How classical units reproject
A unit is a promise about scale. Change how scale is represented and some promises get simpler, some get richer, and a few turn out to have been theorems all along.
Every unit names a place on the dial
Units feel like arbitrary human choices — a metre is a metre because somebody said so. Half true. The size is a choice; the kind is not. Each unit answers a different question about how a quantity behaves when you make everything bigger or smaller, and that answer is a location, not a convention.
Look at where each unit lands below. Some sit near the unit anchor +1: they are quantities that mean something at their own scale and lose meaning when stretched. Others drift toward the infinity pole: they only make sense as an integral over a population or a stretch of time. Nothing on this diagram is a demotion. Placement is a statement about which of the five constraints a quantity is anchored to.
Since the 2019 SI redefinition, all seven base units are fixed by constants (Δν_Cs, c, h, e, k_B, N_A, K_cd). Structurally this already implements the Laegna reading: the units are no longer artefacts but anchor points chosen on a scale axis. What Laegna adds is the classification of which pole each anchor addresses, which determines the correct behaviour of the quantity under renormalisation without case-by-case argument.
What simplifies, what deepens
Three of the seven get noticeably simpler in this reading. Two gain meaning they did not obviously have. And two stop looking like units at all — they look like results.
| m | position · A | Simplifies. Length is the unit anchor itself; every other spatial quantity is a compression of it. |
| s | cycle · O | Simplifies. A second is one closed repetition — a full turn, not a distance along a line. |
| kg | integration · g | Deepens. Mass measures how strongly a region resists re-scaling. This is why it curves the ruler. |
| A | flow · I | Simplifies. Pure linear extension per cycle: I over O. Charge becomes the derived quantity. |
| K | spread | Becomes a theorem. Temperature is the width of the goal basin — a property of a distribution, not a substance. |
| cd | threshold · E | Deepens. Candela is explicitly observer-relative; in Laegna every threshold is, and this unit admits it. |
| mol | ensemble | Becomes a theorem. A count folded into one digit — the base-4 statement that large ensembles need no new magnitude. |
The two that become theorems are worth dwelling on. Kelvin and mole are the units that describe many things at once. In a system where a single digit already carries an octave range, "many" is not a new dimension — it is a shift along the scale axis. So these two stop being independent base units and reappear as consequences of how the dial handles ensembles.
Formally: dimensional analysis over the SI group ℝ⁷ reduces, under the Laegna anchoring, to ℝ⁵ with the ensemble directions (K, mol) expressed as scale translations of the remaining five. This is not the same as natural units, which set constants to 1 and discard the scale information; here the scale is retained as the k parameter and the reduction is invertible. Testable consequence: any dimensionally consistent SI relation must remain consistent after the reduction, and the residual two-parameter freedom must show up as exactly the Boltzmann and Avogadro constants. Verifying this exhaustively across the standard relation set is a finite, tedious, entirely doable piece of work — and it has not been done.
Where the reprojection pays for itself
The point of a reprojection is not tidiness. It is that things which used to need separate explanations start needing only one.
Three examples you can check against what you already know. First: entropy and information have the same units in this reading, because both measure the width of a basin — the fact that they turned out to be the same thing stops being a surprise. Second: action, in joule-seconds, is magnitude times cycle, which is why ħ is a quantum of a closed loop and not of an amount. Third: the fine structure constant is dimensionless, and dimensionless means it is a pure angle — a rotation per event — which is exactly why it refuses to be derived from any unit at all.
Open reprojection tasks, in increasing order of effort: (i) tabulate all derived SI units in Laegna anchor form and check closure under multiplication; (ii) verify the five-parameter reduction against the CODATA relation set; (iii) determine whether the compressed measure reproduces standard renormalisation group flow for at least one toy field theory; (iv) express the Standard Model coupling running in octave-angle coordinates and check for fixed points at rational angles. None of these requires a new idea. All of them require care.
When the numeral already carries the scale
A measurement is a number plus a unit, and the unit is mostly there to remember the scale. What if the numeral remembered it by itself?
In the Laegna grid, digits run along a row by ordinary addition and rows stack by exponentiation. Because four is two squared, a digit is exactly two octaves and a row of four is exactly eight — so the row index is the order of magnitude, carried in the notation rather than in a prefix. Kilo, milli and micro stop being vocabulary and become row numbers. That is a small change with a long shadow: it means scale conversion and arithmetic are the same operation.
Practically this is a mixed-radix fixed-point format whose outer radix is a power of the inner one, so the octave count of a quantity is available without a separate exponent field and without rounding drift. The claim to test is narrow and cheap: implement the format, run a scale-heavy simulation in it and in IEEE-754 doubles, and compare error growth across twenty octaves. It needs no new theory and would settle a real question. The arithmetic it rests on is on the octaves page.