Information, Signal, and Price

Volume II — Foundations: The Paradigm and the Science

THE PLATE

Three colleagues at a whiteboard in a sunlit room, one writing while the others listen. — Plate II.07One Number on a Chalkboard.All of that reaches the next person as one number. The chalk is not a summary of what she knows. It is the only part of it that can be transmitted.

THE LETTER

Somewhere in the next hour you will look at a price and make a decision. You will do it quickly, without effort, and almost certainly well — and you will be using an instrument whose capacity nobody ever quoted you.

This chapter quotes it. Not metaphorically: in bits, computed from a tick size and a volatility, with the arithmetic in lib/verify/II_07.py so you can change the inputs and watch the answer move. The number is smaller than you expect, and the smallness is not a complaint. It is the achievement. Friedrich Hayek's claim in 1945 was that the price system performs a compression no planner could perform, and he was right, and the compression ratio is the reason it works.

But a compression has a shape as well as a size, and the shape is where this chapter earns its place in a book about living systems. Everything a living system does is multi-dimensional — a soil has moisture, carbon, structure, biology and time; a person has hours, health, skill, attachment and consent; a fishery has biomass, age structure, genetic diversity and the weather. A price is one number. The question is not whether one number is enough. The question is what happens to the dimensions it is not pointed at, and the answer turns out to be exact, provable, and independent of how precise the price is.

That result is the middle of this chapter and it changes what an externality is. Not a quantity someone got wrong. A direction the instrument has no component in.

You will also find here the one impossibility result that every serious defence of markets has to hold, which is Grossman and Stiglitz's, and the honest accounting of what it has cost to widen the channel wherever anyone has actually tried. That last section is not encouraging and it is not meant to be discouraging either. It is a budget, and a budget is the most useful thing anybody can hand you.

By the end you will be able to say, of any decision in front of you, how many bits it needs, what a bit of it is worth, and whether the price you are looking at is carrying them.

— The Editors

DISCOVERY

What is already working

Start where the instrument is at its best, because it is very good and the rest of this chapter is unreadable if you have not admitted that first.

Kerala, 1997 to 2001. Robert Jensen followed the sardine fisheries of the Malabar coast as mobile phone coverage arrived in three districts in sequence. Before coverage, a boat had to choose a landing beach before it knew the price there, and the result was the textbook picture of a market that is not one: on a given morning, one beach would have fish with no buyers and a beach fifteen kilometres away would have buyers with no fish. Jensen recorded days when eleven boats dumped their entire catch into the sea while, within a short sail, demand went unmet.

Then the phones arrived, and the measurements are among the cleanest in development economics. Price dispersion across beaches collapsed. Waste — fish landed and discarded — fell to essentially zero. Fishermen's profits rose by about 8 percent, and consumer prices fell by about 4 percent, at the same time. Both sides gained, and nothing was produced to make it happen. The entire welfare gain came from moving one number a few kilometres before a decision was made. That is Hayek's claim, measured, with a control group.

The tin, 1945. Hayek's own example deserves to be stated in his words rather than ours: when tin becomes scarce, "tens of thousands of people whose identity could not be ascertained by months of investigation" economise on tin, and they do it without knowing why, without an order, and without any of them holding more than a fragment of the reason. He called the price "a kind of symbol" in which "only the most essential information is passed on." He is describing a compression, and he is describing it three years before Claude Shannon gave the world the unit it is measured in. The two papers have never been properly introduced. This chapter is that introduction.

Iowa City, since 1988. The Iowa Electronic Markets have run continuously for more than three decades as a regulated academic exchange where contracts pay out on election outcomes. Across a dozen years of elections, Berg, Forsythe, Nelson and Rietz found the market price closer to the eventual result than the contemporaneous polls in roughly three-quarters of the comparisons they could make. A price aggregated the private information of a few thousand traders more accurately than an instrument designed specifically to collect it.

The United States, 1995 to 2004. Title IV of the Clean Air Act Amendments of 1990 did something the rest of this chapter takes seriously: it built a second price channel on purpose, for one attribute, sulphur dioxide. Schmalensee and Stavins, reviewing the whole experiment two decades later, assemble the record: emissions from covered sources fell sharply while generation rose, and the compliance cost came in far below the ex ante estimates — including the estimates made by the programme's own advocates. It is the single best evidence in the record that a missing dimension can be given a channel and that the channel will be used intelligently by people who were not consulted about it.

Chile, since 2016. Ley 20.606 put a black octagonal warning on the front of foods above thresholds for sugar, sodium, saturated fat and calories. It is not a price. It is a channel of a few bits, deliberately kept narrow, sitting beside the price. Taillie and colleagues measured purchases of sugar-sweetened beverages before and after and found a large fall in the volume households bought. The channel worked because it was small. A nutrition table carrying forty numbers had been on the back of the same package for years and had moved almost nothing.

Five cases, five kinds of channel: a phone, a commodity price, an exchange, a permit market, a label. In every one, the gain came from a number arriving where a decision was being made, and in none of them did anyone have to be persuaded of anything. That is the positive core, and it is large. Hold it while we do the arithmetic, because the arithmetic is going to be exacting, and it is exacting in service of that core rather than against it.

THE ARITHMETIC

How many bits, what they cost, and the dimension that precision cannot buy

First: how big is a price?

Shannon gives the unit. A quoted price is a random variable with a support and a resolution; its entropy is the average number of binary questions you would need to identify it. Take a share at one hundred dollars, a daily return standard deviation of 2 percent, and the one-cent minimum increment that SEC Rule 612 requires of a quotation at or above a dollar.

  daily sigma in currency     100.00 x 0.02        =  USD 2.00
  Gaussian constant           sqrt(2 pi e)         =  4.13273
  effective support           2.00 x 4.13273       =  USD 8.2655
  ticks spanned               8.2655 / 0.01        =  826.55
  entropy                     log2(826.55)         =  9.691 bits

A closing price carries about 9.69 bits. One and a fifth bytes. The letter a in this sentence is eight bits and carries more.

And it is stubborn. Take the volatility from a half percent to four percent — an eightfold move, the difference between a utility and a distressed biotech — and the message moves by three bits, from 7.69 to 10.69. Cut the tick by a factor of ten thousand and you buy thirteen more. The size of a price is remarkably insensitive to everything, which is precisely why it is such a good instrument and precisely why it cannot be stretched.

Now the price a person actually meets. Nakamura and Steinsson measured the median frequency of consumer price change at 8 to 11 percent a month excluding sales; Bils and Klenow, on different data a few years earlier, found a monthly hazard near 21 percent including them. Run the binary entropy:

  H(0.09) = 0.4365 bits/month   ->   5.24 bits/year
  H(0.21) = 0.7415 bits/month   ->   8.90 bits/year

One alphanumeric character is log2(36) = 5.17 bits. Two independent measurements twenty years and two datasets apart agree that a shelf price transmits between one and two characters a year about the state of the world it came from. Everything the grower, the shipper, the processor and the retailer know reaches the person buying it at roughly that rate.

Second: the cut.

Here is the move, and it is not about bandwidth at all.

Let the thing being signalled be a vector x in R^k — for a garment, say, carbon, water, land, chemical load, wage floor, hours, freedom of association, child labour, worker exposure, durability, repairability, end-of-life. Twelve dimensions, and each is real, measurable, and consequential. A price is a scalar functional p = w · x. Then:

  attributes k   channels   invisible dims   share invisible   gradient recovered
         12          1             11              91.7%               8.3%
         12          2             10              83.3%              16.7%
         12          3              9              75.0%              25.0%
         40          1             39              97.5%               2.5%

The kernel of w has dimension k − 1. Any two garments differing by a vector in that kernel produce the same price, and I(p ; x⊥) = 0 is an identity, not an estimate. Now spend precision instead:

  tick          bits in the price     invisible dimensions
  0.01                 9.691                    11
  0.0001              16.335                    11
  0.000001            22.979                    11

Three orders of magnitude of precision buys thirteen bits and moves the invisible dimension count by exactly zero.

This is the chapter. An externality is not a quantity somebody got wrong and a better estimate will fix. It is a direction the pricing functional has no component in. You cannot reach it with more decimal places, more liquidity, more traders or more compute, because none of those change the rank of a linear map. Mount and Reiter proved in 1974 that the competitive message space is the smallest that can clear a classical economy — the price system's celebrated efficiency is a statement about the dimension of what it sends. Minimality and blindness are one theorem read twice. Hayek's marvel and Pigou's problem are the same fact, and nobody has to be wrong for both to be true.

Third: Grossman and Stiglitz, and what the channel costs.

In 1980 Sanford Grossman and Joseph Stiglitz showed that a fully informative price cannot exist in equilibrium. If the price reveals what the informed know, nobody pays to become informed; if nobody pays, the price reveals nothing. The market must be partly uninformative, permanently, and the informed must earn back the cost of their information out of trading with people who are not. This is not a criticism of markets. It is the operating condition of one.

So the channel has a bill. Kenneth French computed it for US equities: an average of 0.67 percent of aggregate market value a year from 1980 to 2006, and 101.8 billion dollars in 2006 alone. Thomas Philippon found the unit cost of US financial intermediation sitting near 1.9 percent of intermediated assets and essentially unchanged since 1886, through the telegraph, the telephone, the mainframe and the internet.

Now divide the bill by the bits.

  4,000 listed companies x 252 days x 9.691 bits  =  9,768,479 bits/year
                                                  =  1.164 MiB/year
  USD 101.8 bn / 9,768,479 bits                   =  USD 10,421 per bit

The entire annual closing-price output of the US equity market is about one and a sixth mebibytes, and in 2006 it cost a hundred billion dollars to produce. At six thousand listed companies it is 6,948 a bit; at three thousand, 13,895. The order of magnitude does not move.

State the denominator, because a check with an unstated one is worse than none. That figure looks only at closes. Go to one-minute bars and the bandwidth rises to 2,101 bits a session, 216.8 times as much, and the cost falls to 48.07 a bit. But Grossman and Stiglitz already told us what the extra bandwidth is: the informed are paid out of the noise, so a channel carrying only signal would pay nobody to fill it. The additional bits are not a discount on information. They are the noise the arrangement structurally requires. Bai, Philippon and Savov close the loop from the other end: measured price informativeness in the aggregate has not risen since 1960, through every fall in the cost of compute in the history of computing.

Fourth — the honest negative, and it is the load-bearing part.

If the missing dimensions need channels, and channels are the answer, then the answer has a price and somebody has to pay it. Here is what it has actually cost where it has been tried.

Take monitoring, reporting and verification for one installation at 20,000 euros a year, and carbon at 80 euros a tonne.

  500,000 tCO2e/yr   ->  EUR  0.04 per tonne of measurement
   25,000 tCO2e/yr   ->  EUR  0.80
    5,000 tCO2e/yr   ->  EUR  4.00
    1,000 tCO2e/yr   ->  EUR 20.00

  abatement the signal must enable to pay for itself   20,000 / 80  =  250 tCO2e
  at a 10 percent abatement rate, installation floor    250 / 0.10  =  2,500 tCO2e/yr

The European Union reached the same conclusion in law and set its line an order of magnitude more conservatively: Article 27 of Directive 2003/87/EC lets member states exclude installations under 25,000 tonnes, and the monitoring regulation lets de minimis source streams under 1,000 tonnes use simplified methods. The people who run the largest second price channel ever built have written into it that below a certain scale it is not worth running.

Then the largest deliberate bandwidth purchase in the history of commerce. The European Sustainability Reporting Standards carry on the order of eleven hundred datapoints — an explicit attempt to transmit the whole vector rather than the projection. On 26 February 2025 the Commission proposed removing roughly four-fifths of in-scope companies and stated an expected saving of 6.3 billion euros a year in administrative cost. Divide:

  EUR 6.3 bn / 40,000 companies              =  EUR 157,500 per company per year
  EUR 157,500 / 1,100 datapoints             =  EUR 143 per datapoint per year

That is the measured price of the missing dimensions, and it was measured by a legislature deciding to stop paying it. Not an estimate by an opponent. A revealed price, quoted in a repeal.

Against it, the channel that survived. A farm certification fee of a thousand dollars, against which the USDA cost-share programme pays 75 percent capped at 750, leaves a net cost of 250 dollars for one bit — one word on a label. At a 20 percent farmgate premium the break-even is 1,250 dollars of sales; unreimbursed, 5,000.

Cheap channels get bought and expensive ones get repealed, and that is the whole history of every attempt to widen this pipe. The living-systems approach loses, cleanly and without argument, wherever the measurement costs more than the decisions it changes are worth — which is most of the economy by count of enterprises, and the section that follows is about what to do with that rather than about regretting it.

DREAM

What becomes ordinary

Describe it in the present tense, because a dream in the future tense is a wish.

In the economy that has absorbed this chapter, nobody argues about whether prices are informative. They ask how many bits a decision needs and where those bits are coming from, and the question is as routine as asking what a thing costs. A procurement officer looking at two suppliers with identical quotes does not say they are the same. She says they are price-identical, which tells me they differ somewhere in the kernel, and here are the three dimensions I have decided are worth instrumenting. The phrase is ordinary. It is taught in the second week.

Second channels exist where they pay and nowhere else, and the arithmetic deciding which is which is published rather than argued. Every attribute anyone proposes to measure arrives with two numbers beside it: what the measurement costs, and what the decision it changes is worth. Proposals where the second exceeds the first are funded without ceremony. Proposals where it does not are not rejected on principle — they are pooled, until enough holders share the channel that each one clears, and pooling is a normal commercial act rather than a concession.

The instruments are small. Nobody is trying to transmit a thousand datapoints through a channel that will be read in four seconds by a person holding a trolley. The design discipline runs the other way: given four bits at the point of decision, which four? Chile answered that question with an octagon and it moved a national diet. A generation of designers has since answered it for fisheries, for textiles, for cement, for care work, and the good answers all look alike — narrow, unambiguous, and aimed at one decision that was genuinely in doubt.

Firms keep a channel budget the way they keep a capital budget, and it is reviewed in the same meeting. It has a total, it has a hurdle, and it has a line for retirement: a measurement whose decision has stopped being uncertain is switched off, because a bit spent on something you already know is a bit spent on nothing. Nobody finds this cynical. It is the same discipline that stops a laboratory running an assay whose result never changes what happens next.

And prices are trusted more, not less, because their job has been stated. A price that is understood to be a projection is a reliable instrument. A price expected to carry everything is a disappointed one, and disappointment is where the last century's arguments about markets came from — both the arguments that they are miraculous and the arguments that they are wicked. Both were claims about a channel nobody had measured.

The people who work inside this can say what their work is doing to stocks that have no price. Not because they are asked to feel it, but because somebody decided which three of those stocks were worth a channel and built it. The rest is left honestly unmeasured and said to be unmeasured, which is a far better state than being measured badly and reported confidently.

DESIGN

The structure that gets there

Four moves, in order. They are cheap and they compound.

One: name the vector before you argue about the price.

Take any decision you are actually making and write down the attributes that would change it. Not everything that matters — everything that would change this decision. You will typically get between four and fifteen. That count is k, and it is the first honest number in the exercise, because 1/k is the share of the welfare gradient a single price can be expected to recover for you. At k = 12 it is 8.3 percent. Write that percentage down too. It reframes every subsequent conversation from is the price right to which of the eleven are we flying blind on, and do we mind.

Two: rank the dimensions by value of information, not by importance.

These are different and confusing them is the commonest failure in this whole field. An attribute can be enormously important and worth nothing to measure, because you already know the answer. Ronald Howard settled this in 1966: the value of information is the value of the decision change it produces. A dimension on which you are certain carries no bits and can change nothing, no matter how much it matters. Measure what you cannot guess, and guess the rest. That sentence is a theorem, not a temperament.

Three: buy channels one at a time, cheapest-value-per-bit first, and pool below the floor.

Each channel is a capital decision with a break-even, and the break-even is computable before a euro is spent. Where a single holder cannot clear it, the structure is a pool: a shared measurement vehicle with an internal control system, which the European organic regulation already recognises as a group of operators and which the SO2 market achieved by a different route, letting the market itself allocate measurement effort to where abatement was cheapest.

Four: design the receiving end, not the sending end.

The whole literature argues about producing information and almost none of it about delivery. Christopher Sims named the reason: the constraint that binds is not the world's capacity to emit signals but the receiver's capacity to process them. A person at a shelf has, generously, a few bits of attention. A procurement committee has more but not a thousand datapoints' worth. Design to the receiver's channel capacity and the sender's problem largely solves itself — which is why an octagon moved a country and a nutrition panel did not.

Underneath all four is the governance point. A second channel is a standing institution, not a report, and Elinor Ostrom's design principles apply to it exactly: clearly defined boundaries for what it measures, monitors accountable to those they monitor, graduated sanctions, and a cheap way to resolve a dispute. A measurement regime without those fails in the ordinary way, which is that it is gamed, and then it is disbelieved, and then it is repealed — and the repeal takes the honest measurements down with the dishonest ones.

DESTINY

How it holds when nobody is pushing

Three things make a channel outlive the person who built it, and they are not the three people expect.

It is attached to a decision that recurs. A measurement tied to a one-off decision dies with the decision. A measurement tied to something decided every quarter acquires a schedule, and a schedule is what survives a change of personnel. The organic bit persists because someone re-chooses a supplier every season.

Its cost per bit falls faster than its value per bit. Channels that do not get cheaper get switched off, and the arithmetic in this chapter tells you when: the day the measurement cost crosses the expected value of the decision change. That day is computable in advance, which means it can be planned for rather than discovered.

Somebody is paid out of it. Grossman and Stiglitz, generalised: a channel whose operators earn nothing from its accuracy will not stay accurate. The SO2 market worked because holding an allowance was worth money. The Kerala phones worked because the fisherman keeping the price paid for the airtime out of the gain. A measurement regime funded purely as compliance cost has no party inside it whose income depends on its being right, and that is the structural description of every reporting standard that has quietly degraded into a template.

Now the failure modes, named plainly. A channel fails when it is designed to transmit everything, because the receiver ignores all of it. It fails when it is mandated below its scale floor, because the small holders lobby and the whole regime is reopened. It fails when it measures a proxy rather than the attribute, because the proxy is optimised and the attribute is not — Goodhart's mechanism, and it is the most reliable failure in the set. And it fails when the second channel is presented as a moral obligation rather than an instrument with a return, because the first budget round after the champion leaves will cut it, and nobody will have to defend cutting it.

DELIGHT

What it feels like

There is a specific pleasure in computing the size of something everyone treats as infinite. You do the division, you get one and a sixth mebibytes, and the world does not become smaller — it becomes legible. A hundred billion dollars and a megabyte, sitting on the same line, and both of them true.

Then a better pleasure, which arrives a week later. You are in a meeting and somebody says two options are the same because the price is the same, and you hear the sentence differently now. Not as a claim about the world. As a claim about an instrument, with a known aperture, pointed in a known direction. You are not annoyed. You are curious, in a way you specifically were not before, about what is standing in the part of the room the instrument does not see.

And the third, which is the one worth staying for: the quiet of not having to argue that prices are bad. They are not bad. They are a magnificent, minimal, nine-and-a-half-bit instrument that solved a problem nobody could solve any other way, and knowing exactly what they are frees you to build the second thing beside them without any of the heat. You stop defending or attacking the chalkboard. You start asking what else the market could afford to write down.

OPERATIONALIZE THIS

At the level of finance

The instrument is a Channel Budget Facility: a ring-fenced, pooled measurement vehicle whose covenant is written in value per decision-bit rather than in datapoints. It is the structure that makes a second channel financeable at scales where a single holder cannot carry one.

The structure. A special-purpose measurement entity subscribed by a group of operators — a supply chain tier, a cooperative, a trade body, a landlord's tenants. It buys one shared monitoring, reporting and verification capability, operates an internal control system, and issues each member an attested attribute record. Members pay a subscription; the entity holds the contracts with the verifier.

The mechanics.

Size. The annual cost of the channel, plus 15 percent contingency. Nothing else. The facility buys measurement; it does not buy abatement.
Admission test. A member is admitted only if the expected value of information on the attribute exceeds their share of the subscription. That test, and not a pledge, is the entry condition.
Pooling threshold. Computed before launch: members ≥ channel cost / EVPI per member. At a 20,000-euro channel and a 1,500-euro expected value per member, that is 14 members, at 1,429 euros each. Below fourteen, do not launch — publish the arithmetic and wait.
Retirement clause. Any attribute whose measured posterior has stopped moving for four consecutive periods is retired from the channel and its budget released. A bit spent on something already known is a bit spent on nothing, and the clause is what stops the vehicle accreting.
Governance. Ostrom's terms, written in: defined boundaries, a monitor accountable to members, graduated sanctions for misreporting, and a dispute step that costs less than the subscription.

The balance-sheet treatment. The subscription is an operating expense. Resist the temptation to capitalise a measurement system — its useful economic life is exactly as long as the decision uncertainty it resolves, which is not an asset life your auditor will recognise. Where the attested record carries a market price, as an allowance or a certificate does, that instrument is recognised separately on its own terms. Keep the two apart. Conflating the channel with the commodity it measures is how measurement regimes end up on a balance sheet nobody can defend.

The counterparty. Start with the party who already buys your output and already wants the attribute — a customer, not a regulator. A customer-funded channel has a party inside it whose money depends on its being right, which is the destiny condition from the previous movement. A regulator-funded one does not, and will degrade.

The first ninety days.

Day	Action	Artifact
1–15	Write the attribute vector for one recurring decision; state `k`	The vector, and `1/k`
16–30	Compute EVPI per attribute; rank by value per bit	The ranked list
31–45	Price the channel for the top attribute; compute the pooling threshold	A break-even memo
46–60	Recruit to the threshold; sign the internal control system	Subscriptions ≥ threshold
61–75	Run one period of measurement against a signed baseline	The first attestation
76–90	Publish cost per bit and value per bit, side by side	The one page

The number that decides it. One inequality, on the front page:

      EVPI on the attribute            annual channel cost
   ----------------------------   >   ---------------------
     bits perfect information            same bits, H(p)
        supplies, H(p)

which reduces, as it should, to EVPI > annual channel cost. Worked at the chapter's inputs — a 10 percent prior on the state that would change the decision, a million-dollar loss from getting it wrong, a fifty-thousand-dollar remedy — EVPI is 45,000 dollars, H(0.10) is 0.4690 bits, and the value per bit is 95,950 dollars. Against the 10,421 dollars a bit that general price discovery costs to produce, a bit aimed at one decision is worth about nine times as much.

These are a value and a cost and the comparison is an order of magnitude rather than an identity — but read as an operating rule it is unambiguous, and it is the sentence to take out of this chapter: instrument the decision, not the world.

APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

Think of a decision here that got dramatically better because one number arrived earlier or from somewhere new. What was the number, and who moved it?
Where do we already run a second channel beside price — a certification, an audit, a rating, a direct relationship — and what does it let us do that our competitors cannot?
Which of our people can tell, without instruments, something about quality that our systems cannot record? What are they reading?

Dream — what becomes possible

If every recurring decision here arrived with its attribute vector named, what would change about the meetings we have?
Imagine four bits of attention from our customer at the moment of choice. What four would we want them to have, and why those?
If we could retire every measurement whose answer we already know, what would we do with the time and the budget that came back?

Design — what we build

Which single attribute, if we could attest it credibly, would change a decision somebody makes about us every quarter?
Who else carries the same measurement cost that we do — and what would it take to share the channel with them rather than each build one?
What is the cheapest honest thing we could measure this quarter, and what decision would it change?

Destiny — how it holds

Whose income here depends on one of our measurements being accurate? If the honest answer is nobody, who should it be?
What would tell us early that a measurement of ours had started being gamed, and who would see it first?
Ten years from now, which of our current numbers would we most want to still be trusted — and what would we have to do this year to earn that?

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Ostrom, E. (1990). Governing the Commons: The Evolution of Institutions for Collective Action. Cambridge University Press.

Directive 2003/87/EC establishing a scheme for greenhouse gas emission allowance trading, Article 27 (small installation opt-out), as amended.

Commission Implementing Regulation (EU) 2018/2066 on the monitoring and reporting of greenhouse gas emissions (de minimis source streams).

Directive (EU) 2022/2464 (Corporate Sustainability Reporting Directive) and Commission Delegated Regulation (EU) 2023/2772 (European Sustainability Reporting Standards, Set 1).

European Commission (2025). Omnibus I simplification package, 26 February 2025, and the administrative cost saving stated therein.

Regulation (EU) 2018/848 on organic production and labelling, provisions on groups of operators and internal control systems.

United States Securities and Exchange Commission, Rule 612 (Regulation NMS), minimum pricing increment.

United States Department of Agriculture, Organic Certification Cost Share Program.

Note on figures. Entropy, cost-per-bit, null-space, threshold and value-of-information figures are computed in lib/verify/II_07.py with every input printed beside its unit and its source. Three boundaries are stated in that module and repeated here: the entropy figures describe a quoted price under a Gaussian log-return model and not the order book; the cost-per-bit figures divide one published cost by one modelled bit count and are offered as an order of magnitude; the null-space result is exact and depends on no estimate. The Omnibus per-company and per-datapoint figures divide the Commission's own stated saving by a stated population and are labelled as such. Not verified for this chapter and named as unverified: the per-product cost of the Digital Product Passport under Regulation (EU) 2024/1781, for which no settled measurement existed at the time of writing.