Integrated Table of Quantitative Elements

Let us be honest about the term at the outset. As of September 2026, no standards body — not the International Bureau of Weights and Measures (BIPM), not the National Institute of Standards and Technology (NIST), not the Bureau of Economic Analysis (BEA) — publishes anything called the Integrated Table of Quantitative Elements. Search the official record for the acronym ITQE and you will come back empty-handed. So why coin it?

Because the phrase names something real that has never had a name: a recurring intellectual ambition to take the primitive quantities of a field and lay them into a single grid, so that the relationships among the cells can be read, checked, and — in the best cases — used to predict a cell that is still empty. That ambition has been realized, independently, at least three times in the history of science and accounting. It has simply never been recognized as one idea wearing three costumes.

This deep-dive treats those three realizations as a single research object. The argument it defends is deliberately modest, and it is worth stating plainly before we begin: an ITQE is legitimate only when every cell in it declares four things — its quantity kind, its unit (either SI or an explicitly mapped non-SI accounting unit), its uncertainty or exactness status, and the identity rule that produced it. A cell that cannot declare those four things is not an element. It is a label wearing a lab coat.

That is the whole discipline of the thing. Everything that follows is an argument for why those four contracts are not bureaucratic fussiness but the difference between a table that is science and a table that is merely a poster of numbers.

Three traditions arrived at the same instinct using completely different materials. Chemistry got there first. Dmitri Mendeleev’s 1869 arrangement placed sixty-three known elements in order of atomic weight and chemical affinity, and — this is the part that still astonishes — it left deliberate gaps. Those gaps were not confessions of ignorance; they were predictions. When gallium, scandium, and germanium were later discovered with properties close to what the empty cells demanded, the table stopped being a filing system and became an instrument.

Metrology arrived second, and by a stranger road. After the 26th General Conference on Weights and Measures met in 2018, the International System of Units was redefined so that seven constants of nature were assigned exact numerical values. The old platinum-iridium kilogram in its vault outside Paris was retired. In its place: a number. This, too, is a table of elements — seven of them — but the cells are legal definitions rather than chemical species.

Economics arrived third. Building on François Quesnay’s eighteenth-century Tableau économique, Wassily Leontief published in 1936 an empirical input–output table of the United States economy, asking a question no periodic table ever posed: how much of every industry’s output does every other industry consume? His matrix of coefficients became, through the later work of the U.S. Bureau of Economic Analysis, a public statistical instrument that still underwrites national accounting today.

Here is the honest caveat, and it matters. These three grids share an ambition — integration — and they share a risk — category error. What they do not share is a single algebra. You cannot add an atomic number to a dollar flow and call the result an element. Recognizing the family resemblance is useful; pretending the three are interchangeable is exactly the mistake the ITQE contract exists to forbid.

It is tempting to admire the periodic table for the elegance of its shape. That admiration misses the point. The grid became powerful the moment a vacant cell came to mean something specific: a real substance, not yet found, whose properties were bounded in advance by its position. The table did not merely organize what was known. It made falsifiable claims about what was not.

The ordering index itself evolved, and the evolution is instructive. Mendeleev periodized by atomic weight. But after Henry Moseley’s 1913 X-ray spectroscopy, the true index turned out to be the atomic number Z — the proton count of the nucleus. Mendeleev did not live to accept that his beautiful table had been re-founded on a quantity he had not measured. The lesson for any integrated table is sobering: the axis you order by can be replaced by a deeper one, and the table survives the replacement only because each cell still carries an honest quantity kind.

This is the first design principle an ITQE inherits. Prediction is a property of a register, not of the whole page. Empty chemical cells predict undiscovered species. They do not predict dollar flows or physical constants. A grid that blurred those boundaries — that let the predictive glamour of chemistry leak into columns where it has no license — would be borrowing authority it did not earn.

On 20 May 2019, the definition of measurement itself changed. The revised International System of Units fixed the exact numerical values of seven constants, and from those seven every unit is now derived. Because these are the load-bearing cells of the entire metrological register, they deserve to be quoted exactly:

• The caesium-133 hyperfine transition frequency: Δν_Cs = 9 192 631 770 Hz — this defines the second.
• The speed of light in vacuum: c = 299 792 458 m·s⁻¹ — with the second, this defines the metre.
• The Planck constant: h = 6.626 070 15 × 10⁻³⁴ J·s — the quantum of action, which defines the kilogram.
• The elementary charge: e = 1.602 176 634 × 10⁻¹⁹ C — this defines the ampere.
• The Boltzmann constant: k = 1.380 649 × 10⁻²³ J·K⁻¹ — this defines the kelvin.
• The Avogadro constant: N_A = 6.022 140 76 × 10²³ mol⁻¹ — this defines the mole.
• The luminous efficacy of 540 THz radiation: K_cd = 683 lm·W⁻¹ — this defines the candela.

Notice what none of these seven values carries: a measurement uncertainty. That is not an oversight. It is the entire point of a defining constant. By decree, its uncertainty is zero, and every other quantity in the system is measured against it.

The architectural lesson is severe, and it cuts against sloppy integration. The periodic table integrates species that share a quantity kind. The SI table integrates constants that share a legal role — they define units — even though they do not share a quantity kind at all. A frequency, a speed, and an action are not the same sort of thing. An integrated table that copied only the visual density of these grids, without declaring each cell’s kind and role, would produce a handsome artifact with no scientific content.

Defining constants are exact. Almost nothing else is, and the ITQE contract insists that the difference be printed on the page. The CODATA recommended values are a separate object from the SI defining constants: the 2022 adjustment, published in Reviews of Modern Physics in 2025 and reprinted in part on the NIST wallet card of May 2024, is a least-squares reconciliation of every accepted theoretical and experimental result through 31 December 2022. The exact constants enter as fixed; the rest are fitted.

Consider the Newtonian constant of gravitation, G = 6.674 30(15) × 10⁻¹¹ m³·kg⁻¹·s⁻². Those parenthetical digits are not decoration. They are the standard uncertainty applied to the last figures of the quoted value — a public confession of how well we know the number. Contrast that with the fine-structure constant, α = 7.297 352 5643(11) × 10⁻³, whose reciprocal α⁻¹ = 137.035 999 177(21) is among the most precisely tested pure numbers in all of physics.

Now imagine placing α — a dimensionless constant of nature — in the same column as a quarterly dollar flow, with no conversion rule between them. That is not integration. It is a category error more consequential than any misplaced footnote, because it invites arithmetic that means nothing. The Guide to the Expression of Uncertainty in Measurement (JCGM 100:2008) exists precisely to keep such operations honest: combined standard uncertainty follows from the law of propagation of uncertainty, and coverage intervals require an explicit coverage factor.

So the rule for a living ITQE is uncompromising. Every non-exact cell must print either a standard uncertainty or an explicit statement that it is an accounting identity carrying no metrological uncertainty at all. Silence is not permitted. A number without a stated status is a number you are not allowed to trust.

The economic register is where the temptation to over-claim is greatest, so it deserves the most careful language. Leontief’s input–output tables, formalized in his 1936 paper and the 1941 Harvard monograph The Structure of American Economy, and operationalized in the BEA’s handbook on the U.S. input–output accounts, express the economy as a matrix of make-and-use flows. The element in such a table is not a nuclide. It is an industry or commodity aggregate.

The central quantity is the technical coefficient. Let i index the supplying industry and j the using industry. Then a_ij = z_ij / x_j, where z_ij is the flow from i to j during the accounting period and x_j is the total output of j. The coefficient answers a genuinely useful question — how much input from i does one unit of j’s output require? — under the maintained assumption of fixed proportions during a reference period.

But read that sentence again and notice every hedge in it: accounting period, reference period, fixed proportions. The identity a_ij = z_ij/x_j is an accounting relation, not a law of nature. Change the technology and a_ij changes. Revise the industry classification — and NAICS revisions do exactly this — and coefficients from different vintages stop being comparable. An ITQE that treated a_ij as if it were the speed of light would misunderstand both objects at once: it would grant a survey estimate the permanence of a constant, and strip a constant of the exactness that defines it.

This is why the economic register in a defensible ITQE must date every coefficient, name its classification scheme, and mark its currency or chain-type quantity unit. A number that does not say when and under what scheme it was measured is, in economics, barely a number at all.

Underneath all of this sits a distinction chemists and physicists have kept for over a century, and it is the hinge on which the entire ITQE turns. Intensive quantities do not scale with the size of the system: temperature, pressure, density. Extensive quantities do: mass, volume, entropy, enthalpy, internal energy. Georg Helm introduced the language into physics in 1898; Richard Tolman restated it in 1917. The rule that follows is almost embarrassingly simple to state and endlessly violated in practice: you may not add an intensive cell to an extensive cell and report the sum as a new element. Doing so does not integrate the data. It destroys it.

From this single prohibition the whole architecture of a lawful ITQE unfolds. Physical constants, chemical species, and industry aggregates may share a single page — but only if the page is zoned, with visible gutters between the zones and a standing ban on cross-zone arithmetic unless a published bridge identity licenses it:

• Zone A — Defining constants. Exact by decree; uncertainty zero. The seven SI constants live here.
• Zone B — Adjusted constants. CODATA values with parenthetical standard uncertainties; G and α live here.
• Zone C — Chemical and physical quantities. Keyed by atomic number, with standard atomic weights and derived properties in coherent SI units.
• Zone D — Economic flow coefficients. Each dated, classified by NAICS vintage, and carrying a currency or chain-type quantity unit.

These four zones are not a finished table. They are the preconditions under which a finished table would still be science rather than collage. The gutters are the product. Remove them and you are left with a grid that looks authoritative and computes nonsense — which is worse than no grid at all, because it is nonsense that photographs well.

So what does a defensible Integrated Table of Quantitative Elements actually look like? Not a new list of numbers — an architecture. Picture four stacked registers, printable on one sheet only if the gutters between them stay visible. Register I reprints the seven SI defining constants with their exact values and the units they define. Register II reprints selected CODATA 2022 adjusted values with their parenthetical uncertainties. Register III reprints a current IUPAC-consistent periodic grid keyed by atomic number. Register IV reprints a stated-vintage make-and-use extract from the national accounts, with every coefficient dated.

Each register carries a header naming its issuing body and the date it was accessed. The living copyright year in a document footer never substitutes for those access dates — a subtle but important honesty. Owning the monograph is not the same as owning the speed of light, and the table must never let its footer pretend otherwise.

Within this design, prediction stays inside its register, exactly as the periodic table taught us. Empty chemical cells predict undiscovered species. Empty economic cells predict missing survey detail, not new laws of motion. Register I has no empty cells at all — defining constants are complete by construction. That asymmetry is a result worth stating, not a defect to apologize for.

Honesty also requires naming what is unfinished. A fully typeset Register II would need the complete 2022 CODATA tables, not merely the wallet-card subset. Standard atomic weights are intervals for many elements, so a cell that prints a single figure without its interval misleads. Industry classifications drift, breaking coefficient comparability across NAICS vintages. And no peer instrument yet maps marketing or information-retrieval quantities onto SI without resorting to metaphor — which is precisely why such quantities may enter a fifth register only after a unit and an uncertainty rule exist for them, and not one day before.

That, in the end, is the whole contribution. The ITQE does not replace the periodic table, the SI Brochure, the CODATA adjustment, or the national accounts. It is a disciplined overlay that states the conditions under which those instruments may share a page without committing a category error. Keep the gutters. Date every register. Explain every symbol at first use. It is an editorial and institutional discipline, not an oracle — and that modesty is exactly what makes it trustworthy.