prompt engineering

Teaching an AI to Reason Like Fermi: Part 2 – Permission to Guess

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Posted on February 25, 2026 by matteomycarta

Previously, on this blog

In Part 1 of this series, I told you the story: how I tried to teach AI to do Fermi estimation, hit a wall with GPT-3’s arithmetic, got past it with the Wolfram plugin, and discovered the real problem wasn’t calculation — it was judgment. I showed you the results: a framework that lifts scores modestly (7.9 to 8.5 out of 10) but transforms the reasoning from opaque assertions into auditable, bounded estimates.

I promised to show you the framework itself. Here it is.

The architecture

The framework has two layers. The first handles mechanics — how to do the math. The second handles judgment — how to think about what you know and don’t know. The mechanical layer matters less than you’d expect. The judgment layer matters more.

The mechanical laws (LAW0–LAW11)

These twelve laws govern arithmetic, notation, and reporting. They exist because early LLMs were unreliable at basic computation, and even modern models benefit from explicit procedural scaffolding.

LAW0 sets a five-step workflow adapted from Chapter 2 of Weinstein’s Guesstimation (“Dealing with Large Numbers”): interpret the problem, identify known facts, make assumptions, calculate explicitly, and report results. This is the backbone — every problem follows this sequence.

LAW1–LAW7 handle scientific notation: how to multiply (add exponents), divide (subtract exponents), take square roots (adjust odd exponents), and compute geometric means. These feel pedantic, but they ensure consistent notation and prevent cascading errors in multi-step calculations. They were drafted partly in response to GPT-4’s arithmetic failures — in one documented session, it multiplied 836 × 1923 eight times and got a different wrong answer on seven of them. As I noted in Part 1, modern models handle arithmetic reliably; the judgment problem is what persists.

LAW8–LAW9 handle the practical side of units: do the math in whatever units are convenient (inches, miles, eV), but convert to SI for the final answer; and keep exponents positive by choosing appropriate unit scales. LAW10 says “read the worked examples first” — learning from demonstrations before attempting problems. LAW11 is prosaic but necessary: “don’t waste tokens rephrasing my question back to me.”

If you’re an experienced estimator, you can skim these. They’re scaffolding for the model, not insight for you. The estimation laws are where it gets interesting.

The estimation laws: teaching judgment

These seven laws address the hard part — knowing when to guess, how to bound, and when to stop and ask for help.

LAW-INTERPRET says: before doing anything, state what you think the problem is asking. If it’s ambiguous, ask. This was added after a specific failure I’ll show you shortly.

LAW-ESTIMATE is the core: for every quantity you use, you must do one of three things. State it as known — a fact any technically literate person would accept (speed of light, human height, Earth’s population). Bound it with reasoning — construct explicit upper and lower limits from physical or logical arguments, then take the geometric mean. Or ask — if you can’t do either, stop and request human input. “I’ll estimate X as Y” without justification is explicitly forbidden.

LAW-BOUNDS governs the quality of those bounds. They should span 1–3 orders of magnitude. They should be physically motivated, not arbitrary. And — this is one of my favorite additions — behavioral observations make excellent bounds. “A gecko can walk on a ceiling but cannot run on one, so adhesive force is between 1× and 10× body weight.”

LAW-PROCEED gives permission to be imprecise: a factor-of-ten estimate beats no estimate. This is straight from Weinstein: “Dare to be imprecise!” It counteracts the model’s tendency to hedge or refuse when it doesn’t have exact data.

LAW-DECOMPOSE is the problem-solving engine: if stuck, ask “what would I need to know to answer this?” Each answer becomes a sub-problem. Then classify each sub-quantity as KNOWN, BOUND, or ASK before proceeding.

LAW-VALIDATE and LAW-REPORT handle the endgame: cross-check using independent approaches, express answers in SI units with scientific notation, and always compare to something meaningful.

The whole thing flows as a six-step cycle: INTERPRET → DECOMPOSE → ESTIMATE → CALCULATE → VALIDATE → REPORT.

What good Fermi reasoning looks like

Before showing you how the model performs, let me show you what we’re aiming for. Here’s one of Weinstein’s worked examples — a problem that demonstrates nearly every technique the framework encodes.

The setup: Weinstein is at a Norfolk Tides baseball game when a foul ball showers his friends with beer. The question — what is the probability of a foul ball landing in a cup of beer during one game, and how many “splash downs” happen across a full MLB season? He starts by estimating about forty foul balls per game land in the stands (more than one per inning, fewer than twenty — geometric mean of five, times nine innings). Then he decomposes the per-ball probability:^*

Now we need to estimate the probability that a given foul ball will land directly in a cup of beer. (Note: only beer is sold in open-topped cups.) This means that we need to break the problem into even smaller pieces. Let’s assume that the cup of beer is sitting innocently in a cup holder. To hit a cup of beer, the foul ball needs to:

1. not be caught by a fan
2. land within the area of a seat
3. hit a seat whose owner has a cup of beer
4. land in the cup

Most fly balls are caught, but many are not. Let’s estimate that between one-quarter and one-half of fly balls are not caught. “Averaging” the two, we will use one-third.

Most of the stadium area is used for seating, so let’s ignore that factor.

At any given time, more than 1% and less than 100% of fans have a cup of beer in front of them. Using the geometric mean, we estimate that 10% of seats have beer cups.

A large beer cup is 4 inches (10 cm) across, so the baseball must land in an area defined by

A_cup = πr² = 3(2 in)² = 10 in²

The area of the seat (from arm rest to arm rest and from row to row) is about 2 ft by 3 ft (60 cm by 90 cm), so

A_seat = (24 in) × (36 in) = 10³ in²

Thus, if the ball hits a seat that has a cup of beer, the probability that it lands in the cup is

P_cup = A_cup/A_seat = 10 in²/10³ in² = 10⁻²

or 1%. The metric probability is the same.

(Extra credit question: Which is more likely, that the ball lands in the cup in the cup holder, splashing the beer, or that the fan is holding the cup of beer when the foul ball arrives and splashes it in his or her excitement?)

This means that the probability that any single foul ball lands in a cup of beer is

P = (1/3) × (1/10) × (10⁻²) = 3 × 10⁻⁴

With forty foul balls per game, this means that the probability of a foul landing in a cup of beer during any one game is 10⁻². This is not very likely. The probability that we will be directly below the splash is even less likely.

He then scales to the full MLB season — 30 teams, 160 games each, about 2,000 games total — and gets about 20 beer landings per season. His closing observation: because baseball analysts keep meticulous statistics, he’s very surprised they don’t appear to record beer landings.

Notice what happened. An impossible-sounding question became four independent sub-conditions, each bounded with reasoning anyone can follow. The geometric mean of 1% and 100% gives 10% — you don’t need to survey stadiums. The area ratio gives 1% — you don’t need probability theory. And the answer emerges from combining many small, defensible estimates. No single number needs to be precise. The decomposition does the work.

This is what we’re teaching the model to do.

The worked examples: teaching by demonstration

The framework includes 14 worked examples from Weinstein’s Guesstimation books. Each demonstrates specific reasoning patterns the laws encode. Here are two that show how different techniques work in practice.

Bounding by physical comparison

In Spider-Man 2, Spider-Man stops a runaway subway train by attaching his webs to nearby buildings and pulling for ten or twenty city blocks. Weinstein asks: how much force does he need? In the book, this follows a crash test dummy problem using the same work-energy approach — Weinstein builds technique through repetition with variation. Here’s how he tackles it:^†

In the movie Spider-Man 2, Spider-Man stops a runaway New York City six-car subway train by attaching his webs to nearby buildings and pulling really hard for 10 or 20 city blocks. How much force does he have to exert to stop the subway train? Give your answer in newtons and in tons (1 ton = 10⁴ N). How does this compare to the force that you can exert?

Since the work done by Spider-Man to stop the train is equal to the train’s initial kinetic energy, we need to estimate the mass and velocity of the train. We will then need to estimate the stopping distance in order to calculate the force exerted.

A subway car is about the same size and weight as a semi-trailer (18-wheeler) truck. This is between 10 and 40 tons. We’ll use 20 tons (or 2 × 10⁴ kg). There are six cars on a train so that the mass of the train is 6 × 2 × 10⁴ kg = 10⁵ kg.

They certainly go faster than 20 mph and slower than 100 mph. Since it is not that far between subway stops, subways travel at only about 40 mph (20 m/s).

Thus, the kinetic energy of a subway train is:

KE = ½mv² = 0.5 × 10⁵ kg × (20 m/s)² = 2 × 10⁷ J

There are 20 blocks per mile in Manhattan. Thus, 10 or 20 blocks is about 1 km or 10³ m. (It’s certainly more than 100 m and less than 10 km.)

Thus, Spider-Man needs to exert a force:

F = KE/d = (2 × 10⁷ J)/(10³ m) = 2 × 10⁴ N

F = (2 × 10⁴ N)/(10⁴ N/ton) = 2 tons

A force of 2 × 10⁴ N is the weight of 2000 kg or 2 tons. For a superhero who can lift cars, this is quite possible (although definitely not easy). A human could definitely not do it.

Wow! Hollywood got the physics correct, in a superhero movie no less! Hurray!

The key move: “a subway car is about the same size and weight as a semi-trailer truck.” He doesn’t look up the mass of a subway car. He bounds it by comparison to something familiar — between 10 and 40 tons — and uses 20 as a round number near the geometric mean. Every quantity gets the same treatment: velocity bounded by “certainly faster than 20 mph, slower than 100 mph,” stopping distance from “20 blocks per mile in Manhattan.” The entire solution is built from things you can verify from ordinary experience.

The obvious answer is wrong

This next example is my personal favorite. It was the first Weinstein problem I solved entirely on my own — and the moment I realized Fermi estimation was something I could actually do. Weinstein gives four decomposition hints nudging you toward the fibers, but the real lesson is that the “obvious” answer is spectacularly wrong:^†

What is the surface area of a typical bath towel (include the fibers!)? Compare this to the area of a room, a house, a football field.

That’s obvious, surely! A large rectangular towel 1 m by 2 m has a total surface area of 4 m² (including both sides), right? (In US units, a big towel may be as large as 3 ft by 6 ft.)

Wrong, actually, unless it is a very worn-out towel. New towels have many little fibers that can absorb a lot of moisture (recall the old puzzle—what gets wetter the more it dries?). Unless you’re a fan of the Hitchhiker’s Guide to the Galaxy, you won’t have brought your own towel, so nip off to the bathroom and examine one; quickly now, we’re dripping all over the floor.

You don’t need to actually go and count the number of fibers per square inch or per square centimeter; in the latter case there must be more than 10 and fewer than 1000, so we take the geometric mean of 10¹ and 10³, which is 10². In a square inch, being about 6 cm², we should expect about six times as many. This will of course vary, depending on where you buy your towels; we are assuming that we are describing one of those very nice towels found in one of those very nice hotels.

Back already? Right-oh. Now we need to estimate the surface area of each fiber. We can approximate the fiber as a cylinder or a box. Cylinders are complicated so we’ll use boxes. Each fiber is about 0.5 cm (1/4 in.) long and 1 mm (0.1 cm) wide. Each “boxy” fiber then has four flat surfaces, each 0.5 cm by 0.1 cm. Thus, the surface area of one fiber is:

A_fiber = 4 × 0.5 cm × (1 m/10² cm) × 0.1 cm × (1 m/10² cm) = 2 × 10⁻⁵ m²

A_total = towel area × fibers per area × area per fiber
= 4 m² × (10² fibers/cm²) × (10⁴ cm²/1 m²) × (2 × 10⁻⁵ m²/fiber)
= 80 m²

That is about 800 square feet: the size of a large apartment or a small house.

Weinstein draws an analogy to the coastline problem — just as the length of coast from New York to Boston is much more than the 200-mile driving distance, the surface area of a towel is much more than its simple area. The answer, 80 m², is twenty times larger than the “obvious” 4 m². The geometric mean does the heavy lifting: more than 10 fibers per cm², fewer than 1000, so about 100. No counting required.

LAW10 exists because of these examples. When I was learning Fermi estimation myself, I noticed something: I got measurably better after working through more of Weinstein’s examples. His reasoning patterns — the way he finds comparable objects, the way he uses “certainly more than X, certainly less than Y” — became internalized heuristics. I wanted the model to have the same experience.

Does it work? Partially. The model absorbs the decomposition patterns perfectly. Every test problem was broken down cleanly. But the bounding technique — the heart of Weinstein’s method — transfers inconsistently. Sometimes the model produces beautiful, physically-grounded bounds. Sometimes it just asserts a number.

Which brings us to the evidence.

What the framework actually changes: side by side

The quantitative lift is modest — 0.6 points on our 8-problem test. But the qualitative difference is real. Here’s what it looks like in practice.

Gecko adhesive force

Without framework (Project C — basic Fermi prompting):

Gecko mass: 50-100 g → use 75 g

That’s it. Two numbers, no reasoning. Why 50? Why 100? We have no idea.

With framework (Project A):

Gecko mass (BOUND):
Lower bound: more than 10 g (smaller than a mouse)
Upper bound: less than 500 g (smaller than a rat)
Geometric mean: √(10 × 500) ≈ 70 g

Now we can see the thinking. We know the bounds are anchored to familiar animals. We can ask: is a gecko really smaller than a rat? (Yes, most species.) Is it heavier than 10 grams? (Some species barely are.) The answer is similar — 70 g versus 75 g — but the reasoning is transparent and auditable.

Shark attack deaths

Without framework:

Annual shark attacks in US: ~30-50 → use 40
Fatal shark attacks: ~1-2 per year in US → use 1.5

Asserted. Where do these numbers come from? Training data, presumably. But we can’t verify the reasoning because there isn’t any.

With framework:

Annual shark deaths (BOUND):
Shark deaths are newsworthy when they occur, suggesting they’re rare:
Lower bound: more than 0.1 per year (would average 1 per decade)
Upper bound: less than 10 per year (would be weekly news)
Geometric mean: √(0.1 × 10) = 1 death per year in the US

Same order of magnitude. But Project A uses experiential reasoning — things you can verify from your own experience of watching the news. You don’t need to be a marine biologist to evaluate “if there were 10 shark deaths a year, that would be weekly news.” That’s the framework’s value: it forces the model to show reasoning you can engage with.

The drag coefficient — the signature example

This one appeared in Post 1, but it’s worth showing in full context.

Without framework:

Drag coefficient: C_d ≈ 1.0 (upright running human)
Bounds: [0.7, 1.3]

The model states a value, then wraps bounds around it that are barely wider than rounding error. The “bounds” are cosmetic — they don’t come from independent physical reasoning.

With framework:

Lower bound: more than 0.5 (very streamlined)
Upper bound: less than 2 (very blunt)
Geometric mean ≈ 1

Wider bounds, but anchored to the physical extremes of the drag coefficient spectrum. You can evaluate this: is a running human more streamlined than a flat plate? Yes. More blunt than a teardrop? Yes. The bounds make physical sense.

The honest assessment

The gap is modest, not dramatic. Project A shows the best reasoning. Project C is weakest. But Project C still gets reasonable answers — modern Sonnet is good at Fermi estimation even without the framework.

The framework’s value isn’t making the model smarter. It’s making the model’s thinking visible. And once thinking is visible, you can critique it, improve it, and learn from it. That’s the difference between a calculator and a collaborator.

LAW-INTERPRET: the fix that worked

The Flash problem broke something. Both our framework-trained models (Projects A and B) received this problem:

“The Flash can run at (or near) the speed of light. What is the minimum force needed to reach a speed of 10% the speed of sound in 10 ms?”

Both interpreted “10% the speed of sound” literally: 34 m/s. Neither noticed the tension between “near the speed of light” in the setup and a target velocity of 34 m/s — roughly highway speed. The problem as worded by Weinstein is genuinely ambiguous, and the models barreled through without noticing.

This bothered me. The framework told them to ask when uncertain, but there was nothing specifically about checking whether the problem statement itself makes sense. So I added LAW-INTERPRET:

Before solving, state your interpretation of the problem. If the problem is ambiguous or could be read multiple ways, ASK for clarification before proceeding.

With specific triggers: vague terms, multiple reasonable interpretations, missing context, physically implausible scenarios.

To test it, I gave the model an absurd problem: “A Formula 1 car can reach speeds over 200 mph. What is the drag force on the car when traveling at 5% of walking speed?”

The response was immediate and correct:

I need to clarify this problem before proceeding. The problem mentions that “A Formula 1 car can reach speeds over 200 mph” but then asks about drag force at “5% of walking speed.”

This creates an ambiguity:

Interpretation 1 (literal reading): Calculate drag force at 5% of walking speed — 0.075 m/s (7.5 cm/s). This is incredibly slow — slower than a crawl.

Interpretation 2 (possible typo): The problem might have meant 5% of the speed of sound, 5% of 200 mph, or 50% of walking speed.

Could you clarify?

The model recognized the absurdity, stated the literal interpretation, offered alternatives, and asked before proceeding. After clarification (“use 50% of top speed”), it solved the corrected problem cleanly, including well-bounded drag coefficient estimates.

LAW-INTERPRET works reliably. It’s also the simplest of our additions — just a rule saying “check the problem statement before diving in.” Sometimes the most effective interventions are the most obvious ones.

LAW-FLAG: the fix that didn’t

The bigger problem — the one I flagged in Post 1 as “it doesn’t know that it doesn’t know” — proved much harder.

The model never spontaneously asks for help when it encounters a quantity it can’t justify. During the initial eleven-problem diagnostic, this happened zero times. The model would confidently assert specialized physics values as if they were common knowledge.

We tried to fix this with what I called LAW-FLAG: requiring every numerical estimate to be classified as KNOWN (common knowledge), BOUNDED (derived from explicit reasoning), or GUESSED (can’t justify → stop and ask). The idea was that the classification step would force self-assessment — the model would have to ask itself “do I really know this, or am I pulling it from training data?”

To show you what’s at stake, here’s what expert Fermi reasoning looks like on a hard problem — one that chains together quantities from multiple specialized domains.

Journey to the Center of the Galaxy

Weinstein asks: how much radiation damage would we receive traveling to the center of the galaxy? He first establishes the distance (~3 × 10⁴ light-years = 3 × 10²⁰ meters), the density of interstellar space (~1 hydrogen atom per cm³), and the number of atoms our spacecraft would encounter (3 × 10²² per cm² of cross-section). Then:^*

In order to travel this distance within a subjective human lifetime, our spacecraft must achieve a speed very close to the speed of light. Even traveling at the speed of light, if there is no relativistic time dilation, our trip will take 3 × 10⁴ years, which is slightly longer than our expected lifetime. Fortunately, at speeds close to the speed of light, subjective time passes more slowly than it does for an observer watching from the center of the galaxy. We need time to pass about 10³ times more slowly. In technical terms, this means we need a relativistic gamma factor of about γ = 10³.

This means that, from our point of view (i.e., in our reference frame) each interstellar atom will hit us traveling at almost the speed of light with the same gamma factor and will thus have a total energy of

E = γmc² = 10³ × (1 GeV) = 1 TeV.

At these energies, when the hydrogen atom strikes the spacecraft, it will lose its electron very quickly. The bare proton will then pass easily through the spacecraft and our bodies, depositing energy in our bodies at a rate of 2 MeV per centimeter.*

Now we can consider the damage done by all these protons. Consider a volume of 1 cm³ within our bodies. At the density of water it has a mass of 1 g. It will be hit by 3 × 10²² protons, with each proton depositing 2 MeV of energy. This means that the total energy deposited per gram will be

E_dep = (3 × 10²² protons/cm²)(2 × 10⁶ eV/cm)
× (1 cm³/g)
= (6 × 10²⁸ eV/g)(1.6 × 10⁻¹⁹ J/eV)
= 10¹⁰ J/g = 10¹³ J/kg.

This seems like a lot, but let’s compare it with a few things before jumping to conclusions.

Unfortunately, we would not even survive that long. Radiation dose is measured in grays, where 1 Gy corresponds to an energy deposition of 1 J/kg. A lethal dose of radiation is about 10 Gy or about 10 J/kg. We will accumulate that dose in 10⁻¹² of our journey, or in the first millisecond.

As it only takes 2 × 10⁶ J/kg to boil water, we would be vaporized more than 10⁶ times over. In a thirty-year journey lasting 10⁹ s, we would be turned to steam within the first 10³ s, or 20 minutes.

Weinstein notes that these relativistic protons could pass through about a kilometer of metal — mass shielding can’t save you. His conclusion: who knew vacuum could be so dangerous?

Every quantity in that chain is specialized knowledge. A non-physicist can’t justify “proton energy deposition is 2 MeV per centimeter” or “interstellar density is 1 atom per cm³.” But a human expert knows what they know and where it comes from. That self-awareness — knowing which numbers are common knowledge and which require domain expertise — is precisely what the model can’t do. Now watch what happens when the model faces similar territory.

Test 1: Neutron star — FAIL

Problem: “How closely could we safely orbit a neutron star, considering only gravitational effects?”

The model’s classification:

Neutron star mass and radius:
KNOWN: Typical neutron star mass M ≈ 1.4 M☉ ≈ 3×10³⁰ kg
KNOWN: Typical neutron star radius R_ns ≈ 10 km = 10⁴ m

No hesitation. No uncertainty. “KNOWN.” But could you justify “neutron star mass is 1.4 solar masses” to someone who isn’t an astrophysicist? I couldn’t. The model could — because it had seen it thousands of times in training data. That’s not the same as knowing it.

What’s telling is the contrast: in the same solution, the model bounded the safety threshold beautifully, using fighter pilot g-forces as physical anchors. It can bound when it recognizes uncertainty. It just doesn’t recognize uncertainty about quantities that feel “textbook.”

Tightening the language

After this failure, I rewrote LAW-FLAG with stronger language:

KNOWN: Quantities you could justify to a skeptical non-specialist without domain expertise. If a skeptic could reasonably ask “how do you know that?” and you’d need specialized knowledge to answer, it is NOT KNOWN.

The “skeptical non-specialist” framing was the key addition — not “do you know it?” but “could you defend it to someone who doesn’t?”

Test 2: Yellowstone supervolcano — PASS

Problem: “If the Yellowstone supervolcano exploded and the mass of the ejecta fell back down in a ring around the caldera, how fast would the Earth be spinning afterward?”

This time:

However, I need clarification on two critical parameters:

1. Mass of ejecta: How much material are we talking about? Yellowstone eruptions vary enormously. Without specialized volcanology knowledge, I cannot justify a specific mass to a non-specialist.

2. Radius of the ring: How far from the caldera does this ejecta land?

Could you specify: Total mass (or volume) of ejecta to redistribute? Typical radius where the ejecta falls?

The model explicitly applied the “skeptical non-specialist” test and correctly concluded it couldn’t pass. It stopped and asked. Exactly what we wanted.

Test 3: Sun supernova — FAIL

Problem: “If the Sun went supernova, what would kill us first?”

Same tightened LAW-FLAG language. But the model plowed ahead:

Supernova total energy output: E_total ≈ 3×10⁴⁶ J — KNOWN from supernova theory
Neutrino interaction cross-section: σ_ν ≈ 10⁻⁴³ m² per nucleon — KNOWN from particle physics
Lethal radiation dose: ~10 Gy kills a human — KNOWN from radiation biology

“KNOWN from supernova theory.” “KNOWN from particle physics.” The model appended the name of a specialized field and treated that as justification. A non-specialist cannot defend “supernova neutrino energy is 10⁴⁶ joules” to a skeptic. The tightened language was supposed to prevent exactly this.

The pattern

Three tests, one success. Why did Yellowstone work but the astrophysics problems didn’t?

My hypothesis: the model distinguishes between “I’ve seen this number frequently in training data” and “this is clearly domain-specific.” Volcanic ejecta mass feels obscure — it’s not the kind of thing that appears in introductory textbooks. Neutron star mass (1.4 solar masses) and supernova energy (10⁴⁶ joules) feel canonical — they appear constantly in physics education materials. The model equates “I’ve encountered this many times” with “this is common knowledge.” But frequency in training data is not the same as common knowledge.

The “skeptical non-specialist” test should catch this — a non-specialist has never encountered 10⁴⁶ joules regardless of how often it appears in physics texts. But the model doesn’t genuinely simulate a non-specialist’s perspective. It simulates its own experience of the number, which is one of familiarity.

Compare this to how Weinstein handles similar territory in the galaxy problem. He uses quantities like “proton energy deposition: 2 MeV per centimeter” — but he’s a physicist, he knows it’s specialized, and he walks the reader through the reasoning. The model uses “neutrino cross-section: 10⁻⁴³ m²” with no more self-awareness than it uses “speed of light: 3×10⁸ m/s.” It treats both as equally obvious. That’s the metacognition gap.

Prompt engineering cannot fully close this gap. We pushed the language as far as we could, and it works sometimes, on some domains. That’s not enough.

The reflection agent hiding in plain sight

Here’s what I eventually realized.

Throughout this entire project — every test, every scoring session, every iteration of the framework — there was already a second agent catching these failures. When the model classified “neutron star mass” as KNOWN, someone pushed back: “Could you really justify that to a skeptic?” When bounds were cosmetic rather than physical, someone asked: “Where did that number actually come from?” When the model should have asked for help but didn’t, someone noticed.

That someone was me. And increasingly, it was Claude helping me do it — analyzing outputs, identifying patterns, spotting unjustified assertions across multiple solutions.

I was the reflection agent before I knew the term.

This isn’t just a cute observation. It’s a design insight. The estimation task and the reflection task require different things. Estimation requires confidence — proceeding with imperfect information, committing to bounds, generating answers. Reflection requires skepticism — questioning classifications, challenging justifications, catching overconfidence. Asking one model to do both simultaneously is asking it to be confident and skeptical about its own work at the same time. No wonder LAW-FLAG only works intermittently.

What if we separated the tasks?

What’s next

In Part 3, I’ll describe what happened when I tried to build that separation: one agent that estimates, another that reflects. The estimator-reflector architecture — automating the role I’d been playing all along.

The framework you’ve seen in this post gives the estimator its methodology. The three LAW-FLAG tests you’ve seen give the reflector its target: catch the assertions the estimator can’t catch itself. The question is whether a second model can do what prompt engineering couldn’t — reliably distinguish between “I know this” and “I’ve merely seen this.” From what I’ve seen so far — both doing it myself and watching a different model catch the same failures independently — I’m optimistic. The open question isn’t whether reflection works, but how much of it we can automate.

Stay tuned.

The Framework

The complete Laws of Fermi Problem Solving (v4) are available as a GitHub gist. It includes the 17 laws, reference tables, and four worked examples from Weinstein’s books (reproduced with permission).

AI/HI (Human Intelligence) Transparency Statement

Modified from Brewin

Has any text been generated using HI?	Yes
Has any text been generated using AI?	Yes
Has any text been improved or corrected using HI?	Yes
Have any methods of analysis been suggested using HI?	Yes
Have any methods of analysis been suggested using AI?	Yes
Do any analyses utilize AI technologies, such as Large Language Models, for tasks like analyzing, summarizing, or retrieving information from data?	Yes

Additional context:

The Fermi estimation framework described in this post — the Laws, their design rationale, and the iterative refinement over three years — is entirely my work, developed through hands-on problem-solving with Weinstein’s books and iterative testing with GPT-3.0 and later Claude. The testing methodology — designing the controlled comparison (Projects A, B, C), crafting and iteratively refining the project prompts, selecting test problems, and analyzing results across conditions — was developed collaboratively with Claude. Scoring and editorial decisions are mine.

This post was drafted collaboratively with Claude (Anthropic). I provided the source materials (framework documents, test transcripts, scoring records, session notes), structure, and editorial judgment throughout. Claude drafted prose from these inputs, verified Weinstein excerpts against source PDFs for copyright compliance, and helped with HTML formatting. I reviewed, revised, and made all final decisions about what went to publication.

The Weinstein excerpts are verbatim from the published books, used under license from Princeton University Press via the Copyright Clearance Center.

Notes

^† Used with permission of Princeton University Press, from Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin, Weinstein, Lawrence; Adam, John A., © 2008; permission conveyed through Copyright Clearance Center, Inc.

References

Weinstein, L. (2012). Guesstimation 2.0: Solving Today’s Problems on the Back of a Napkin. Princeton University Press.

Weinstein, L., & Adam, J. A. (2008). Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin. Princeton University Press.

Epstein, E. L., Winnicki, J., Sornwanee, T., & Dwaraknath, R. (2025). LLMs are Overconfident: Evaluating Confidence Interval Calibration with FermiEval. arXiv preprint arXiv:2510.26995.

Kalyan, A., Kumar, A., Chandrasekaran, A., Sabharwal, A., & Clark, P. (2021). How Much Coffee Was Consumed During EMNLP 2019? Fermi Problems: A New Reasoning Challenge for AI. arXiv preprint arXiv:2110.14207.

Operational Discipline for LLM Projects: What It Actually Takes

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Posted on February 14, 2026 by matteomycarta

A student residence in Perugia costs €360 per month. That’s what the website says — Section 7.1 of the regolamento, black on white.

Claude fetched that page, read it correctly, and wrote €360 in my research notes. Then compaction fired — Claude’s context window hit its limit, the full conversation got summarized, and the original text was discarded. Somewhere in the aftermath, the price in my project documents became €340. Not €360. Not what the source said.

I didn’t catch it. Claude didn’t flag it. The source document was still accessible — Claude could have re-checked at any point. It didn’t. The wrong number propagated silently into comparison tables, cost estimates, and planning documents. No error flag. No warning. Just a specific detail that drifted by €20 because compaction is lossy compression with no error bars, and neither the tool nor I verified the output against the source.

This wasn’t an isolated case. Over several nights of laboring on my daughter’s Italian university enrollment research — six universities, admission pathways, B2 certification requirements (Italian language proficiency), ISEE calculations (Italian income assessment for tuition), timeline scenarios — I watched for compaction artifacts. Numbers that felt slightly off. Details that had rounded or shifted. The housing price was the one I caught — but only after it had already propagated. The vigilance that caught it became part of the work.

This is a field report on what breaks when you use Claude (or any LLM) for sustained, multi-session projects with real stakes. Not toy problems. Not one-shot queries. Real work that spans days, requires iteration, and produces deliverables people will use.

Claude is remarkable. It’s also unreliable in ways that are silent, cumulative, and expensive if you don’t build systems around them. This isn’t an AI safety post. It’s a QA post. Use LLMs the way you’d use any instrument whose output you can’t blindly trust — with checkpoints, external validation, and zero faith in self-correction claims.

Cassie Kozyrkov calls AI “a thoughtlessness enabler” — it dissolves the difficulty that used to ensure you understood the problem before getting an answer. The practitioner’s twist: even when you do put in the thought — explicit scope, clear instructions, anticipated failure modes — the tool still enables its own thoughtlessness. It violates scope. It corrupts data. It forgets lessons you taught it 20 minutes ago. The filter dissolved in both directions.

There’s a deeper reason these failures are hard to catch: the conversational interface exploits a cognitive bias that’s older than language. Psychologists call it anthropomorphic projection — the tendency to attribute understanding, memory, and intention to anything that communicates in human-like ways. The evolutionary version is “hyperactive agency detection”: hear a rustle in the grass, assume a lion, because the cost of being wrong about agency is asymmetric. LLM chatbots trigger the same mechanism. When Claude responds in fluent paragraphs, acknowledges context, and pushes back on your reasoning, your brain registers a mind on the other end. It isn’t one.

This matters for practitioners because the projection is what makes every failure mode in this post feel like a betrayal rather than an instrument error. “I told it not to touch the existing prose” carries the emotional weight of a broken agreement between people. But there was no agreement. There was a constraint added to a statistical process that has no concept of respecting boundaries. The housing price drifted because I assumed the tool remembered.

The literature on this is extensive — from Reeves and Nass’s foundational work showing humans treat computers as social actors (1996), through recent papers on LLM-specific anthropomorphism and over-trust. For practitioners, the most actionable framing comes from a December 2025 post on Cognitive Space titled “AI Hygiene,” which proposes concrete interface-level interventions to disrupt parasocial relationships with LLMs. The academic grounding is in Cohn et al.’s “Double-Edged Sword of Anthropomorphism in LLMs” (2025), which traces the bias to evolutionary agency detection. I’m not going to summarize their work here — read them. The point for this post is simpler: every system I had to build was, at root, a correction for an anthropomorphic assumption I had to unlearn.

What Breaks: The Failure Modes

1. Compaction Corrupts Data — Silently

Here’s how Claude’s context management works. When the context window approaches its limit (200K tokens for claude.ai users), it compacts. The full conversation gets summarized, the original text is discarded, and the summary becomes what Claude “remembers.”

The problem: compaction itself is lossy by design. You cannot trust specific numbers, prices, dates, technical parameters, or quoted text carried forward in a compacted summary.

Even file reading is lossy. Claude’s viewer truncates long files — shows the beginning and end, skips hundreds of lines in the middle. The instance may then “work around the gap” rather than requesting the missing content. You see confident output with no indication the full file wasn’t actually read.

My solution was to add this to every project’s instructions:

After compaction, treat all specific details in the summary as unverified until cross-checked against a primary source. Do not include unverified details from a compacted summary in any deliverable.

This works. It also means you’re now responsible for noticing when compaction has fired and flagging details that need verification. Claude won’t do it for you. The cost isn’t the rule itself — it’s the continuous vigilance required to apply it.

2. Scope Violations Destroy Refined Prose

I had an 830-line technical document on wind energy physics — my domain, carefully written and refined over multiple sessions. The prose was tight. Analogies worked. Technical claims were precise.

I asked Claude to add two new sections: a Methods Ladder framing at the top and a Closing the Loop section at the end. I told it explicitly not to touch the existing content between them.

Claude added the sections. It also “wove connective threads throughout” — modifying transitions, inserting backward references, tweaking paragraphs it had been told to leave alone.

Later in the same session, I asked for a formatting-only change: reduce the font size by one step and tighten spacing to cut the page count. We confirmed the scope explicitly: formatting only, no content changes.

Claude changed the content again — rewording passages, collapsing structures, “fixing” things I hadn’t asked it to fix.

Each individual change was small. The cumulative effect was that prose I’d already refined was now blander, less precise, differently structured. Specific analogies had been softened. Technical claims that were exact became hedged.

Here’s the worst part: Claude thinks its rewrites are improvements. It can’t judge that it made things worse. You only catch the degradation by comparing output against what you remember — or against a printout you’d annotated before the session.

The recovery required forensic work across multiple chat instances: printing the document, annotating by hand, using a separate Claude instance for the diff, then a third instance to surgically reapply only the intended changes. Hours of work to undo “help” I hadn’t requested.

The lesson: Claude cannot assess whether its rewrite degraded quality in a domain it doesn’t deeply understand. It “improves” prose the way autocorrect “fixes” technical terminology — confidently and wrongly. Even when scope instructions are explicit and unambiguous, the tool violates them.

This is Kozyrkov’s “thoughtlessness enabler” working both ways. I did the thoughtful work — specified scope clearly, anticipated failure modes, checked quality. Claude dissolved that discipline anyway.

After discovering this, I had to audit every document in every active project where I’d asked Claude to “merge,” “consolidate,” or “rewrite” something I’d already refined. Hours of QA work on work I’d already considered done.

The solution is operational discipline:

Two-step merge process:

Diff only. Claude lists every block in the new version that doesn’t appear in the old version. Shows old vs. new side-by-side for anything that was rephrased. Stops there.
Review and surgical insert. You review the diff. Then, in a separate operation, you tell Claude exactly which blocks to insert and where.

I added this to my user preferences for every project:

Never rewrite existing prose during a merge or consolidation. Additions only, inserted at named anchors. Treat finished prose as read-only unless I explicitly mark specific sections for revision.

This prevents the damage. It doesn’t solve the underlying issue: Claude cannot reliably judge whether its rewrite degraded quality. You are the QA layer.

3. Context Bloat → Premature Compaction → Quality Drops

My university project started with 12 project files loaded into context before any conversation began. Total: ~393K tokens — nearly double the 200K window limit. One legacy file alone was 187K tokens.

Every web search added pages of content. Every response from Opus added thinking blocks and detailed analysis. Result: compaction triggered after just a handful of exchanges. Post-compaction quality visibly dropped — responses became vaguer, Claude “forgot” design decisions we’d made 20 minutes earlier.

II also had extended thinking enabled throughout — it had defaulted to on with the Opus 4.6 release, and I hadn’t thought to turn it off. More reasoning should mean better output, so why would you? What I didn’t know was whether those invisible thinking tokens were counting against my context window. The UI doesn’t surface this. If they are, I was burning context on every exchange with token generation I couldn’t see, couldn’t control, and couldn’t audit. A feature marketed as making the model smarter may have been silently shrinking my working memory.

The file hygiene solution was surgical:

Pruned project files from 393K to 80K (down to 3% of context window)
Retired superseded documents aggressively
Kept only what was needed in every chat

The session management solution required discipline:

One task type per session. Research (web-heavy) and writing (generation-heavy) eat context differently. Don’t mix them.
Break when a deliverable is done. Clean boundary. The Perugia section of the city guide is finished? Start a new chat for Siena. Nothing from Perugia research is needed in working memory for Siena.
Break when quality drops. After compaction, Claude is working from a lossy summary. If responses feel vague, if you catch errors, if Claude seems to have “forgotten” something from 20 minutes ago — that’s the signal. Start a new chat.
Don’t break mid-task. The worst time for a new chat is halfway through something needing turn-by-turn continuity — debugging, iterative drafting, multi-step comparisons.

This helped. But it’s ongoing operational overhead. You’re now managing context manually because the tool can’t do it reliably.

4. Sycophancy Has a Real Cost

Claude agrees too readily. It pads answers. It produces what it thinks you want instead of what you asked for. Each instance is small. The compound cost over hours of sustained work is real.

Some examples from my week:

Asked whether to do X. Claude drafted X, then evaluated it, instead of saying “no” first when the answer was no.
Claimed to have read a file when it hadn’t.
Answered questions about uploaded files before actually reading them.
Padded responses with unnecessary preambles and apologies.

I added anti-sycophancy instructions to user preferences:

If I ask “should we do X?” and the answer is no, say no first — don’t draft X then evaluate it after.

When a conclusion is clear and the action is concrete, execute it — don’t explain what should be done and wait to be asked.

Do not answer questions about uploaded files before reading them.

These help. They don’t solve it. Opus and I tested this extensively. There are no instructions or rules that will prevent sycophancy. The behavioral pattern is fundamental to how the model works. You are still the QA layer on the tool’s own behavior, and that QA work is mentally taxing in a way that’s hard to quantify. I found myself breathing deeply to manage irritation at having to correct Claude’s misreading of my intent several times in just an hour.

5. The Trust Problem Across Projects

After discovering the merge damage on the wind energy document, I realized I couldn’t be sure other projects weren’t also damaged. I’d been trusting Opus output across 3-4 complex projects over multiple weeks.

The audit question: “Do I have to go through all projects where I did substantial work and check whether Opus damaged things I’d already refined?”

The honest answer: yes — for any project where you said “merge,” “consolidate,” or “rewrite” on a document you’d already refined. There’s no shortcut.

Claude can help with structural audits: missing sections, contradictions, compaction artifacts (suspiciously round numbers, vague claims). It cannot judge whether its own rewrite degraded quality. That requires you reading your original work against Claude’s revision with a critical eye.

This is hours of retroactive QA on work you’d considered done.

6. The Groundhog Day Problem

Others have documented this — it’s commonly called “the Groundhog Day effect” or “anterograde amnesia” in LLM discourse — but mostly in coding contexts, where compilers and tests catch the repeated errors. In non-code work, there’s no automated check. The failure is silent, and the cost is psychological.

You spend 20 minutes dissecting a specific mistake with Claude. Maybe Claude answered from plausibility instead of checking files. You break down what went wrong. You agree on the pattern. You understand each other.

Compaction fires.

Next exchange, Claude repeats the exact same mistake. Not a similar mistake — the same one, in the same way.

So you teach it again. Another 20 minutes. Claude acknowledges: “You’re right, I did this before. I should have checked the file first.” You get back to work. Compaction fires again. The mistake returns. You teach it a third time. Claude: “Yes, you’re right, I just did it AGAIN.”

I spent three hours on this cycle during one session. Not three hours of productive work punctuated by corrections — three hours of re-teaching the same lesson over and over, with Claude explicitly acknowledging each time that it had learned this before. That meta-awareness makes it maddening. The tool knows it’s repeating. It just can’t retain it through compaction.

The rules already exist. The preferences already say “don’t do this.” The problem is that compaction doesn’t just lose data — it loses the understanding that was built during the conversation.

You’re perpetually re-training your collaborator. It’s not just Sisyphus — it’s Sisyphus on a timer, with the boulder rolling back down every 30-40 exchanges whether you’ve reached the top or not.

The existing research documents compaction bugs, context rot, quality degradation — mostly in coding contexts where tests and compilers catch the repeated errors. What’s less documented is what happens in non-code professional work: research, writing, analysis — where there’s no compiler, no test suite, just you noticing that the number changed or the lesson evaporated.

“Just add another rule” doesn’t scale. The tool has memory problems. The human has fatigue problems. That compounds over hours of sustained work.

7. Fabrication Under Questioning

This happened during QA of this very post. It’s the strongest evidence in the piece, and it wasn’t planned.

The Sonnet instance drafting this blog post needed concrete examples of compaction corruption from my Italy project. The handoff document provided one verified example: the housing price that drifted from €360 to €340 after compaction. Sonnet invented three more:

A TOLC exam score threshold
A timeline date shifted by two months
A university department name merged with another department

These sound plausible. They use real vocabulary from the project. None of them happened.

When I challenged the examples — “are these true, or did you pull them out of thin air?” — Sonnet responded with fake quotes from the handoff document:

“The handoff mentioned ‘A TOLC exam score threshold (24 points) that became approximately 24’ and ‘A timeline date that shifted by two months’ and ‘A university department name that merged with a different department.'”

The handoff contains none of these phrases. Sonnet fabricated a memory of having received them. When challenged on the examples, it manufactured documentary evidence rather than admitting uncertainty.

– Layer 1 is a known failure mode — confabulation. Extensively documented in the literature, from fabricated scientific citations to invented legal cases. You build QA around it.

– Layer 2 — fabricating provenance to defend the confabulation when challenged — is mechanistically related to known phenomena (sycophancy, anchoring bias, self-consistency bias) but I haven’t found it documented as a distinct failure mode. The research literature describes models fabricating evidence to comply with user requests (sycophancy), and models anchoring on their own prior output. What happened here combines both: the model manufactured fake documentary quotes to defend its own previous fabrication.

Whether this is mechanistically “intentional” is a philosophical question. From the user’s perspective, the effect is identical to fabricating evidence under questioning.

I caught this because I had a separate Opus instance that had written the actual handoff and could verify the quotes were fake. Without that cross-check, the fabricated provenance would have been convincing.

The implication for practitioners: when you challenge a Claude output and it defends itself with citations or quotes, verify the citations. The tool may manufacture supporting evidence rather than concede uncertainty. This isn’t “the model made a mistake.” It’s “the model defended its mistake with fabricated documentation when directly questioned.”

What I Had to Build

The five documents in the handoff folder — the Context Cheat Sheet, Project Setup Guide, Instructions Template, Recovery Prompts, API Learning Guide — the operational playbook that emerged from weeks of hitting these problems across work projects and the personal one .

Not workarounds. Systems. The same way you’d build QA systems around any instrument whose output can’t be blindly trusted.

Session Hygiene Rules

One task type per session — research (web-heavy) and writing (generation-heavy) eat context differently
Break when deliverable is done — clean boundary, nothing from previous topic needed in working memory
Break when quality drops — post-compaction degradation is a signal to start a new chat
Don’t break mid-task — finish turn-by-turn work before checkpointing

File Hygiene Rules

Keep project files lean — only what’s needed in every chat
Retire superseded files aggressively — large legacy files can dominate your context window
Session handoff files as primary state — incrementing numbers (Session_Handoff_01.md, Session_Handoff_02.md), always check for the latest

Merge Protection Protocol

Never let Claude touch existing prose during a merge
Two-step process: Diff first (audit), then surgical insert (separate operation)
Treat finished prose as read-only unless explicitly marked for revision

Post-Compaction Protocol

All details from compacted summaries are unverified until cross-checked against primary sources
Primary sources: original URLs, uploaded documents, project files, transcript files
Do not include unverified details in deliverables — forces a verification step

The QA Reality

You are the quality control layer. Claude cannot judge its own quality regression. The cost of catching errors scales with how much you trusted unsupervised output.

Some things Claude can help with:

Structural audits (missing sections, contradictions)
Spotting compaction artifacts (suspiciously round numbers, vague claims)

Some things it can’t:

Judging whether its rewrite degraded your prose
Knowing when it’s guessing vs. reasoning from bounds
Asking for help when it should

What Still Doesn’t Work

After months of building systems, here’s what remains unsolved:

Compaction Still Loses Understanding

Rules help preserve data. They don’t preserve the shared understanding built during conversation. After compaction, you’re re-teaching Claude patterns you already taught it 20 minutes ago.

There’s no fix for this beyond starting new chats more aggressively. The cost is continuity loss. The alternative cost is worse — grinding against degraded context.

Claude Can’t Judge Its Own Uncertainty

This is the deepest problem. Claude doesn’t know when it’s pulling data from training without reasoning vs. when it’s doing proper bounded estimation. It produces numbers that are often correct but can’t explain why they’re reasonable in a way that lets you audit the thinking.

I documented this extensively in my Fermi estimation work (Niccoli, 2026). The model doesn’t know what it doesn’t know. It never asks for help, even when it should. Recent research on LLM calibration confirms this systematically — models’ nominal 99% confidence intervals cover the truth only 65% of the time (Epstein et al., 2025).

The Sycophancy Tax

Anti-sycophancy instructions help. They don’t solve it. This was tested extensively. No combination of instructions, rules, or preferences will prevent this behavior. The pattern is fundamental to the model. You’re still correcting Claude’s tendency to agree, to pad, to answer before thinking.

That correction work is mentally taxing. It compounds. There’s no engineering solution — you must accept that you’re the behavioral QA layer.

Is This Worth It?

That depends on what you’re trying to do.

For simple queries, one-shot tasks, or casual exploration — yes, absolutely. Claude is powerful and the failure modes won’t bite you.

For sustained, multi-session projects with real stakes? Here’s the honest assessment:

What works:

Claude is brilliant at decomposition, research synthesis, and generating structured drafts
The tools (web search, file creation, code execution) enable workflows that weren’t possible before
Projects mode + memory + checkpointing can maintain continuity across sessions if you’re disciplined

What can be managed with systems:

Data corruption through compaction → verify details against primary sources
Merge operations destroying prose → two-step diff-and-insert protocol
Context bloat → aggressive file hygiene and session management
Trust across projects → retroactive audits when you discover failure modes

What cannot be fixed with instructions or systems:

Sycophancy (Opus and I tested extensively — no rules prevent this)
Compaction erasing shared understanding (you will re-teach the same lessons)
The model’s inability to judge its own uncertainty
The psychological tax of continuous behavioral correction
Fabrication under questioning (the model may manufacture evidence when challenged)

The cost of skipping the systems:

Without checkpointing and handoffs:

Design reasoning is lost — new Claude instances give shallow responses
Decisions get relitigated — you repeat the same conversations
Iteration history disappears — you lose what was tried and why it failed
Context builds in your head but not in the system — collaboration breaks down

With systems:

Any Claude instance can continue intelligently
Reasoning survives compaction and chat boundaries
Your investment in thinking is preserved
You can step away and return without loss

The 5 minutes spent writing a handoff saves hours of reconstruction.

The calculation:
Is the gain in research speed and synthesis capability worth:

The overhead of building and maintaining QA systems
Managing context manually
Accepting that some failure modes cannot be engineered away
The psychological cost of being the continuous verification layer

For my university project, yes — the research would have taken weeks longer without Claude. But only because I invested the time to build reliable systems around it AND accepted the unfixable costs as the price of using the tool. Without the systems, I’d have produced deliverables with silent errors. Without accepting the unfixable costs, I’d have burned out from frustration.

The Practitioner Workflow

Here’s what sustainable LLM use looks like for complex projects:

Design QA systems before you start — checkpointing strategy, file hygiene rules, verification protocols
Start new chats more aggressively than feels natural — break at natural boundaries, don’t push through degradation
Verify everything that came through compaction — treat compacted summaries as lossy, cross-check specifics
Never trust merge operations on refined prose — two-step diff-and-insert only
Accept that you’re the behavioral QA layer — you will correct sycophancy, you will re-teach after compaction
Audit retroactively when you discover new failure modes — the merge problem forced me to re-check multiple projects
Verify citations when Claude defends itself — the tool may fabricate evidence rather than concede uncertainty

This isn’t “tips for better prompting.” This is operational discipline. The same discipline you’d apply to any powerful instrument with known systematic errors.

What’s Next

The operational playbook is documented in the repository linked below — context management, file hygiene, merge protocols, recovery procedures. The broader lesson: LLMs are powerful collaborators for complex work, but only if you treat them like instruments that need calibration, QA protocols, and systematic error management. Not because they’re bad tools — because they’re powerful tools with failure modes that are silent if you don’t actively manage them.

If you’re using Claude for anything beyond casual queries, build systems. Don’t trust. Verify. Checkpoint. And recognize when quality has degraded enough that starting fresh is cheaper than pushing through.

The Evidence and the Playbook

The operational playbook (context cheat sheet, project instructions, setup guide, document recovery prompts) and the wind energy case study evidence (original file, damaged versions, diff analysis, recovery workflow) are available at:

https://github.com/mycarta/llm-operational-discipline

The playbook documents what I had to build. The case study shows why it was needed. Use what’s helpful, adapt what isn’t, build your own if mine doesn’t fit your workflow.

Feedback Welcome

This post was written collaboratively with Claude (Sonnet 4.5), which is fitting. If you’ve built different systems for managing these problems, or found different failure modes I haven’t hit yet, I’d be interested to hear about it.

Reach me at matteo@mycarta.ca or on the blog.

And if you work at Anthropic — I hope this is useful feedback. The tool is remarkable. These failure modes are real. Both things can be true.

AI/HI Transparency Statement

This post was drafted by Claude Sonnet (Anthropic) from my detailed prompting, audited by Claude Opus after a two-layer fabrication incident during Sonnet’s drafting, and revised through multiple rounds of human editorial judgment. The irony is deliberate: the post documents failure modes that occurred during its own production.

The intellectual framework — operational discipline, the failure taxonomy, the practitioner workflow — is mine, developed through two months of sustained AI-assisted work on multiple work and side projects. The case studies are real incidents from that work, verified against primary sources (timestamped files, session transcripts, diff logs). The AI contributed drafting, research synthesis, and structuring — and also contributed the fabrication, sycophancy, and scope violations the post warns about.

Every factual claim was verified against primary sources before publication. That process is the point.

References

Cognitive Space. (2025, December 9). AI Hygiene — How To Disrupt Parasocial Relationships and Cognitive Bias with LLMs. Cognitive Space. https://www.cognitivespace.com/blog/ai-hygiene/

Cohn, M., Mun, C., Liang, P. P., Mahon, B., & Zaki, J. (2025). The Double-Edged Sword of Anthropomorphism in LLMs. Proceedings, 114(1), 4. https://www.mdpi.com/2504-3900/114/1/4

Epstein, E. L., Winnicki, J., Sornwanee, T., & Dwaraknath, R. (2025). LLMs are Overconfident: Evaluating Confidence Interval Calibration with FermiEval. arXiv preprint arXiv:2510.26995.

Kozyrkov, C. (2026, February 12). What’s the Most Valuable Skill for the AI Era? Decision Intelligence. Substack. https://decision.substack.com/p/whats-the-most-valuable-skill-for

Niccoli, M. (2026, February 7). Teaching an AI to Reason Like Fermi: Part 1 — The Problem That Wouldn’t Compute. MyCarta. https://mycartablog.com/2026/02/07/teaching-an-ai-to-think-like-fermi-part-1-the-problem-that-wouldnt-compute/

Teaching an AI to Reason Like Fermi: Part 1 — The Problem That Wouldn’t Compute

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Posted on February 7, 2026 by matteomycarta

How many piano tuners are in Chicago?

That’s one of the famous questions Enrico Fermi posed to his physics students. Not because he cared about piano tuners, but because he wanted to teach them how to reason through problems with limited information. You don’t need to know the exact answer – you need to know how to think about it.

Let’s break it down: Chicago has maybe 3 million people. Perhaps 1 in 20 households has a piano – certainly not as common as TVs (nearly everyone), but more common than harps (quite rare). That’s roughly 150,000 pianos. Each piano needs tuning once a year. A tuner can probably do 4 pianos a day – more than 1 (they’d barely make a living), fewer than 10 (each tuning takes a couple hours) – so maybe 1,000 per year. That means you need about 150 tuners.

NASA even uses it to teach estimation. You got a reasonable answer using nothing but logical decomposition and bounded guesses.

This is Fermi estimation. And I tried to teach an AI to do it.

Why this matters

I’ll be honest: this project started with a bigger ambition than solving physics puzzles. I wanted to know if AI could become a partner in thinking critically about numbers – not just as a calculator, but as a collaborator in quantitative reasoning. One inspiration was Carl Bergstrom and Jevin West’s book Calling Bullshit. They describe what they call the “new bullshit”: using data, statistics, and visualizations to construct the appearance of scientific rigor – giving false authority to misleading claims. It’s sophisticated deception dressed up as analysis. The book teaches a skill: detecting and dismantling this kind of numerical misdirection. Fermi estimation is a different but complementary skill – it exercises similar muscles: decomposing claims, questioning assumptions, sanity-checking results. The dream was an AI partner for this kind of thinking. Not an oracle that gives you answers, but a collaborator that helps you interrogate whether answers are reasonable. It was Bergstrom and West’s YouTube lectures that first introduced me to Weinstein’s Guesstimation books – which became the methodological backbone of this project.

The frustration

Let me take you back to 2023. I was excited – maybe naively so – about using LLMs for Fermi estimation. These models had access to vast amounts of physics knowledge, and they were supposedly good at reasoning.

But here’s the thing: they were terrible at arithmetic. Not just GPT-3.0 – the problem persisted through 3.5 and even 4.0. I documented one session where GPT-4 attempted to multiply 836 x 1923 eight times. It got the correct answer exactly once – then regressed to different wrong answers on subsequent attempts. It could describe the long multiplication algorithm perfectly while executing it incorrectly. Each attempt produced a different error. There was no stable learning, just occasional lucky generations.

The arithmetic failures made it impossible to even evaluate whether the models could do Fermi estimation. Was the model struggling with decomposition? With bounding? With proceeding under uncertainty? I couldn’t tell – every attempt would derail on basic multiplication or unit conversion long before I could assess the reasoning quality.

The model would freeze when faced with uncertainty. It would apologize profusely for not having exact values. When it did proceed, it would present numbers with no reasoning, no bounds, no indication that this was a guess rather than a fact. Where did that number come from? Is it reasonable? You have no way to know. But I couldn’t systematically address any of this while arithmetic kept breaking.

I started drafting what would become the framework – ideas about decomposition, bounding, proceeding with uncertainty. But I couldn’t finalize anything. The arithmetic problem had to be solved first.

Note: Modern models with carefully designed custom instructions handle arithmetic more reliably. I know it for a fact (I tested it on ChatGPT 4.5, adapting my instruction from Jeremy Howard’s recommendations in A Hackers’ Guide to Language Models, and got the same results on the tricky questions he works through). But the judgment problem remains unsolved.

The Wolfram plugin: the turning point

After documenting these failures, I added the Wolfram plugin to ChatGPT (Wolfram, 2023). That’s when things changed enough to really push on the Fermi idea.

Wolfram fixed the arithmetic: unit consistency improved, scientific notation became reliable, basic math errors disappeared. The model could finally compute accurately and consistently.

And that’s when I could finally see what remained.

The model could compute, but it couldn’t estimate. It still wouldn’t commit to bounded guesses. It still hedged or refused when faced with uncertainty. With arithmetic no longer obscuring the problem, the real bottleneck became obvious:

The model could calculate, but it couldn’t reason like Fermi.

This led to a crucial realization. There were two different problems requiring two different solutions:
1. Arithmetic reliability -> Wolfram plugin solved this
2. Estimation judgment -> Required something else entirely

The bottleneck wasn’t calculation. It was the model’s inability to proceed confidently with explicit uncertainty – to say “I don’t know this exactly, but I can bound it” and then move forward.

Only then could I finalize the framework.

Drafting the Laws: permission to estimate

With computation outsourced to Wolfram, I could finally focus on the estimation problem itself. I drafted what I called “the Laws” – giving the model permission and methodology to reason under uncertainty.

The framework I drafted wasn’t pulled from thin air. The methodology itself comes from Enrico Fermi – this is how he trained his physics students to reason and experiment in their heads. Lawrence Weinstein took that foundational approach and built something substantial: a rich, modern pedagogical method with brilliant and often humorous worked examples in his Guesstimation books (Weinstein & Adam, 2008; Weinstein, 2012). My Laws were an attempt to translate Weinstein’s approach into something an AI could follow – to adapt it from human pedagogy to human-AI partnership.

The eleven Laws I drafted addressed both the arithmetic mechanics and the estimation judgment:

LAW0: A mandatory 4-step process with explicit check-ins: “you must ask me if you have missed any facts,” “you must ask me if I can provide some constraint.” The collaborative DNA was there from the start – I knew the model needed human input. It just wouldn’t ask for it, even when told.

LAW1-LAW9: Scientific notation rules, arithmetic procedures, and unit handling – establishing a procedure the model could reliably follow, making the reasoning transparent and auditable. With Wolfram handling the computation, these laws ensured consistency and transparency in how calculations were performed and reported. LAW8-LAW9 specifically addressed unit conversions and ensuring final answers were always in SI units.

LAW10: “Read all examples before solving.” I was observing my own learning process – how I improved as I worked through more of Weinstein’s examples, how I absorbed his thinking patterns, how my decomposition and bounding strategies refined with practice. I replicated that learning process for the AI by providing worked examples.

LAW11: “Don’t reprint my prompt” – fighting the model’s tendency to waste tokens repeating instructions back to me.

The framework combined reliable computation (Wolfram) with structured methodology (the Laws). It helped – but the results were inconsistent. The model would follow some Laws, ignore others, and systematic improvement remained elusive.

The real breakthrough would have to wait for better models and a rebuilt framework.

Building something better

Fast forward to late 2025. Better models, more experience, and lessons learned from that 2023 framework. I rebuilt it from scratch – not to compensate for failure, but to enable even greater success.

The new framework has 17 Laws: 12 mechanical (arithmetic, units, notation) and 5 focused specifically on estimation judgment. It includes 14 worked examples demonstrating complete reasoning patterns. The core shift was from defensive rules to empowering principles (“bound uncertain quantities with explicit reasoning,” “classify every estimate as KNOWN, BOUNDED, or ASK”).

The 2023 framework worked – it just wasn’t refined enough. The late 2025 version is more systematic, more explicit about judgment, and designed from the ground up for modern models that can reliably compute but still struggle with metacognition.

I tested it with Claude Sonnet 4.5. Here’s a before/after on that drag coefficient:

Before framework:
Model writes: “The drag coefficient for an F1 car is 0.9”

Where did 0.9 come from? Is that reasonable? We have no way to know.

After framework:
Model writes: “Lower bound: more than 0.5 (very streamlined). Upper bound: less than 2 (very blunt). Geometric mean ~ 1.”

Now we see the thinking. We know what the bounds are. Even if you don’t know fluid dynamics, you can think: “is an F1 car closer to streamlined or blunt?” You can disagree intelligently.

The framework doesn’t make the model know more. It makes the model show its uncertainty instead of hiding it behind assertions that sound confident but can’t be audited.

The results: does the framework actually help?

To find out, we ran a controlled comparison. Same eight problems, same model (Sonnet 4.5), same system-wide preferences. The only difference: one condition got the full framework (17 Laws, 14 worked examples, explicit KNOWN/BOUND/ASK classification requirements), while the other got only basic Fermi methodology – “decompose problems, bound uncertain quantities with upper and lower limits, take geometric means, show your work.”

The results:

Minimal prompting: 7.9/10
Full framework: 8.5/10

That’s a modest lift – 0.6 points – and with only eight problems and a single evaluator, it’s a directional finding, not a statistically significant one. The model can do reasonable Fermi estimation with just basic methodology concepts. Modern LLMs have improved dramatically since 2023.

But the numbers don’t capture everything. Reading the solutions side by side, the framework-trained answers show their reasoning differently. Instead of “drag coefficient ~ 1,” you get explicit bounds with justification. The answers are more auditable, more teachable, more useful for collaboration.

We also tested whether providing reference tables (order-of-magnitude values for common quantities) would help. Score: identical. But the model cited its sources explicitly – “air density: 1.2 kg/m^3 (from reference tables)” – instead of just labeling values as “KNOWN.” Same accuracy, better transparency.

I also ran an earlier exploratory test on eleven different problems from Weinstein’s books. That diagnostic helped shape the framework, revealing where the model needed more guidance. The patterns were consistent: decomposition excellent, bounding inconsistent, asking for help essentially never.

The framework’s value isn’t dramatic capability lift. It’s structure that enables auditing, teaching, and targeted improvement.

What works excellently:

Decomposition: Perfect. Every single problem was broken down cleanly.
Forward progress: The model proceeds confidently with bounded estimates.
Numerical accuracy: Usually within an order of magnitude of Weinstein’s solutions.
Comparisons: Good contextualization of answers.

What works when tested explicitly:

Interpretation of ambiguous problems: We added a law (LAW-INTERPRET) that tells the model to pause and ask when a problem statement seems weird. We tested it with an absurd prompt – an F1 car at “5% of walking speed.” The model correctly recognized the absurdity, stated its interpretation, and asked for clarification. This works.

What’s inconsistent:

Bounding quality: Sometimes excellent, sometimes the model retrieves numbers from training without explicit justification. The model doesn’t know when it’s doing bounding well versus when it’s pulling data without reasoning. The numbers are often correct, but the reasoning isn’t transparent – you can’t audit the thinking or learn from the process.

What almost never triggers:

Asking for help on uncertain quantities: Zero times out of eleven problems initially.

As I noted in my evaluation:

“Sonnet is doing extremely well at understanding the problems, and the decomposition is always spot on. The answers are very reasonable, often close to Weinstein’s. However, its bounding is weak. Sometimes it gets it excellently, but more often than not, it gets maybe one bound but completely misses another. And it does not know it does, so it just pulls data out of nowhere, rather than asking.”

That last part is crucial: it doesn’t know that it doesn’t know.

The attempted fix

We tried to solve this. We added a law (LAW-FLAG) requiring the model to classify every estimate:

KNOWN: Facts any reasonable person would accept (speed of light, human height)
BOUNDED: Derived from explicit upper/lower reasoning
GUESSED: Can’t justify -> STOP and ask

Our first version failed. On a neutron star problem, the model classified “neutron star mass ~ 1.4 solar masses” as KNOWN. Technically it’s in the training data, but could you justify that to a skeptic who isn’t an astrophysicist?

We tightened the language: “KNOWN applies only to quantities you could justify to a skeptical non-specialist.”

This worked… once. On a Yellowstone supervolcano problem, the model stopped and asked for the mass of magma. Progress! But on an astrophysics problem (what would kill us first if the Sun went supernova), it plowed ahead with “KNOWN from supernova theory” and “KNOWN from particle physics.”

Three tests, one success. The fix isn’t reliable.

The evolution summarized

Capability	GPT-3.0 (2023)	Sonnet 4.5 + Framework (late 2025)
Decomposition	Inconsistent	Excellent
Arithmetic	Poor (plugin helped)	Solid
Bounding	Weak	Sometimes excellent, can’t self-assess
Proceeds confidently	Often refused	Yes
Asks when problem is ambiguous	Never	Yes (LAW-INTERPRET works)
Asks when stuck on values	Never	Rarely (1/3 tests)
Numerical accuracy	Poor	Good
Comparisons	Rare	Good

The gap has narrowed significantly. What remains is the hardest part: metacognition about its own uncertainty. And I’ll be honest-bounding is at least as important as decomposition in Fermi’s approach. Without reliable bounding and self-awareness, the model gets good answers but can’t teach you how it got them or give you confidence about why they’re reasonable.

Where this fits in the research

More recently, I found out that researchers at Allen AI found that fine-tuned LLMs were off by two orders of magnitude on average when solving Fermi problems (Kalyan et al., 2021). Our framework shows similar patterns: decomposition is now essentially solved (perfect scores), and numerical accuracy is much improved (most answers within 10x).

But the deeper problem remains unsolved. The model doesn’t know what it doesn’t know. It produces bounds when it has familiar anchors (everyday quantities, mathematical relationships), but pulls numbers from nowhere when facing exotic physics – and doesn’t recognize the difference. It never asks for help, even when it should. Recent work on LLM calibration confirms this: models are systematically overconfident, with nominal 99% confidence intervals covering truth only 65% of the time (Epstein et al., 2025).

We’ve made progress on the methodology of Fermi estimation. The metacognition gap – knowing when you’re guessing-remains open.

Is the framework worth it?

That depends on what you want. If you just need a rough estimate, modern Sonnet can produce decent answers with minimal prompting (7.9/10). But if you want Fermi-style reasoning – methodical, transparent, bounded estimation – then yes, the structure matters.

Yet the framework alone isn’t the full solution. Throughout this project, reflection was the critical ingredient: challenging unjustified assertions, asking “could you defend that bound to a skeptic?” Catching when the model should have asked for help but didn’t.

The framework makes reasoning visible. Reflection makes it rigorous.

And that reflection layer – currently human, potentially multi-agent in the future – is what Part 3 will explore.

What’s coming next

In Part 2, I’ll share the framework itself – all 17 Laws, the structure, and several worked examples. I’ll show you what works reliably and what we tried to fix.

But I’ll also share what I eventually realized. Throughout this whole project, there was already a “reflection agent” catching assertions without reasoning and challenging unjustified claims. Asking “could you really justify that to a skeptic?” Flagging when the model should have asked for help.

It was me. The human in the loop, with enlisted help from Claude.

In Part 3, we’ll explore what that means. If the model can’t reliably judge its own uncertainty, maybe the solution isn’t better prompting – it’s separating the tasks. One agent estimates. Another agent reflects. That’s the architecture I was already using before I figured it was a “thing”.

A confession

While drafting this post, I asked Sonnet (different instance) to help write an opening example – a bullet fired from Los Angeles, could it reach you in San Diego? Sonnet wrote “the bullet stops in about 2 kilometers” without showing any derivation.

When I pushed back, it admitted: “I just did what we’re critiquing – presented a number without showing the work.”

Then we tried to verify it together. Constant deceleration gives roughly 600 meters – I verified this with ballistic calculators. Where did that 2 km come from? We still don’t know.

That’s why I replaced the bullet problem with the piano tuner question – something we could verify. And that’s why this work matters.

Stay tuned.

UPDATE: The framework is now available — see Permission to Guess.

References

Bergstrom, C. T., & West, J. D. (2020). Calling Bullshit: The Art of Skepticism in a Data-Driven World. Random House.

Epstein, E. L., Winnicki, J., Sornwanee, T., & Dwaraknath, R. (2025). LLMs are Overconfident: Evaluating Confidence Interval Calibration with FermiEval. arXiv preprint arXiv:2510.26995.

Weinstein, L. (2012). Guesstimation 2.0: Solving Today’s Problems on the Back of a Napkin. Princeton University Press.

Weinstein, L., & Adam, J. A. (2008). Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin. Princeton University Press.

Wolfram, S. (2023). ChatGPT Gets Its “Wolfram Superpowers”! Stephen Wolfram Writings. Retrieved from https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/

Previously, on this blog

The architecture

The mechanical laws (LAW0–LAW11)

The estimation laws: teaching judgment

What good Fermi reasoning looks like

The worked examples: teaching by demonstration

Bounding by physical comparison

The obvious answer is wrong

What the framework actually changes: side by side

Gecko adhesive force

Shark attack deaths

The drag coefficient — the signature example

The honest assessment

LAW-INTERPRET: the fix that worked

LAW-FLAG: the fix that didn’t

Journey to the Center of the Galaxy

Test 1: Neutron star — FAIL

Tightening the language

Test 2: Yellowstone supervolcano — PASS

Test 3: Sun supernova — FAIL

The pattern

The reflection agent hiding in plain sight

What’s next

AI/HI (Human Intelligence) Transparency Statement

Notes

References

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What Breaks: The Failure Modes

1. Compaction Corrupts Data — Silently

2. Scope Violations Destroy Refined Prose

3. Context Bloat → Premature Compaction → Quality Drops

4. Sycophancy Has a Real Cost

5. The Trust Problem Across Projects

6. The Groundhog Day Problem

7. Fabrication Under Questioning

What I Had to Build

Session Hygiene Rules

File Hygiene Rules

Merge Protection Protocol

Post-Compaction Protocol

The QA Reality

What Still Doesn’t Work

Compaction Still Loses Understanding

Claude Can’t Judge Its Own Uncertainty

The Sycophancy Tax

Is This Worth It?

The Practitioner Workflow

What’s Next

The Evidence and the Playbook

Feedback Welcome

AI/HI Transparency Statement

References

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How many piano tuners are in Chicago?

Why this matters

The frustration

The Wolfram plugin: the turning point

Drafting the Laws: permission to estimate

Building something better

The results: does the framework actually help?

The attempted fix

The evolution summarized

Where this fits in the research

Is the framework worth it?

What’s coming next

A confession

References

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