete. It describes a model that was right about the world but assigned too little probability to its extremes. What actually happened was different and more fundamental. LTCM’s models were not describing the wrong tails of the right distribution. They were describing the wrong distribution entirely — and they had no mechanism to know it.

Why the Universe Requires Stochastic Processes

To understand what went wrong at LTCM with precision, it helps to start one level deeper than models — at the structure of the problem they are trying to solve.

Financial systems evolve through continuous time across continuous state spaces. Asset prices, interest rates, credit spreads, and volatility surfaces are not discrete objects that can be listed or enumerated. They are real-valued processes whose possible trajectories at any moment form an uncountable infinity — a space so dense that between any two possible paths, infinitely many others exist. This is not a computational limitation that more processing power resolves. In 1874, Georg Cantor proved that the infinity of real numbers is strictly larger than the infinity of counting numbers — uncountable rather than countable — which means no list, however long, can exhaust it. The space of possible financial trajectories has the same structure. It cannot be enumerated by construction.

This is why financial modeling requires stochastic processes. A stochastic process does not attempt to list possible outcomes. It assigns probability across an uncountable space by specifying a probability measure over entire families of paths — a mathematical architecture that describes how likelihood is distributed across trajectories through time rather than across isolated points. When a model specifies a stochastic process, it is saying: here is the full structure of uncertainty, not as a list of scenarios but as a distribution over the space of all possible paths. This is the most powerful tool available for navigating a world that cannot be enumerated. It is also a tool that only works if reality cooperates — specifically, if the process generating the future is the same process that generated the past.

The Assumption Stochastic Processes Cannot Escape

To estimate a stochastic process from historical data — to calibrate its parameters, fit its distributions, build its probability architecture — you must assume that the process generating your historical data is the same process that will generate your future. That the correlations you observed are stable. That the volatility structure, the tail behavior, the relationships between variables reflect something durable about the world rather than something specific to the period you happened to measure. This assumption has a name: stationarity.

A stationary process is one whose statistical properties do not change over time. The mean, the variance, the correlation structure — all remain stable as the system evolves. If stationarity holds, historical estimation is legitimate. The past reliably describes the distributional architecture of the future, and the stochastic process you have built is a genuine map of the territory you are navigating. If stationarity fails, you are fitting a model to a regime that no longer exists and projecting it onto one that operates differently. The map describes a territory that has changed underneath it.

The deeper problem is that stationarity cannot be confirmed from within the model. Statistical tests exist and are useful, but they test against historical data — the same data the model was built on. They can detect that the past was stationary. They cannot detect a regime shift that has not yet occurred. They cannot tell you that the process you have specified will remain the process operating in the moment when your position needs the model to be right most urgently. This is the boundary condition of every stochastic model ever built: it is estimated from one regime and deployed into the future, which may or may not be the same regime.

Two Ways Stationarity Fails

When stationarity breaks down in practice, it breaks down in two distinct ways that are often conflated but are categorically different in their implications.

The first is tail underestimation. The stochastic process has the right architecture — it is describing the right world — but its parameters are wrong. The tails of the distribution are thinner in the model than in reality. Extreme events are possible within the model’s framework but assigned too little probability. This is the standard critique of financial risk models and it is a real problem. It can be partially addressed through better calibration, fatter-tailed distributions, and more conservative parameter estimation. The model is describing the right process. It is just miscalibrated at the extremes.

The second failure mode is regime shift. The process itself changes. The model is not describing the wrong tails of the right distribution — it is describing the wrong distribution entirely, because the mechanism generating outcomes has shifted in a way that was not present in the historical data and therefore has no representation in the model’s architecture. This is not a calibration problem. It is a structural problem. You cannot assign probability to a mechanism your model has no language to describe, and no amount of recalibrating the existing model resolves it, because the problem is not inside the model. It is outside it.

LTCM encountered the second failure mode, not the first. Their models were well-specified for a world in which liquidity existed, arbitrage relationships were mean-reverting, and correlations across asset classes remained within historically observed ranges. What happened in August 1998 was not that the tails of that world were fatter than expected. It was that the world itself changed. When Russia defaulted, every major leveraged player in the market began unwinding simultaneously. The resulting feedback loop — forced selling depressing prices, depressed prices triggering further margin calls, margin calls forcing further selling — was a systemic mechanism that had never operated at that scale before. It was not in the historical data. It was therefore not in the model. Not in the tails. Not anywhere. The correlations that broke down were not extreme realizations of the relationships LTCM had modeled. They were the product of a mechanism the model did not contain.

This is the distinction that matters. Tail risk says: this event was possible within my model but I assigned it too little probability. Process risk says: this event was not possible within my model because my model did not contain the mechanism that generated it. The first problem is a better model away from being solved. The second problem cannot be solved from inside the model at all, because the model has no language for what it does not know it is missing.

What This Means for How Systems Are Built

The intellectual chain that leads here has a precise structure. The universe of possible financial trajectories is uncountable, so enumeration is impossible and stochastic processes are the best available tool. But stochastic processes must be estimated from historical data, which requires stationarity. And stationarity fails in two ways — miscalibration, which better modeling can address, and regime shift, which it cannot. Process risk is therefore not a refinement of market risk or tail risk. It is a different category of risk that sits underneath them, generated by the boundary condition every stochastic model carries intrinsically: it was built in one regime and will be used in the next one, which may be a totally different regime.

The practical implication is not that models should be abandoned. It is that the systems built around models need to remain viable when the model turns out to be describing the wrong process. Position sizing, leverage, liquidity buffers, and optionality all look different when the objective is not to optimize within a known process but to survive the discovery that you have been in the wrong one. LTCM was not undone by bad mathematics. It was undone by a system with no capacity to absorb the discovery that its mathematics, however correct, was describing a world that had already changed.

The question was never only whether the model is correct. It was whether the process the model described is the one actually operating — and whether the system built around it can survive finding out that it is not.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

Models Are the Math of “Once Upon a Time” was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

In March 2021, Archegos Capital Management collapsed in a matter of days, triggering over $20 billion in losses across the banks that had financed it. The positions Archegos held weren’t obviously wrong. Several of the underlying stocks recovered meaningfully in the months that followed. But the path through which those positions had to survive — a sudden margin call, forced liquidations, a collapsing equity base — was incompatible with the system that was supposed to outlast it.

This is not a story about bad predictions. It is a story about the difference between a process and the path it takes to get there. That distinction is easy to overlook, and very expensive when you do.

Start with $100. If it grows at 10% per year, the path is uneventful. After one year, it is $110. Then $121. Then $133. Then $146. After five periods, it reaches $161. Each step builds on the previous one. The compounding feels intuitive because the sequence is smooth. Nothing forces a decision. Nothing interrupts the trajectory. The path and the outcome are aligned.

Now take a different sequence, starting from the same $100. The first move is -40%. The value drops to $60. Then it increases by 30% over the next four years: $78, then $101, then $131, then $171. After five periods, it ends 6% above the first path.

If the only thing that mattered were the final number, the second sequence looks better. If the only thing that mattered were average returns, the two sequences would look nearly identical — or even favor the second.

But that is not how the process is experienced.

For four out of five years, the second path is economically worse. It only catches up at the very end. Any constraint applied before that moment — capital requirements, reporting obligations, liquidity needs, investor patience — would treat it as the inferior path. The recovery happens, but only after the damage has already forced its consequences.

Consider the difference in a human register for a moment. A person who takes on a large mortgage and then loses their job faces this exact structure. The long-run math might still work. But the path through unemployment, missed payments, and foreclosure can destroy the position before the recovery ever arrives. The process was fine. The path was not survivable.

This is the first place where the representation begins to diverge from the experience. Both sequences can be summarized with a small set of statistics. But the system does not unfold as a summary. It unfolds one step at a time.

What a Return Process Actually Isx

A return process is simple in structure: each period produces a return, and that return is applied to the current value. The sequence of returns defines the path. There is no mechanism that averages them out in real time. Each realization becomes the starting point for the next one.

What matters in practice is not that the process is random. It is that the sequence is where constraints are triggered. Capital is deployed at a point in time. Losses and gains occur in a particular order. Financing, liquidity, and decision-making are all tied to realized values, not expected ones.

Once you understand this, the earlier example stops being a curiosity. The two paths are not variations around a common center. They are different trajectories through the same space — one of which passes through a region where the system’s constraints become binding before the recovery arrives.

When the Same Process Becomes Unliveable

Now introduce leverage.

Take the same path — beginning with a 40% decline — but fund the position with $100 of equity and $100 of debt, for $200 in total assets. Assume the debt is fixed at $100 and carries a 5% annual cost.

Year 1: A 40% loss on $200 reduces assets by $80, from $200 to $120. After the $5 financing cost, total assets sit at $115. With debt still at $100, equity falls to $15.

The position does not enter Year 2 at the original 2x leverage. It enters at 7.7x. Losses compress the system immediately. Recovery takes time.

That was not a decision. It was the direct consequence of the path.

Year 2: A 30% return on $115 adds $35, bringing assets to $150. After the $5 financing cost, assets end the year at $145. Equity rises to $45.

Years 3 through 5: The recovery continues. Total assets reach $298 by the end. Equity rises to $198. The ending value is higher than in the unlevered case.

On paper, the process recovers. The sequence of returns has not changed.

But the system has.

After Year 1, 85% of the equity has been wiped out. The position is no longer operating under the assumptions that defined the original model. It is operating under a different capital structure, with a different level of fragility, and under tighter constraints than when it began.

In practice, most positions do not survive that transition. A drawdown of that magnitude triggers consequences before the recovery takes place. Margin requirements tighten. Covenants are breached. Liquidity disappears. The position gets closed — not because the long-run thesis was wrong, but because the path made it unfinanceable before it became unprofitable. This is precisely what happened to Archegos.

Where the Model and the Experience Diverge

Most models are built to describe the process, not the path. They summarize the distribution, estimate parameters, and project outcomes. They compress the system into something that can be evaluated and compared.

But the compression removes the conditions under which the process actually operates.

The model assumes continuity. It assumes capital remains available. It assumes the process runs long enough for the expected outcome to emerge. The path determines whether any of those assumptions hold.

A leveraged strategy can show a positive expected return and acceptable risk metrics while still being exposed to sequences that terminate it early. A deal can produce an attractive projected return while still being sensitive to the timing of cash flows — delays in early periods create liquidity pressures that the projection doesn’t capture. A business can have stable long-term growth assumptions while still being vulnerable to early shocks that alter its trajectory permanently.

In each case, the issue is not that the model is wrong. The calculations can be precise. The assumptions can be reasonable. The divergence arises because the model describes the process, while the outcome is governed by the path — and the path determines whether the system itself remains intact.

From Averages to Paths

Two earlier columns — An Average May Not Exist in Finance and The Lie of the Average — explored why summaries mislead: sometimes the average doesn’t exist at all, and sometimes it exists but describes no one’s actual experience. Both problems remain. But even when they are resolved, this one appears underneath them.

A process can be well-defined. The average can exist. The model can be correct. And the outcome can still be entirely different from what the model suggests — because the model describes where the process might go, while the path determines whether it gets there.

This changes the question you should be asking. Not only: is this process attractive in expectation? But: is the path required to realize that expectation compatible with the constraints of the system I’m actually operating?

Finance is typically optimized in expectations. Decisions are framed in terms of averages, projections, and long-run behavior. But the system is experienced through realizations, one period at a time, under constraints that do not wait for the average to emerge.

A smooth path produces a steady outcome. A volatile path can produce a higher final number — and still destroy the system before it arrives there.

The average describes the process.

The path determines whether the system survives long enough for the process to matter.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

The Path Is the Outcome was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

Exactly one year ago I started writing out of something very practical. I was working across several projects at the same time, fully invested in them, and naturally those projects had to be presented to different audiences, clients, vendors, partners, investors. At some point I needed a way to express what I was building so that someone else could actually understand it without me sitting next to them walking through every assumption. The natural path in finance and startups is to build decks, but their limitation shows up quickly. A deck only works if someone is there explaining it. Left on its own, it gets skimmed, reduced to a few lines, and whatever required real reasoning disappears between slides. It’s not that people don’t pay attention, it’s that the format doesn’t demand it. Writing demands it. A memo assumes the reader will sit with it, follow the argument from beginning to end, and make sense of it without external context. That constraint forces ideas to be developed in full and expressed clearly enough to stand on their own.

That was incentive enough to start, so the first pieces came directly from what I was working on, because that was the material I had in front of me every day. Writing forces a level of precision that execution alone doesn’t demand. You can move a project forward while carrying ideas that are still half-formed, because reality will give you feedback quickly enough to adjust. Writing doesn’t give you that flexibility. Once something is on the page, it either holds or it doesn’t, and when it doesn’t, there is nowhere to hide. That challenge made me realize the exercise was worth continuing, and the topics expanded naturally with my work and my professional experiences throughout the year. I stopped writing only about what I was building and started writing about the underlying systems as my intellectual curiosity expanded. It also changed how I work. I stopped relying on decks in professional settings and moved to memos as the default way to communicate anything that matters, and in some cases the column itself became part of that process, when a piece captured an idea better than any presentation could.

When the Work Becomes the Material

As I kept writing I was forced to think more clearly about what I was already doing. The topics drifted as the work drifted, and over time the writing started to move away from describing projects and into the things behind them. A lot of that came initially from working on Compago and spending a great deal of time designing a fintech product around how people actually manage money in today’s world. Most financial systems assume stability. Income is supposed to be predictable, expenses are supposed to follow some order, and credit decisions are made as if those patterns hold. In reality, especially in places like Mexico, none of that is true for a large part of the population. Income is irregular, expenses move constantly, and financial decisions are made in very tight margins, compounded by low financial literacy and a weak rule of law. That gap is what fintech should be solving, but most of what is called innovation in finance is just a faster way to run the same logic. The interface gets better and better, but the underlying system doesn’t.

The next columns came from trying to make sense of these systems, because the work I was doing depended on them working. Thinking about loan agreements that can adjust over time instead of being fixed at the moment they are signed, or questioning whether credit can actually be priced correctly when the inputs themselves are unstable. These are the structural flaws that need addressing in places where lack of financing isn’t just an inconvenience but a drag on economic development itself. And at the same time, we have to actually design solutions that move things forward instead of simply repackaging existing practices.

Another set of observations started to take shape in parallel, and those had less to do with products and more to do with the movement of people and capital between Mexico and the U.S. Being back living in the United States, I realized that although growth in Mexico is still there, it now comes with more constraints, more uncertainty, and more friction. For Mexican businesspeople, expanding into the U.S. or directly acquiring businesses here is no longer just an option for scaling, it’s becoming a way to protect what they have already built. You can see it in how financial decisions are made, in how capital is being allocated, and in the kinds of opportunities people are willing to pursue.

The Language of Deals

As the year moved forward, opportunities in M&A started to come up again and I went back to dealmaking. By then I had already been writing consistently, and deals stopped feeling like isolated events. They became material. Although most dealmaking is confidential, and terms and situations remain so even after closing, every transaction provided an enormous amount to think about. Every deal behaves as a dynamic system in the mathematical sense of the word. Everything interacts, and it does so continuously. You adjust one piece and the rest respond immediately, sometimes in ways that are not obvious until you are already too far into the process.

Writing started to follow that same line. The usual way of talking about deals wasn’t enough to describe what was actually going on, so I ended up reaching for a different language, one that was closer to how these situations actually behave. Constraints, trade-offs, feasible ranges, systems that move as conditions change. Once you look at deals that way, it becomes difficult to reduce them back to simple negotiation or isolated terms. They behave like systems, and they have to be understood as such if you want to work through them without losing the thread.

Forty Plus Columns and What They Actually Are

The journey I’ve described produced more than forty columns over the course of the year. Looking at them together, the arc is clearer than it was while living it. The first pieces I published were the most grounded in where I was standing. Writing about building four ventures at once, about a non-linear career, about what it means to build a fintech platform from scratch, these were a way of introducing myself as the person behind the work, establishing the context that would make everything that followed legible. If you are going to argue for a different way of thinking about credit infrastructure or dealmaking, it helps for the reader to understand what you have actually built and where that thinking comes from.

From there the writing moved into the territory I knew most directly. A long sequence of pieces on credit and financial access, What If We Got Lending Right, We’re Lending People Into Poverty and Calling It Progress, the SMB piece, the Compago rationale, were a sustained argument for a different financial architecture, approached from multiple angles across multiple pieces. Each one made the case that the system as designed was built for conditions that don’t reflect how most people actually manage money, and that the path forward isn’t iteration on the existing model but a genuine rethinking of the infrastructure underneath it. Teaching AI How to Lend, If We Can Anticipate Buyers We Can Also Anticipate Borrowers, and The Living Credit Contract pushed that argument further, into what a credit system designed around real financial behavior, rather than assumed stability, would actually look like.

The lens then widened to the movement of capital and people between Mexico and the United States. Latinos in U.S. Finance, Mexican Capital Is Fueling America’s Next Growth Center, Thirty Years After NAFTA Mexico Will Start Owning in America, White-Collar Nearshoring Is North America’s Next Integration, these pieces came from what I was observing directly, living in Houston at the intersection of two economies that most financial strategies still treat as separate. None of it was prediction. It was description of flows already in motion, visible to anyone paying close enough attention. Financial Markets Are the Most Democratic System We’ve Ever Built belongs to this period too, a historical argument that the transformation of capital markets from a closed system serving insiders to one accessible to ordinary people is one of the most underappreciated shifts in modern economic life, and that the tools most people already have access to are more powerful than they realize.

As the year moved toward dealmaking, the writing narrowed again but the same thread carried through. The Illusion of Precision in Valuation and The Hockey Stick Test both came from watching how confidently people treat numbers that rest on assumptions defined by art more than science. That observation opened a deeper question: if the standard tools of financial analysis rest on unstable foundations, what is the more precise language for describing how financial systems actually behave? The mathematical sequence that followed, The Lie of the Average, An Average May Not Exist in Finance, Modeling an Uncountable World, The Local Maximum Trap, The Cost of Perfect Optimization, was an attempt to build that language, piece by piece. Not deal pieces, but foundational ones: about distributions that don’t converge, about optimization problems that resist exact solutions, about the traps that appear when a system keeps improving toward the nearest peak while a better one sits out of reach. Once that vocabulary existed, applying it to transactions was the natural next step. Valuation Is Not the Deal, The Mathematics of Dealmaking, and The Mathematics of a Moving Deal are where it found its full application, treating transactions not as static agreements to be negotiated but as dynamic systems to be navigated as conditions change.

From Cacophony to Symphony and Websites and Slide Decks Are Still Boring sit slightly outside the main clusters but belong to the same underlying question. The first tries to describe a market with all the instruments and none of the score, and to argue that coherence in Mexico’s digital finance requires national objectives, not just more players. The second is the piece that explains, in the most direct terms, why my column exists at all.

Looking at the full year, the work moved from personal context to system critique to geographic observation to mathematical formalization, but the original intention never changed. Writing was a way to communicate complex work to people who weren’t in the room. It turned out to also be a way of developing the ideas themselves, of finding out what actually held once it had to stand on its own. That is the part I didn’t anticipate when I started, and the part that made it worth continuing.

I don’t expect anyone to read all of it, and I don’t take for granted the time it takes to sit through one piece. Each column is written with the assumption that it has to stand on its own and be worth the time it asks for. Some will land more than others depending on what the reader is looking for, but the standard is the same. It’s a way to work through ideas that show up in practice and to put them in a form that can be examined.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

A Year in Writing was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

On Monday, the deal works. A buyer is prepared to underwrite an $80 million valuation on an $8 million EBITDA business with roughly 4x leverage, and the model clears the required return with enough margin to support conviction. The lender is comfortable with the structure under base and downside cases, the seller is aligned with the mix of cash at closing and residual exposure, and the process advances with the sense that the essential terms are already in place. By Wednesday, the same deal no longer works. No headline term has been formally changed, no one has walked away, and yet the structure no longer satisfies the conditions required by each participant. Diligence surfaced a concentration risk that increases volatility, the lender recalibrated its tolerance for leverage under stress, and the buyer’s return compressed once both effects were incorporated. The deal changed because it crossed a boundary that was always present and is only visible once the system updates.

A transaction is often treated as a fixed object that is negotiated into agreement. In practice, what participants call “the deal” is a provisional structure that only holds as long as the underlying inputs remain within a narrow range. The seller is not agreeing to a number in isolation, the buyer is not committing to a return in isolation, and the lender is not approving leverage in isolation. Each of those positions depends on a set of assumptions that are continuously revised as new information enters the process. What appears stable early in a transaction is often a temporary alignment of conditions rather than a resolved outcome.

Negotiations are described as converging toward a fixed set of terms, but the reality is that the system itself is moving while that convergence is attempted.

The Feasible Set Moves

In the prior column, I formalized a transaction as a structure x that must satisfy three simultaneous constraints: the seller’s value function VS(x) must clear a minimum threshold, the buyer’s value function VB(x) must clear its required threshold, and the lender’s value function VL(x) must satisfy its downside conditions. The feasible set F is the intersection of all structures where those constraints hold simultaneously. But VS(x) was written as VS(P(x),C(x),T(x),-R(x)), treating background conditions as fixed. That formulation is correct when the problem is static. But cash at closing depends on what the financing market will support at the moment of closing, not only on what the structure specifies. Execution certainty depends on what diligence has revealed by the time a commitment is required. Retained risk depends on how the risk profile of the business has been reassessed as information accumulates. Price is the one argument that is directly specified by the structure, while the others are jointly determined by structure and environment.

This requires making the environmental dependence explicit. So define θ(t) as the vector of underlying conditions that participants cannot control through negotiation but that continuously redefine how any structure is evaluated: expected cash flows CF(t), financing market conditions κ(t), business risk assessments ρ(t), and accumulated diligence findings δ(t). The full expression of the seller’s value function then becomes:

VS(x, θ(t)) = VS(P(x), C(x, κ(t)), T(x, κ(t), δ(t)), -R(x, ρ(t), δ(t)))

The same logic applies to the buyer and lender. The buyer’s value function, expressed in terms of return, depends on CF(t) and κ(t). The lender’s value function, expressed through coverage constraints, depends on CF(t) under downside scenarios and on ρ(t). The feasible set is therefore time-indexed:

F(t) = {x : VS(x, θ(t)) ≥ VSmin, VB(x, θ(t)) ≥ VBmin, VL(x, θ(t)) ≥ VLmin}

The prior formulation asked whether x satisfies all constraints under current conditions. The dynamic formulation asks whether x(t) ∈ F(t) holds continuously along the entire path from initiation to closing. A structure that cleared every constraint at t=0 can fail at t=1 without any term being renegotiated, because θ(t) moved the boundaries of what each participant finds acceptable. The deal that appeared agreed in principle had already crossed a boundary before anyone recognized it.

This is a dynamic system in the mathematical sense: outcomes depend on how variables evolve over time, not only on where they start or end.

Sensitivity Determines What Actually Matters

Not all variables in θ affect the feasible set equally. The boundary of F(t) is defined by constraint surfaces, the hypersurfaces in the space of possible structures where each participant’s value function exactly equals its minimum threshold. The question of which variables matter most is a question about the gradient of those constraint functions with respect to the components of θ.

Leverage is a high-gradient variable because it propagates through multiple constraints simultaneously. Consider how a reduction from 4x to 3.5x leverage affects the system. At 4x, an $8 million EBITDA business supports $32 million of debt, but at 3.5x, debt capacity falls to $28 million. That $4 million reduction increases the required equity at the same purchase price, which compresses the buyer’s IRR. Simultaneously, the reduced debt service load changes the coverage ratio, which shifts the lender’s constraint surface. A single variable moves the boundaries of VS(x,θ(t)) and VL(x,θ(t)) at the same time. In gradient terms, leverage has large partial derivatives with respect to multiple constraint functions. It sits near the intersection of constraint surfaces, so small movements in it affect feasibility across the entire system. This is why small adjustments to leverage often feel disproportionately consequential in live processes: they are moving multiple constraint boundaries at once.

Customer concentration behaves differently. It does not primarily shift the base-case value of θ. It widens the distribution of possible θ trajectories. Specifically, it increases the variance of future cash flow paths under stress. A lender’s constraint is evaluated not at the base case but under downside scenarios, so widening the distribution of outcomes tightens VL(x,θ) even when the expected case is unchanged. The mechanism is not a shift in the gradient but a change in which region of the θ distribution the constraint is evaluated against. These are mathematically distinct failure modes, and conflating them leads to misdiagnosis of why a deal broke.

Rate of Change Breaks Deals

The dynamic problem can be stated precisely. A transaction involves a structure x(t) that is being adjusted by participants and a feasible set F(t) whose boundaries are moving as θ(t) evolves. The condition for a live deal is that x(t) ∈ F(t) holds continuously, not just at signing, but throughout the process. Feasibility is not a test passed once; it is a condition that must hold along the entire trajectory.

What determines whether that condition holds is the relationship between two rates. The first is the rate at which θ(t) moves the feasible boundary over time, which we can represent informally as dF/dt, the speed at which the environment is moving the set of acceptable structures. The second is the rate at which x(t) can be reconfigured in response to that movement, call this dx/dt, the adaptation rate of the structure. The deal survives if dx/dt is sufficient to keep x(t) inside F(t) as the boundary moves. If dF/dt exceeds the system’s ability to adapt, the trajectory exits the feasible region.

This is the precise content of the claim that “timing breaks deals.” It is not that the magnitude of the shock was too large in some absolute sense. It is that the shock arrived faster than the structure could be reconfigured. A cash flow deterioration of a given magnitude, identified at the beginning of a process, allows the buyer to revise underwriting, the lender to recalibrate leverage, and the seller to adjust expectations, all while feasible points still exist. The same deterioration, arriving in the final weeks after financing has been committed and price expectations anchored, moves the feasible boundary faster than the structure can follow. The deal exits F(t) not because conditions were worse, but because the rate exceeded the adaptive capacity of the system.

This also explains path dependence. Two transactions can arrive at identical final conditions and still produce different outcomes. If one path allowed for gradual recalibration while F(t) was moving, the structure tracked the feasible region successfully. If the other path anchored commitments early and encountered the same movement late, the structure could not adapt in time. Final conditions are the same; the trajectory is different; the outcomes diverge.

The deal is path-dependent in the precise sense that the integral of its trajectory, not just its endpoint, determines whether feasibility is maintained.

What Better Dealmakers Track

A deal that appears viable at a point in time may already be on a trajectory that leads outside the feasible region. The structure can hold under current assumptions and still fail because the variables that sustain it are moving in a direction or at a speed that cannot be absorbed. In that sense, the question is not only whether the deal works, but whether it can continue to work as the system evolves.

The practical implication of this framework is a reorientation of attention. Most process management focuses on x(t), the structure, the terms, the negotiated positions. But x(t) can only remain inside F(t) if the boundaries defined by θ(t) are understood and anticipated. A dealmaker who tracks the structure without tracking the environment is navigating by position alone, without reading the movement of the terrain. The high-sensitivity variables identified by the gradient analysis, leverage, cash flow concentration, financing conditions, are the components of θ that move the constraint surfaces most quickly. Addressing them early is the only way to preserve the adaptive capacity that path dependence requires.

The structure that closes is not the one that looked best at any single point in time. It is the one whose trajectory remained inside a feasible region that was moving from the start.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

The Mathematics of a Moving Deal was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

In my previous column, I wrote about Arrow’s Impossibility Theorem and the structural reason no decision rule can satisfy every participant’s preferences at the same time. In a transaction, that shows up immediately. Sellers prioritize price, buyers prioritize return, lenders prioritize protection. Each group evaluates the same set of outcomes differently, and there is no mechanism that preserves all of those rankings simultaneously. Something has to give. That limitation is not the result of poor negotiation or incomplete information. It is embedded in the structure of the problem.

But transactions still happen. Companies are sold every day, often in competitive processes involving multiple bidders, financing sources, and advisors. If perfect alignment is impossible, the relevant question is what determines whether a deal exists, and if it does, where it lands. In practice, people talk about valuation gaps, market conditions, leverage availability, and execution risk. Those are real forces, but they are descriptions of a deeper structure. The process is not arbitrary. It follows a logic that can be written down with precision, and once written, it becomes clear that dealmaking is already behaving like a mathematical system whether anyone in the room represents it that way or not.

Every Deal Begins as a Set of Preferences

A transaction is often described as a negotiation between parties with different views of value. But “value” is not a single variable. In reality, every participant is balancing several objectives at once. A seller does not care only about headline valuation. The seller may also care about certainty of closing, how that value is paid (cash, stock, earnouts, or a combination), the extent of post-closing liability, the preservation of the company’s identity, and the speed of execution. Buyers face the same structural problem. Purchase price matters, but so do expected return, financing constraints, integration risk, and the quality of the asset under downside scenarios. Lenders add a third layer, where the focus is not price or upside, but coverage ratios, collateral, and the probability of default, all under downside conditions rather than base-case optimism.

The structure can be written formally. Let x represent a potential transaction structure. That includes price, cash at closing, deferred payments, earnouts, leverage, rollover equity, indemnities, and timing. Each participant evaluates x through its own value function:

VS(x) for the seller, VB(x) for the buyer, and VL(x) for the lender. These are not abstractions added after the fact. They are a precise way of expressing how each participant already evaluates a deal. Each function captures how that participant values different combinations of outcomes. A seller prefers higher price and lower retained liability. A buyer prefers higher return, and lower risk. A lender prefers structures that maximize the likelihood of repayment under stress.

A participant does not need to love a structure for it to remain in play. It only needs to clear a minimum acceptable threshold. The seller will accept only structures where VS(x)≥VSmin, the buyer requires VB(x)≥VBmin, and the lender requires VL(x)≥VLmin. The set of transaction structures that satisfy all of these conditions simultaneously is:

This set is what practitioners refer to, without formal notation, when they say a deal is financeable, executable, and acceptable to both sides. The critical point is that a transaction exists if and only if this set is non-empty. If F=∅, there is no deal. Once that condition is met, each participant solves their own problem inside that set. From the seller’s perspective:

That is the core structure. There is no global optimum that simultaneously satisfies every participant’s ranking of outcomes. There is only a feasible set, and within that set each participant pushes toward the point most favorable to them. The transaction is the result of that interaction.

The Optimization of Dealmaking

Once a feasible set exists, the nature of the problem changes. The seller is no longer optimizing in isolation, because the seller’s ideal outcome may lie outside what the buyer and lender can support. The seller is not searching for an unconstrained maximum, but for the best outcome among those that can actually survive the system.

To see how this works in practice, it helps to make the seller’s objective more explicit. Suppose the seller evaluates outcomes based on four dimensions: price P(x), cash at closing C(x), execution certainty T(x), and seller’s retained risk or liability R(x). Then:

where

subject to

and

where IRR(x)=return on investment and L(x)=leverage ratio, and

where COV(x)=coverage ratio and D(x)=default risk.

The seller prefers higher P, higher C, higher T, and lower R. But those dimensions do not move independently. A buyer may be willing to increase P(x), but only by replacing cash at closing with an earnout, which increases total potential value while reducing certainty of what will actually be received. A seller may reduce retained liability only by accepting a lower purchase price or a narrower buyer pool. A lender may cap leverage, which reduces the cash available at closing and forces the buyer to rebalance price, structure, or both. The transaction that emerges is therefore not the maximum of any one variable in isolation. It is the point in F where further improvement in one dimension would either violate a constraint or worsen another dimension materially. This is what practitioners are describing when they say a deal has been pushed to its limit.

Finding the Set and the Intersection

Consider again a business generating $8 million in EBITDA. A seller may believe the business supports a 10x (ten-times) multiple, implying an enterprise value of $80 million. A buyer evaluating the same company under a 20% return requirement and approximately 4x leverage would begin by determining how much debt the business can sustain. At 4x, an $8 million EBITDA business supports roughly $32 million of debt. The rest of the purchase price must be funded with equity, and that equity must generate the target return. Once those constraints are modeled, taking into account expected cash flow, growth, and exit assumptions, the maximum price the buyer can justify may be closer to $64 million. That difference is often described as a negotiation gap, but that description is incomplete. What it actually reflects is the distance between two sets of acceptable outcomes defined by different objective functions. In their initial form, those sets may not intersect.

If the seller’s acceptable set begins at $80 million under preferred terms, and the buyer’s acceptable set ends at $64 million under realistic financing constraints, the initial intersection may be empty. That does not end the process, it defines the problem. The work of dealmaking is to test whether changes in structure can move those acceptable regions into overlap. If leverage can expand, if contingent consideration can be introduced, if rollover equity is acceptable, or if risk can be redistributed in a way both sides can tolerate, then the set may become non-empty. If those adjustments still fail to produce overlap, then no transaction exists under the available conditions.

When the intersection does exist, the problem takes a different form. The objective is no longer to create a deal from nothing, but to move within the feasible set toward the most favorable available point. Structure becomes the mechanism through which that movement occurs. An earnout can increase total consideration while transferring part of the performance risk to the seller. A rollover can reduce the buyer’s upfront capital requirement while preserving exposure to future upside. Seller financing can bridge differences between available debt and desired proceeds. Each of these adjustments changes the mapping from x to VS(x), VB(x), and VL(x), which is another way of saying that it changes how each participant evaluates the transaction. None of these mechanisms eliminates the underlying constraint, and none creates a global optimum. What they do is make it possible to search for a point inside the feasible set where the structure holds and the deal can close.

The final transaction is therefore not the maximum of any single variable, nor is it a compromise in the casual sense of the term. It is the point x ∈ F where further movement in one direction pushes the structure outside what the other participants can accept. That is the boundary practitioners reach when they say a deal has been fully negotiated. In formal terms, it is a boundary point of the feasible set. In practice, it is the moment when another turn of price, leverage, or protection breaks the deal.

What Better Dealmakers Actually See

Most people experience transactions through language, not structure. One side says the business is worth more, while the other says the risks are higher than expected. A lender says leverage cannot go that far. In ordinary conversation, that sounds like disagreement, posturing, or pressure. In mathematical terms, it is the system revealing its constraints. That distinction matters because it changes how the process is understood. If every difference in position is interpreted emotionally, the process becomes theatrical very quickly: a buyer might look like a lowballer, a seller might look unrealistic, and a lender might look obstructive. But once the problem is seen structurally, the issue is no longer who is right, but whether a feasible set exists under each party’s conditions, even if pushed by financial engineering. And if it does, the question becomes how the structure can be moved toward the most favorable point inside it.

That is where mathematics becomes useful to the practitioner. It does not turn dealmaking into a machine, and it does not eliminate judgment. It forces clarity about preferences, constraints, and tradeoffs. It makes it easier to separate structural limitations from noise and ego, and it can turn a bad process into a disciplined one. Arrow explains why no transaction can satisfy every participant’s ranking of outcomes at once. The optimization framework explains how to navigate this system and push it toward a viable closing. A deal exists only where acceptable regions overlap, and the final structure is the result of pushing as far as possible within that overlap. We may not speak in functions and sets when we conduct a sale process, but that is the logic we are following all the same.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

The Mathematics of Dealmaking was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

Most people replace makeup brushes without ever asking a more important question:

Was this brush supposed to fail this quickly?

A brush sheds, loses its shape, starts applying unevenly, or simply feels different than it once did. The default response is to replace it. What’s rarely examined is whether that outcome was normal — or avoidable.

Makeup brushes are not designed to last forever. But they are also not designed to degrade as quickly as many do. The difference lies in how they are built, how they are used, and how we interpret their lifespan.

Understanding that difference changes not just how often you replace brushes, but how you evaluate them in the first place.

Why some brushes last — and others don’t

There is a wide gap between how long makeup brushes are expected to last and how they actually perform over time.

Some brushes begin to shed, lose their shape, or apply unevenly within months. Others maintain their structure and control for years of consistent use. At first glance, that difference can feel unpredictable. In reality, it is not.

Brush longevity is not driven by price alone. It is driven by how the brush is built.

Fiber quality determines whether bristles retain their softness and resilience or begin to break down with repeated use. Density and shape control whether the brush holds its form or gradually loses precision. Construction — how fibers are anchored, how the ferrule is fitted, how the handle is balanced — determines whether the brush remains stable or starts to loosen over time.

These are mechanical factors. They are not subjective, and they are not random.

Many brushes are designed to meet a price point, not a performance standard. When materials or construction are compromised, the effects don’t always show immediately. They appear gradually — first as subtle inconsistencies, then as clear degradation.

By contrast, a well-constructed brush is designed to maintain its performance over time. Not indefinitely, but consistently — long enough that the difference becomes obvious only in hindsight.

Understanding this distinction changes how lifespan should be interpreted. The question is no longer whether a brush “lasts,” but how long it continues to perform at a high level before it begins to decline.

What actually wears out in a makeup brush

When brushes fail, they don’t all fail in the same way. The underlying mechanics are relatively consistent.

Shedding is often the first sign of structural weakness. It usually reflects poor fiber anchoring or low-quality adhesive.

Shape loss follows. Fibers that once held a precise form begin to splay or collapse, affecting control and application.

Ferrule loosening is less common but more severe. Once the connection between the handle and the brush head weakens, stability is compromised.

Handle imbalance can develop over time, particularly if materials expand, contract, or loosen with repeated cleaning and use.

None of these issues are random. They are tied directly to how the brush was designed and assembled.

What a “normal” lifespan actually looks like

There is no single lifespan that applies to every brush, but there are realistic ranges based on construction quality.

Lower-quality brushes often show performance decline within a few months to a year. Shedding, uneven application, and shape loss tend to appear relatively early.

Well-made synthetic brushes, when engineered properly, can maintain consistent performance for one to three years or more under regular use.

Higher-end or artisanal brushes can last longer, but their longevity is highly dependent on care and usage patterns.

Some brushes last far longer than expected. Others fail early despite a high price tag. But these ranges reflect what most users should reasonably expect.

Care extends life — but it doesn’t create quality

Cleaning and maintenance matter. Regular washing removes product buildup, preserves fiber softness, and reduces unnecessary stress on the brush.

But care has limits.

Maintenance can extend the lifespan of a well-made brush. It cannot fundamentally fix a poorly constructed one. A brush with weak anchoring or low-quality fibers will degrade, regardless of how carefully it is treated.

This shifts the focus away from maintenance alone and back to what matters most: how the brush was built from the start.

The real metric: cost per use

A makeup brush is not a one-time purchase. It is a tool used repeatedly over time — and its value is defined by how long it continues to perform at a high level.

What matters is not just how long it exists, but how long it continues to perform at a high level.

This changes how value should be evaluated.

A lower-priced brush that needs to be replaced every few months often ends up costing more over time than a well-made brush that maintains its structure, softness, and control for years.

The difference isn’t always obvious upfront. It becomes clear only when you consider how many times a brush needs to be replaced — and how consistently it performs between those replacements.

Seen this way, the question shifts from “How much does this brush cost?” to:

“How long will this brush perform the way I need it to?”

That is a more useful question — and a more demanding one.

Where performance-first design matters

Some brushes are built to meet a price point. Others are built to meet a performance standard.

The difference shows up over time.

Design decisions around fiber quality, density, anchoring, and balance determine not just how a brush feels on day one, but how it behaves after months and years of repeated use. Whether it holds its shape. Whether it sheds. Whether it continues to apply product evenly.

When these elements are treated as performance variables rather than cost constraints, the result is not just a brush that feels good initially, but one that maintains that level of performance over time. That consistency is engineered.

Rethinking what you’re actually buying

When you buy a makeup brush, you are not just buying an object. You are buying a period of reliable performance.

Some brushes deliver that performance briefly. Others sustain it.

The difference is not always visible at the moment of purchase. But it becomes obvious over time — often in ways that feel subtle at first, and then impossible to ignore.

Understanding how long a brush should actually last doesn’t just help you replace it at the right time.

It helps you choose better from the start.

The goal is not for a brush to simply last, but to continue performing the way it was designed to — over time.

Written by BERRY GOODS

This article was originally published on the BERRY GOODS journal.

How Long Should Makeup Brushes Actually Last? was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

The Limits of Aggregation: Arrow’s Impossibility Theorem

In 1951, Kenneth Arrow formalized a problem that had been implicit in economics and political theory for decades. When multiple individuals hold different preferences over a set of alternatives, is it possible to construct a rule that converts those preferences into a single, consistent decision for the group? The question appears technical at first glance, but it sits at the center of how committees decide, how institutions allocate resources, and how markets coordinate outcomes across participants with conflicting objectives.

Arrow approached the problem with precision. Each individual is assumed to have a complete ranking over a set of alternatives. A decision rule, or social welfare function, takes those individual rankings and produces a collective ordering. Formally, the problem can be written as a function that takes individual preference orderings and produces a collective ranking:

where each Pi represents the entire preference ordering of individual i, and P is the resulting group ordering. The ambition was modest in appearance: define a rule that respects a small set of conditions that most people would consider reasonable. If every participant prefers one alternative over another, the collective decision should reflect that agreement. The ranking between two alternatives should not depend on irrelevant third options. And no single participant should determine the outcome regardless of everyone else’s preferences.

Each of these conditions is intuitive on its own. Together they resemble the minimal structure of a fair decision system. Arrow proved that once the number of alternatives reaches three, no rule can satisfy all of them simultaneously. Any mechanism that aggregates preferences must violate at least one of these conditions. The limitation does not arise from poor design or insufficient information. It is embedded in the structure of the problem itself. The moment different participants rank outcomes differently, the system cannot produce a collective decision that preserves every property we would like it to have.

The result reshaped welfare economics because it replaced a search for the correct decision rule with a recognition that no such rule exists under those constraints. The implication is not that decisions cannot be made, but that any decision system necessarily embodies a tradeoff. Some condition must give way for the system to function.

Incompatible Preferences

The force of Arrow’s theorem becomes clearer when the problem is viewed without formal notation. Consider a group evaluating three alternatives. Different individuals rank those alternatives differently, but the group must produce a single ordering. It is natural to assume that a sufficiently well-designed rule will preserve the consistency of those rankings. Arrow’s contribution was to show that the expectation is misplaced. Once three or more alternatives are involved, the conditions that define a “reasonable” aggregation rule conflict with one another.

This conflict is not the result of irrational behavior. Even when every participant is perfectly rational, the aggregation of their preferences introduces contradictions. A rule that respects unanimous agreement may fail to preserve consistency when irrelevant alternatives are introduced. A rule that avoids those inconsistencies may concentrate decisive power in a way that violates the principle of non-dictatorship. The system cannot satisfy all conditions simultaneously because the conditions themselves are incompatible when applied together.

Here is the simplest intuition. Think of the voting paradox discovered by the Marquis de Condorcet, and imagine three voters and three alternatives: A, B, and C. Voter 1 ranks them A≻B≻C, voter 2 ranks them B≻C≻A, and voter 3 ranks them C≻A≻B. Now compare the options two at a time. Between A and B, voters 1 and 3 prefer A, so the group prefers A over B. Between B and C, voters 1 and 2 prefer B, so the group prefers B over C. Between C and A, voters 2 and 3 prefer C, so the group prefers C over A. The collective preference becomes A≻B, B≻C, and C≻A. The group prefers A to B, B to C, and C to A. The ranking is no longer transitive; it collapses into a cycle.

The implication is structural. The limitation does not disappear with better data, more sophisticated modeling, or more cooperative participants. It persists because it is built into the logic of preference aggregation. The expectation of a perfectly consistent and fair decision rule is therefore not simply unrealistic. It is mathematically impossible under the assumptions that define the problem.

The Structure of Every Deal

Transactions in finance operate inside the same logic. A sale process brings together participants whose objectives are not aligned but whose decisions must ultimately produce a single outcome. A seller may rank outcomes according to valuation, certainty of closing, and preservation of what has been built. A buyer evaluates the same set of possibilities through the lens of expected return, strategic fit, and risk. Lenders, when involved, focus on downside protection and the stability of cash flows. Each group enters the process with a distinct ordering of outcomes.

The transaction is the mechanism that collapses those rankings into a single outcome. At first glance it is tempting to assume that a sufficiently well-structured deal can satisfy every participant simultaneously. Experience shows otherwise. Increasing valuation tends to introduce elements of uncertainty such as earnouts or deferred payments. Eliminating uncertainty tends to compress valuation. Strengthening financing terms improves certainty of closing while imposing constraints elsewhere in the structure. Each adjustment improves one dimension of the outcome while weakening another.

Consider a company generating $8 million in EBITDA entering a sale process. The seller may believe the business supports a ten-times (10x) multiple, implying an enterprise value of $80 million. A buyer evaluating the same company may conclude that the appropriate multiple is closer to 8x EBITDA, or $64 million, based on its return requirements and risk assumptions. The difference between those positions is not abstract; it is a $16 million gap that must be resolved before a transaction can exist.

The disagreement is often framed as a question of valuation, but the structure of the problem reveals something deeper. The seller’s preference ranking places maximum weight on price. The buyer’s ranking places maximum weight on risk-adjusted return. If the buyer finances a portion of the transaction with debt and targets a specific internal rate of return, the amount of cash that can be paid at closing is constrained by those parameters. The conflict is therefore not simply a matter of perspective; it reflects the interaction of different preference systems operating under real constraints. Every deal is an optimization problem with incompatible constraints. The mistake is believing those constraints can be removed.

Designing at the Boundary

The work of a sell-side advisor takes place within this structure of incompatible preferences. The objective is not to construct a transaction that satisfies every participant equally, because the system rarely allows that outcome. The objective is to define the seller’s priorities with precision and design a structure that pushes those priorities as far as the constraints of the system permit.

In the example above, the gap between $64 and $80 million does not disappear through argument. It is addressed through structure. A portion of the purchase price may be tied to future performance through an earnout linked to revenue or EBITDA milestones. The seller may roll over a percentage of equity into the combined entity, preserving exposure to future upside while reducing the buyer’s upfront capital requirement. Seller financing may be introduced to bridge the gap between what the buyer can pay at closing and the valuation the seller seeks. Each of these mechanisms reshapes the set of feasible outcomes without eliminating the underlying tension between price and risk.

These structures do not resolve the disagreement in the sense of eliminating it. They convert the disagreement into a form that allows the transaction to exist. The seller’s objective approaches the boundary defined by the buyer’s constraints, but it does not override those constraints. Negotiation becomes a process of understanding where that boundary lies and designing within it. Once the constraint is visible, the discussion shifts from abstract valuation to concrete structure.

Choosing the Tradeoff

Arrow’s theorem forced economists to recognize that certain decision systems cannot satisfy all the conditions we would like them to meet. Transactions in finance encounter the same limitation in practice. When multiple stakeholders rank outcomes differently, no structure can maximize every objective simultaneously. The expectation that a deal can deliver the highest valuation, complete certainty, minimal risk, and full alignment across all participants is not simply optimistic, it is inconsistent with the structure of the problem.

Again, the consequence is not that decisions cannot be made. Decisions are made every day, and transactions close under a wide range of structures. The consequence is that every transaction embodies a choice among tradeoffs. One objective defines the outcome, and the others adjust around it. The role of the advisor is not to eliminate those tradeoffs but to understand them early and design the structure so that the chosen objective is achieved as fully as the system allows.

In practice, no deal satisfies everyone. The structure won’t allow it. The outcome is determined by which constraint binds and which objective is allowed to dominate. Once that is clear, the rest is execution.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

Valuation Is Not the Deal was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.

The average is one of the most widely utilized concepts in statistics: collect many observations, compress them into a single number, and the apparent randomness of individual outcomes reveals an underlying order. If the average temperature in a city is seventy-five degrees, most days will orbit somewhere around that value. If the average exam score in a class is eighty percent, the distribution of results will cluster modestly above and below that center. The average becomes a gravitational point around which reality fluctuates. It promises that beneath the chaos of individual variation lies a stable description of the system.

In many domains this intuition works beautifully because the mathematics cooperates. Human heights cluster tightly around a central value. Measurement errors in engineering follow distributions where large deviations become exponentially unlikely. Manufacturing tolerances drift slightly but rarely catastrophically. In systems like these, the probability of extreme observations declines so rapidly that their contribution to the average becomes negligible. As more observations accumulate, the sample mean converges toward a stable value. The average becomes not merely convenient but meaningful, a reliable summary of the process generating the data.

Because this logic works so well in everyday contexts, we instinctively extend it to far more complicated systems. Businesses forecast average growth rates, investors estimate average returns, economists model representative agents whose behavior stands in for entire populations. The average becomes the lens through which uncertainty is reduced to something intelligible. Faced with a world of innumerable possibilities, the promise of a single stabilizing number becomes irresistible. Yet mathematics offers an unsettling counterexample: an average is not always achievable. Entire families of distributions exist where this comforting property breaks down, where the very concept of an average stops behaving the way our intuition expects. In those systems the average is not merely difficult to estimate or sensitive to the data; mathematics cannot even define one.

Distributions Without a Mean

The phenomenon emerges from a class of probability distributions known as heavy-tailed distributions. Unlike the familiar bell curve, where extreme outcomes become exponentially rare, heavy-tailed distributions decay much more slowly. Large events still become less likely as they grow in magnitude, but they do not disappear quickly enough to stop influencing the system. The tail of the distribution retains enough probability mass that rare observations remain capable of dominating the overall outcome.

One of the simplest examples is the Pareto distribution, introduced by the Italian economist Vilfredo Pareto while studying the distribution of wealth in nineteenth-century Europe. Pareto noticed that a small fraction of individuals controlled a disproportionate share of economic resources, a pattern that appeared repeatedly across countries and datasets. When he examined the numbers mathematically, he discovered that wealth followed a power-law distribution rather than the tidy bell curve economists often assumed. In a power-law world, large outcomes shrink in frequency only gradually, leaving the tail thick with potential extremes.

The mathematics of the Pareto distribution reveals an important threshold. The expected value of the distribution exists only when the tail parameter exceeds one. Mathematically, the mean of a Pareto distribution is

where Xm represents the minimum possible value of the variable and α determines how quickly the probability of extreme outcomes declines. Large values of α mean the tail decays rapidly and extreme events fade quickly; small values of α mean the tail decays slowly and rare events remain influential. When the parameter α>1, the distribution possesses a finite mean. When α≤1, the expected value diverges and the distribution has no finite mean. In practical terms, the average is not merely hard to calculate; it is mathematically undefined. No matter how many observations are collected, the sample mean never stabilizes because rare extreme events keep dragging it upward. The system contains outcomes so large that they overwhelm the accumulation of ordinary observations. In plain terms, when α≤1 the average does not exist.

An even stranger mathematical region appears when 1<α<2. In that interval the average exists, but the variance does not. The distribution has a center, yet its fluctuations are theoretically unbounded because rare extreme observations dominate the variability of the system. Empirical studies of wealth distributions, firm sizes, and financial returns often place real systems somewhere in this peculiar territory where the mean is finite but the variability refuses to behave. The mathematics is not claiming that extreme events occur frequently, but that even rare events can shape the entire structure of a system when their magnitude grows quickly enough.

The Financial Systems Governed by the Tail

Once this mathematical structure is understood, its fingerprints begin to appear throughout finance. Many financial systems display precisely the asymmetry that Pareto observed in wealth distributions. Most outcomes cluster around mediocrity, while a small number of extreme events dominate the totals. The center of the distribution describes the typical experience, but the tail determines the aggregate result. The parameter α can be understood as a measure of how quickly extreme events fade away. When α is large, the probability of very large outcomes declines rapidly and the system behaves in a familiar way: extreme observations become negligible and averages remain stable. When α is small, the tail decays slowly and rare events remain influential even when they occur infrequently. In that regime the mathematics tells us something profound about financial systems: a handful of extreme outcomes can dominate decades of ordinary activity, and α becomes a measure of how strongly the tail governs the system.

Venture capital offers a vivid example. A typical venture portfolio contains dozens of investments. Most fail outright, a handful produce modest gains, and occasionally one company grows into a global platform business whose valuation dwarfs the rest of the portfolio combined. The distribution of outcomes becomes profoundly skewed: the success of the entire fund depends less on the typical investment than on the rare outlier that succeeds spectacularly. The mathematics of heavy tails explains why the venture industry tolerates such a high rate of failure. In a power-law world, a single extraordinary success can dominate the entire distribution.

Public equity markets display a similar pattern. Research by Hendrik Bessembinder has shown that between 1926 and 2016, just 4% of publicly listed companies in the United States accounted for the entire net wealth created by the stock market. The majority of stocks delivered returns comparable to short-term Treasury bills, and many destroyed value altogether. Yet a small minority of firms generated extraordinary gains that dominated the aggregate outcome. Investors experience the market through the lens of averages and indices, but the long-term wealth creation of the system is driven by a narrow set of extreme winners.

Financial crises follow the same logic. Markets can appear stable for years or even decades, fluctuating comfortably around their expected behavior. Then a rare systemic shock arrives, a sudden collapse in liquidity, a cascading failure of leverage, or a crisis of confidence, and the entire system reorganizes in a matter of days. Decades of calm become historical footnotes compared with the magnitude of the tail event. These episodes reveal the same underlying structure: the system spends most of its time near the center of the distribution, but its defining moments occur far out in the tail.

When Modeling Meets the Tail

Seen in this light, the Pareto distribution reveals a third limit in our attempts to model complex systems. Earlier explorations of mathematical structure have shown that combinatorial problems such as the Traveling Salesman explode so quickly that perfect optimization becomes computationally impossible, while Cantor’s work on infinity demonstrates that continuous systems may contain uncountably many possible outcomes. Heavy-tailed distributions introduce a different kind of boundary. Even when we abandon enumeration and compress uncertainty into averages, mathematics may still resist our attempt to summarize the system.

Some distributions are structured in such a way that rare events dominate everything else. In those environments the average either becomes unstable or disappears entirely as a meaningful concept. The center of the distribution exists, but it does not tell the story that determines the system’s fate. The mathematics is not failing; it is revealing that the system cannot be understood by focusing on its typical behavior. Its defining features lie in the extremes.

In finance this insight has practical consequences. If investors optimize portfolios based on average returns, they may miss the rare events that shape long-term outcomes. If risk managers focus on ordinary fluctuations, they may underestimate the shocks that threaten the survival of institutions. Entrepreneurs building venture portfolios quickly learn that success depends less on the median investment than on the extraordinary one that transforms the entire distribution. When systems are governed by heavy tails, the center describes how the system usually behaves, but the tail determines how the story ends.

— Carlos E. Mora
I wake up, I build, I repeat. No guarantees.
I work like it’s all on the line, because it is.
Family is the only true legacy.
Your name is your currency, and it must be earned daily.

An Average May Not Exist in Finance was originally published in All on the Line on Medium, where people are continuing the conversation by highlighting and responding to this story.