Coherence means that the equations, not convention, determine how the units fit together.

Measurement began as a collection of largely unrelated human conventions—feet, hours, weights, temperatures. Physics progressively revealed that a useful system cannot merely assign names to quantities: its units must close under the equations of Nature. If force is mass × acceleration, the unit of force should emerge from the units of mass, length, and time without an arbitrary correction factor. If energy is force × distance, its unit should follow from the same system. This is the essential meaning of a coherent system of units.

The deeper importance of measurement is that measured quantities do not remain isolated: once expressed quantitatively, they constrain one other. Galileo’s analysis of falling bodies made this especially clear: distance, time, velocity, and acceleration form a connected system rather than a collection of independent observations.

In geometry and mechanics, length and time first appear as distinct primitive measurements. Newtonian mechanics binds them through velocity and acceleration, while mass introduces another independent dimension. Force, momentum, energy, and the other mechanical quantities then arise as combinations of these underlying dimensions.

Electromagnetism made the demand for coherence still more explicit. Gauss, Weber, Maxwell, and others found that electrical and magnetic quantities could not be treated as an unrelated collection of measures: the equations themselves impose relations among them. The nineteenth-century absolute systems of units grew from the effort to make systems of measurement reflect the algebraic structure of physical laws.

Dimensional analysis made this principle systematic. An equation whose dimensions do not balance can be rejected before any numerical measurement is made. Coherence therefore does more than simplify notation: it constrains which mathematical expressions can represent physical relations at all. Dimensional consistency is a requirement of physical law.

With Giorgi's MKS proposal, and eventually the International System of Units, coherence became an explicit organizing principle. Derived units are constructed as products of powers of the base units without any additional numerical conversion factors. Force, energy, power, electric potential, resistance, and the other derived quantities therefore do not require independent dimensional conventions; their units follow from the dimensional structure of the system.

Relativity deepened this picture. Space and time are no longer independent backgrounds for physical events: x and ct belong to a common spacetime geometry, joined by the invariant speed of light. In other words, the meter is realized through the second together with the fixed value of c. Therefore, our unit of space is defined through our unit of time and an invariant of Nature. What once appeared to require two independent standards is revealed as a constrained relationship. Likewise, every constant of Nature encodes a fixed relationship in the dimensions of physical reality. Taken together, they constrain how the measurable dimensions of physical reality fit into one coherent system.

A coherent system is therefore more than a common language for measurement: its units reproduce the dimensional structure of physical laws. The quantities governing persistent atomic constructions reveal a minimal coherent basis.

There are coherent units (or bases) of atomic logic: the second, meter, coulomb, kelvin, and kilogram. Together, these provide the dimensional basis to describe the internally consistent constructive logic of persistent forms (atoms).

Combining these coherent units—without introducing any scalar conversion factors—generates the composite units of physics and chemistry. Their dimensional relations therefore arise from a single common basis, allowing the units to work together as one internally consistent system for describing the action parameters for atomic exchanges.

There are also derived units of atomic logic, the: megahertz, femtometer, electron-volt, megaelectron-volt, and the gigaelectron-volt. Each is obtained from the coherent base system by a power of ten rescaling, without introducing an independent dimensional conversion factor. Equivalently, each uses a unit coefficient multiplying a power of ten, while preserving the same underlying dimensional relations. That is, every derived unit has a value of 1 raised to some power of 10—making them coherent with the rest of the units of atomic logic.

The mole is different in kind from these dimensional units: it represents an amount found to be special in atomic constructions rather than an additional physical dimension. The candela is a unit of luminous intensity—an amount of lumens (a unit based on the sensitivity of the human eye) divided by 4 steradians (the geometric unit of solid angle).