Haute Lumière
Commerce · V.02 · MMXXVI · daylight
Work as Metabolism
Volume V — Labour, Value, Flourishing
THE PLATE
THE LETTER
You have probably met the claim that shorter hours raise productivity, and you may have met it as a headline rather than as a finding. It is one of the few propositions in economics that arrives pre-agreed: everyone who wants it to be true already believes it, and everyone who does not, does not.
This chapter treats it as a measurement problem, because that is what it is.
There is a body a person brings to work and a rate at which that body converts food into effort, and both are known to two decimal places. There is a curve relating hours worked to output produced, and it was estimated properly, from factory records, more than a century ago, and re-estimated with modern econometrics in 2015. There is a large literature on shift length and error rates drawn from hospitals, roads and industrial injury statistics, and it is unusually clean, because injuries are counted whether or not anyone is running a trial.
And there is a set of recent four-day-week pilots which are none of those things, and whose reported results are far weaker than the reporting of them.
So: what does work cost a person, in units? What happens to output as hours rise, and where exactly does it turn? Why do output per hour and output per worker point in opposite directions, and which one is your firm actually paid for? What is recovery, physiologically, and what does it cost to skip? And what would it take to settle the four-day-week question, given that nothing run so far has settled it?
Those five questions have answers, and three of them have numbers. The fourth has a number with an interval so wide it is itself the finding. The fifth has a sample size, which we compute.
You will leave this chapter able to put a figure on something most organisations manage entirely by feel.
— The Editors
DISCOVERY
What is already working
The most striking thing about the evidence on working time is how old the good evidence is, and how well it has held.
The Health of Munition Workers Committee, Britain, 1915–1918. Facing a shell shortage, the British government put a committee of physiologists and industrial doctors inside munitions factories with permission to change the schedule and count the output. They were not testing a theory of flourishing. They were trying to produce more shells. What they found, repeatedly, was that the very long weeks then being worked — seventy hours and more — were producing less than shorter ones, and that introducing rest pauses raised weekly output rather than lowering it. The Committee's memoranda are among the most carefully instrumented industrial studies ever conducted, because output was physically countable, the workers were doing the same operation week after week, and the schedule changed while everything else held still.
John Pencavel returned to those records in 2015 and estimated the relationship properly. His result is the backbone of this chapter, and we do the arithmetic on it below.
Sidney Chapman had already predicted it. In 1909, in the Economic Journal, Chapman set out the theory of the hours of labour and drew the distinction that still does all the work: the length of day that maximises output today is longer than the length of day that maximises output sustained, because a day long enough to draw down the worker's capacity borrows from tomorrow's output at an interest rate nobody books. Chapman also predicted that competitive firms would not find the sustainable length on their own, because each firm captures today's output and shares tomorrow's cost with every other employer that person will ever have. He was describing a stock with a regeneration rate, thirty years before anyone used those words for it.
Kellogg, Battle Creek, from 1930. W. K. Kellogg moved the plant to four six-hour shifts a day, explicitly to employ more people through the Depression. The six-hour day survived at Kellogg, in departments and by vote, until 1985 — more than half a century, which makes it by a wide margin the longest-running shorter-hours experiment in industrial history. The company's own figures, reported in Benjamin Hunnicutt's history of it, claimed a substantial fall in accident rates and in unit overhead. Those are company figures rather than an evaluation, and we treat them as such. What is not in doubt is the duration, and duration is a kind of evidence: an arrangement that a firm keeps voluntarily for fifty-five years is not costing it what its critics said it would.
The hospitals, from 2004. Christopher Landrigan and colleagues put interns in intensive care units on two schedules — the traditional one with shifts beyond twenty-four hours, and an intervention schedule that eliminated them — and counted errors by direct observation rather than by self-report. On the traditional schedule the interns made 35.9 percent more serious medical errors, and 5.6 times more serious diagnostic errors. Steven Barger's companion work found that the odds of a motor vehicle crash driving home after an extended shift were 2.26 times those after a normal one, and that each extended shift worked in a month raised that month's commuting crash risk by 16.2 percent. This is the cleanest evidence in the whole field, and it is clean because the outcome was counted by an observer with a clipboard and by a police report, not by asking anybody how they felt.
Aviation, and then rail. Both industries now run formal Fatigue Risk Management Systems: duty limits set from circadian science rather than from custom, with the operator required to demonstrate an equivalent level of safety if it wants to depart from the prescriptive limits. The ICAO manual and the United States' prescriptive flight and duty limits are the mature examples. The point worth taking is structural rather than aeronautical: these are the only two sectors that have successfully priced fatigue, and they did it by making the limit a licence condition rather than a preference. Nobody negotiates it per firm, so nobody is competitively punished for observing it.
Svartedalens, Gothenburg, 2015–2017. Sixty-eight nurses at an elderly care home moved from eight-hour to six-hour shifts at full pay, with a comparable home as a control. The city evaluated it properly, published the costs, and did not extend it. We do that arithmetic below too, and it is the most useful single case in the chapter precisely because it is the one with a control group and an honest cost line.
Six cases, a century apart, and one pattern runs through all of them: wherever somebody has actually counted output and counted hours at the same time, the relationship has turned out to be curved. Not linear, not flat — curved, with a top. The disagreement in this field has never really been about whether the curve bends. It is about where.
THE ARITHMETIC
What works, what does not, and where the line sits
First, what a day of work costs a body.
The unit is the MET. One MET is resting metabolism: 3.5 millilitres of oxygen per kilogram per minute, which for arithmetic is one kilocalorie per kilogram per hour. For a seventy-kilogram person that is 1,680 kcal a day at complete rest.
Take the Compendium of Physical Activities values for two jobs and an eight-hour shift.
MET gross kcal net of resting
office / desk 1.5 840 280
construction 5.5 3,080 2,520
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ratio of net work energy 9.0 xNine to one. Now put the labourer's whole day together — eight hours at 5.5 MET, eight asleep at 0.95, eight at 1.6 for the commute and the evening:
64.4 MET-hours x 70 kg = 4,508 kcal/day
physical activity level = 2.68 x BMR
sustained human ceiling = 2.5 x BMR = 4,200 kcal/day
overshoot = 308 kcal/day (7.3 % over)That ceiling is not a guess. Caitlin Thurber and colleagues, measuring energy expenditure across events lasting from days to months — including athletes running across the United States — found that sustained total expenditure converges on a limit near 2.5 times basal metabolic rate, apparently set by the alimentary tract's capacity to absorb, not by the muscle's capacity to work. Beyond that, the body funds the difference from itself.
Check it by a second route that shares none of the same assumptions. Classical work physiology holds that an eight-hour day is sustainable at roughly a third of maximum oxygen uptake. For a median forty-year-old worker with a VO₂max of 40 mL/kg/min, a third is 13.2 mL/kg/min, which is 3.77 MET. Site work at 5.5 MET is 1.46 times that. Two instruments, different physiology, same verdict: a full eight hours of heavy manual work sits above what a body holds week after week, which is why those trades have always organised themselves around recovery, and why their workforces thin with age.
Second, the cut this chapter is built on.
Now do the same arithmetic for cognitive work. The brain takes about 20 percent of resting energy — 336 kcal a day, 14.0 kcal an hour — and that is its baseline, running whether you are solving a problem or watching the ceiling. Task-evoked increases above that baseline are small; an upper bound of five percent is generous.
hard thinking adds 0.05 x 14.0 = 0.70 kcal/h
a full 8-hour day of it = 5.60 kcal
as a share of the day's energy = 0.33 %
against the labourer's day = 450 x lessFive and a half kilocalories. Less than a teaspoon of milk.
And it is cognitive work — not the 2,520-kilocalorie day on the site — that most reliably ends in burnout, and whose practitioners most reliably describe themselves as exhausted. The entire industrial apparatus for measuring, pricing and rationing work was built around energy expenditure, in the century when energy expenditure was the binding constraint. It is no longer the binding constraint for most of the workforce, and the measuring apparatus has not moved. Work is not a fuel problem. It is a recovery problem, and recovery is the quantity nobody has on a dashboard.
Third, the productivity-of-hours curve, properly.
Pencavel's estimate on the munitions data gives the shape: below a threshold near 49 hours a week output is proportional to hours; above it output rises at a decreasing rate; and output at seventy hours differs little from output at fifty-six.
To locate a peak you need a functional form. Write log output as a quadratic in log hours, so the elasticity is ε(H) = b + 2c·ln H, and pin it at two points: elasticity one at 49 hours, and elasticity 0.35 at 56 hours.
2c = (0.35 - 1.00) / (ln 56 - ln 49) = -0.65 / 0.13353 = -4.8678
H* = 49 x exp(-1 / -4.8678) = 49 x 1.22806 = 60.2 h/weekTotal output peaks at about sixty hours a week. Output per hour peaks where the elasticity is still one — at or below 49. The gap between the two maxima is 11.2 hours a week, and that gap is where every argument about working time has actually been conducted for a hundred and fifteen years. Both sides have been right about their own number.
Now the confidence interval, which is the honest part. Move the two anchors across plausible ranges — threshold anywhere from 46 to 52 hours; elasticity at 56 hours anywhere from 0.15 to 0.55 — and the peak moves:
lowest 56.7 h/week
central 60.2 h/week
highest 71.2 h/week
width 14.5 hoursFourteen and a half hours wide, which is two working days. And note what kind of uncertainty that is: it is not sampling error. Pencavel's published standard errors are respectably narrow — around the elasticity, which is what the data identify. The peak is not what the data identify. Near the top the curve is almost flat, and a nearly flat curve has a barely located maximum. Anyone quoting a precise optimal working week is reporting their functional form, not their evidence.
Fourth, the honest negative, and it is a large one.
Output per worker is output per hour times hours per worker. In logs the identity is exact:
d ln(Y/N) = d ln(Y/H) + d ln(H/N)A five-day week to a four-day week is a 20 percent cut in hours. To hold output per worker constant, output per hour must rise by 1/(1 − 0.20) − 1 = 25.0 percent. That is not a small claim about anything.
And here is the difficulty. A typical trial cuts about 40 hours to 32 — both below the 49-hour threshold, in the region where the only well-identified productivity-of-hours curve we possess says the elasticity is one. In that region the curve predicts output per worker falls by exactly the hours: −20 percent. The trials report that it does not.
Three readings are live, and this chapter does not choose between them. The 1915 curve may not transfer from shell-turning to knowledge work. The measurement in the trials may simply be unable to see a twenty-percent fall. Or there is genuine organisational slack, in which case the finding is not that shorter hours create productivity but that the fifth day was already producing close to nothing, and firms have been buying it anyway.
Fifth, the trials, with their weaknesses stated.
The 2022 United Kingdom pilot ran 61 organisations and about 2,900 employees over six months. 56 of 61 (91.8 percent) continued; 18 (29.5 percent) made it permanent. Revenue was broadly flat — up 1.4 percent, weighted, self-reported. Resignations fell 57 percent and sick days 65 percent, also self-reported. It was self-selected, unblinded, six months long, had no control arm, and over-weighted small service firms.
Iceland's 2015–2019 trials covered about 2,500 workers, roughly 1 percent of the workforce — and were not a four-day week. They cut 40 hours to 36 or to 35 — a 10.0 or 12.5 percent reduction, and most roles had no output measure at all.
Microsoft Japan's August 2019 month reported sales per employee up 39.9 percent against the previous August: one month, one year-on-year comparison, no control, confounded with a simultaneous meeting-length policy.
And Svartedalens, the one with a control group. Sixty-eight nurses went from eight-hour to six-hour shifts — a 25 percent cut each. To hold coverage you need 68 / (6/8) = 90.7 staff, an extra 22.7. The city hired 17, which is 75 percent of the arithmetic requirement.
staff-hours before 68 x 8 = 544 h/day
staff-hours after 85 x 6 = 510 h/day
change = -6.25 %
implied rise in output/hour = +6.67 %A real gain of six and two-thirds percent — for a quarter off everyone's day, a quarter more headcount, and roughly twelve million kronor of extra cost. The trial was not extended. That is the honest shape of the best-controlled result in the literature, and it is nothing like the headline that travelled.
Sixth, what the safety data says, because it was never asking.
hour 9 of a shift relative risk 1.13 +13.0 %
hour 10 1.27 +27.0 %
hour 12 2.00 +100.0 %
any overtime, injury hazard 1.61 +61.0 %
12+ hours a day 1.37 +37.0 %
60+ hours a week 1.23 +23.0 %Hours nine to twelve add 50 percent more time and roughly 100 percent more risk. Across 603,838 people, working 55 hours a week or more carried 1.13 times the coronary heart disease risk and 1.33 times the stroke risk of a 35-to-40-hour week; the WHO and ILO attribute 745,000 deaths in 2016 to that exposure.
And the finding that governs every self-reported trial in this field: Hans Van Dongen and colleagues restricted people to six hours in bed for fourteen nights and produced cognitive deficits equivalent to two nights of total sleep loss — while the subjects rated their own sleepiness as barely changed. People inside a chronic deficit cannot report it. A trial that measures wellbeing by asking is measuring something, but it is not measuring that.
DREAM
What becomes ordinary
In the organisation that has taken this seriously, the working-time question is not a values question. It is a scheduling question with a measurement attached, and it is answered the way any other engineering question is answered.
The roster is designed against a fatigue model rather than against custom. Shift length, shift rotation, the number of successive nights, and the interval before the next start are all set from the same curve, and when somebody proposes a twelve-hour pattern the conversation is about the hundred percent and not about whether people mind. They may not mind. The risk is not a function of whether they mind.
The reporting pack carries two productivity lines, not one, and everyone in the room knows the difference. Output per hour sits beside output per worker, with hours per worker between them, because they are the same identity and a firm that watches only one of them can be made to believe almost anything. When a proposal moves one and not the other, the paper says which.
Recovery has a budget line. Not a wellbeing budget — a recovery budget, which is a different thing and is defended differently. It funds the shift interval, the rest pause, the handover overlap, the genuine holiday that is genuinely taken, and the headcount that makes all of those possible without somebody quietly absorbing them. It is defended the way maintenance is defended: because the asset it maintains is the only one that produces anything.
Nobody in this organisation says that shorter hours raise productivity, because they have read the interval and they know what the evidence will and will not carry. What they say instead is more useful and more defensible: we know what our output per worker is, we know what our hours per worker are, we have measured what happens when we move one, and here is the number.
The people doing the work can see the same two lines. When the firm proposes a change to the schedule, the case for it arrives with the arithmetic already done, and the people whose bodies are the instrument in question are the ones who check it first. That is not consultation. It is quality control, performed by the only people positioned to do it.
And when the four-day week finally is settled, in either direction, this organisation will not have to change its mind about anything. It will simply read the result into a model it already runs.
DESIGN
The structure that gets there
Four components, in sequence. None of them requires a change in the law, and none of them requires believing the four-day-week literature.
One — measure output per worker, from administrative data, before anything else. Not survey data, not manager estimate, not self-report. Whatever your organisation already counts and cannot fudge: units despatched, claims closed, tonnes moved, invoices raised, cases resolved, revenue booked to the team. Twelve months of it, at period frequency, so you have the standard deviation and not just the mean. Everything downstream is arithmetic on this series, and a firm without it cannot evaluate a working-time change no matter how much it spends on the change.
Two — build the hours series beside it, and compute the identity. Actual hours worked, not contracted hours; overtime separated; unpaid overtime estimated and labelled as an estimate. Then publish output per hour, hours per worker, and output per worker on the same page every month, and require any proposal to name which of the three it claims to move. This single page is the cheapest thing in the chapter and it will change more decisions than the rest of it combined, because most working-time arguments are two people each holding one term of an identity.
Three — put a fatigue model on the roster, and make it a licence condition internally. Take the shift-length and successive-night curves as your prescriptive limit: shifts beyond eight hours require a named exception with a stated control, four successive nights require the same, and the exception is signed by someone who is accountable for the injury statistics rather than for the output statistics. This is the aviation move, brought indoors, and it works for the same reason: it removes the limit from the negotiation. No supervisor has to be the one who declines the extra hours, so no supervisor is competing against the one who does not decline them.
Four — and only then — run the working-time change, sized to be measurable. The rule from Chapter I.01 applies with force here, because the effect sizes in this field are small relative to period noise. The effect must exceed roughly three times the period-to-period standard deviation of your output series, or the result will be arguable whatever happens. In most firms that means the pilot must run for a year rather than a quarter, and must cover enough of the workforce that the aggregate is stable.
The governance. One sponsor accountable for output, one accountable for safety, and a written commitment before the trial begins as to what result counts as a failure. That last clause is the whole of the governance. A trial with no pre-specified failure condition cannot fail, and a thing that cannot fail teaches nothing.
And the sequencing rule that saves the most money. Do steps one and two whether or not you ever intend to do steps three and four. A firm that can read its own working-time identity has acquired a permanent capability for the cost of a monthly report. A firm that runs the pilot without them has bought an anecdote.
DESTINY
How it holds when nobody is pushing
Three forces hold it, and each has a named failure mode.
Insurance holds it. Where employer liability or workers' compensation premiums are experience-rated, a fatigue-managed roster shows up in the premium within two renewal cycles, and a premium is reviewed by a finance function every year without anyone advocating for it. The failure mode: in schemes where the premium is not experience-rated, or where the work is done by contractors whose injuries land on somebody else's return, this force is absent entirely, and the economics reverse.
The identity holds it. Once output per hour and output per worker are on the same page, taking one of them off requires an explanation. The failure mode: quiet substitution. A firm under pressure starts quoting whichever of the two is flattering this quarter, and within a year the page is a marketing document. The defence is that both lines are computed by the same query and published together or not at all.
Portability holds it. A limit that travels — a standard, a licence condition, a sector agreement — cannot be competed away, which is exactly Chapman's 1909 point about why firms do not find the sustainable day alone. The failure mode: it is the slowest of the three to build and the only one no single organisation can do by itself.
And the honest failure mode that has killed more shorter-hours schemes than all of these: intensification. Hours fall, output holds, and what actually happened is that the same work was compressed into less time with the breaks removed. Output per hour genuinely rose; recovery genuinely fell; the injury and health consequences arrive two to five years later, outside the evaluation window, and land on a different budget. Any working-time change that does not measure work intensity alongside output is capable of producing exactly the reported result while doing the opposite of what it claims. Measure pace, break-taking and the hours worked outside the counted ones, or do not claim the result.
DELIGHT
What it feels like
The pleasure here is a physical one and it arrives early.
It is the feeling of finishing a working day with something left — not enough to work more, which is the point, but enough to be a person in the evening. Anyone who has had a stretch of properly slept weeks knows the specific quality of it: the day has an edge you can feel, the second half is not a slow negotiation, and you stop being surprised by your own patience.
There is a second pleasure, quieter, for whoever builds the measurement. It is the moment the two productivity lines go on the same page and somebody senior looks at them for a long time without saying anything. They have just seen that the argument they have been having for three years was an argument between two terms of one identity, and that it was never going to resolve, and that it now has.
And there is the plate. The baker on the step with the water, taking the heat off her forearms. Nothing is happening. That is the work. Half the labour of any body is the returning of it to a state that can labour again, and a workplace designed as though that half existed is recognisable from the doorway, before anybody explains anything.
OPERATIONALIZE THIS
At the level of finance
The instrument: a working-time facility, repaid out of verified overtime, injury and replacement savings.
You are not asking for a shorter week. You are proposing to convert a recurring premium-rate expense into a lower straight-time expense, and to finance the one-off cost of the conversion out of the difference. This is an arbitrage on your own payroll, and it is available in any organisation running structural overtime.
The worked plant. Two hundred production workers at $28.00 an hour, each working five overtime hours a week at time-and-a-half, forty-eight weeks a year.
overtime hours 200 x 5 x 48 = 48,000 h/yr
overtime cost 48,000 x 28 x 1.5 = 2,016,000 $/yr
straight-time equivalent = 1,344,000 $/yr
the premium you are paying = 672,000 $/yrAbsorb those hours with new heads at 1,850 productive hours each: 48,000 / 1,850 = 25.9, so 26 heads. Loaded at 1.32 for on-costs, covering the same hours costs 48,000 × 28 × 1.32 = $1,774,080, against $2,016,000 of overtime.
wage line saving 2,016,000 - 1,774,080 = 241,920 $/yr
injury saving 12 x 42,000 x 0.20 = 100,800 $/yr
turnover saving 36 x 9,500 x 0.15 = 51,300 $/yr
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VERIFIED ANNUAL SAVING = 394,020 $/yrBoth assumption rates are deliberately conservative. The injury reduction is set at 0.20 against a published overtime hazard ratio of 1.61. The turnover reduction is set at 0.15 when the UK pilot reported 57 percent — because a self-reported figure is not an input to a facility, it is a hypothesis about one.
The facility.
recruitment and training 26 x 9,500 = 247,000 $
rostering and fatigue system = 60,000 $
workstation, locker, PPE 26 x 4,000 = 104,000 $
contingency 0.15
FACILITY SIZE = 472,650 $
verification 25,000 $ · admin 15,000 $/yrThe balance-sheet treatment. The rostering and fatigue-risk system is a capitalised intangible over its useful life. The recruitment and fit-out costs are period costs in most regimes and should be presented as such rather than argued about — the facility is what spreads them, not the accounting. The line worth taking to your auditors early is the provision: where an organisation carries a self-insured retention on employer liability, a documented fatigue-risk control is directly relevant to the actuarial estimate behind it. That is a conversation about a provision, which they have every year.
The counterparty. Internal treasury first, lending to the plant. External lenders will price this as ordinary working capital and give you nothing for the safety case. Two completed internal facilities give you a track record, and a track record is what converts this into an insurance conversation, which is where the real money is.
The decision inequality.
394,020
---------------------------- = 76.9 % vs WACC 9 %
472,650 + 25,000 + 15,000Say the obvious thing before the treasurer does: the ratio is large because the facility is one-off and the saving recurs. The recurring cost of the 26 new heads is already netted inside the $241,920.
The number that decides it. At $48,000 of contribution per production worker, the plant contributes $9,600,000 a year. So:
break-even fall in output per worker = 394,020 / 9,600,000 = 4.10 %The entire instrument is one bet: that output per worker falls by less than 4.10 percent when the overtime comes out. At a 2.00 percent fall it nets $202,020; at 3.00 percent, $106,020; at 5.00 percent it loses $85,980. Everything else in the paper is bookkeeping.
And this is precisely where the chapter's arithmetic earns its keep. Forty-five hours to forty is a move within the region where the elasticity is one, so the naive reading of the curve says output per worker falls the full 11.1 percent and the facility is badly under water. The reasons to think otherwise — the overtime hours are the fatigued ones, the injuries and replacements are real, the last hour of a forty-five-hour week is not the average hour — are exactly the reasons this must be measured rather than assumed. Write the failure condition into the facility: if verified output per worker falls more than 4.10 percent over two measured quarters, the facility is drawn down no further and the schedule reverts. A pilot that can fail is the only kind worth financing.
The first ninety days.
| Day | Action | Artifact |
|---|---|---|
| 1–20 | Build the twelve-month output-per-worker series from administrative data | The series, with its standard deviation |
| 21–35 | Build the hours series; publish the identity page | The two-line page |
| 36–50 | Cost overtime, injury and replacement; agree the failure condition | Signed baseline and failure clause |
| 51–70 | Draft facility terms; secure the single signature | Facility memo |
| 71–90 | Begin recruitment; first roster on the fatigue model | Measurement log running |
APPRECIATIVE QUESTIONS
Twelve, for a room
Discovery — what is already working
- When has this organisation shortened or restructured hours somewhere and been glad of it? What was the work, who decided, and what did we notice?
- Which team here already protects its own recovery successfully — the handover, the break, the interval before the next start — and how have they made that stick when others have not?
- Where do we already count something honestly enough that we would stake a decision on it? What made that particular measure trustworthy?
Dream — what becomes possible
- If output per worker and output per hour were both on the front page of the monthly pack, what argument that we keep having would simply end?
- Picture the roster we would design if fatigue were priced the way aviation prices it. What is the first thing that looks different on a Tuesday?
- If recovery had its own budget line and was defended the way maintenance is defended, what would we be able to promise people that we cannot promise now?
Design — what we build
- What is the cleanest output series we already own that nobody can fudge — and what would it take to have twelve months of it by the end of the quarter?
- What result would we agree, in advance and in writing, counts as this not working? Who signs that?
- Which single roster rule, adopted as an internal licence condition, would take the most pressure off the most supervisors?
Destiny — how it holds
- What would have to be true for the fatigue model to still be governing the roster after everyone in this room has moved on?
- How would we know, early, if hours had fallen and intensity had simply risen to meet them — and who would be the first to see it?
- If the four-day-week question were settled tomorrow in the direction we did not expect, what in our design would still be worth keeping?
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Note on figures. Every number in this chapter is computed in lib/verify/V_02.py and printed there with its inputs and units. The productivity-of-hours peak is derived from a quadratic-in-logs calibrated to the thresholds Pencavel reports; Pencavel publishes a spline, not this quadratic, and the interval given is calibration uncertainty rather than sampling error. The plant in Operationalize This is an arithmetic worked example, not a reported case, and every input to it is printed.