Physical Units System by Exentation Exponent

Fundamental Assignment

System basis:

T¹ = T (Time)
T² = L (Length)
T³ = M (Mass)

1. Fundamental Exponents

Positive Exponents (Direct Dimensions)

k	n	Tᵏ	Dimension	SI Unit	Physical Meaning
0	1	T⁰	1	—	Dimensionless (pure numbers, radians)
1	2	T¹	T	s	Time, duration, period
2	4	T²	L	m	Length, distance, displacement
3	6	T³	M	kg	Mass, amount of matter
4	8	T⁴	T²	s²	Time squared
5	10	T⁵	L²	m²	Area, surface
6	12	T⁶	M²	kg²	Mass squared
7	14	T⁷	T³	s³	Time cubed
8	16	T⁸	L³	m³	Volume

Negative Exponents (Inverse Dimensions)

k	n	Tᵏ	Dimension	SI Unit	Physical Meaning
-1	3	T⁻¹	T⁻¹	s⁻¹ = Hz	Frequency, temporal rate
-2	5	T⁻²	L⁻¹	m⁻¹	Wave number, linear density
-2	5	T⁻²	L⁻²	m⁻²	Curvature, surface density
-3	7	T⁻³	M⁻¹	kg⁻¹	Inverse specific mass
-4	9	T⁻⁴	T⁻²	s⁻²	Temporal acceleration
-5	11	T⁻⁵	L⁻³	m⁻³	Inverse volumetric density
-6	13	T⁻⁶	M⁻²	kg⁻²	Inverse mass squared

2. Physical Units by Exentation Level

Level k = -1 (n = 3): Temporal Variation

Dimension: T⁻¹ = 1/T

Quantity	SI Unit	Symbol	Applications
Frequency	hertz	Hz = s⁻¹	Waves, oscillations, radiation
Angular velocity	radian/second	rad/s	Rotations, circular motion
Event rate	events/second	s⁻¹	Stochastic processes
Decay constant	inverse second	s⁻¹	Radioactive decay, half-life
Radioactive activity	becquerel	Bq = s⁻¹	Disintegrations per second
Refresh rate	hertz	Hz	Displays, processors

General interpretation: “How many times per unit of time”

Level k = -2 (n = 5): Spatial Variation

Dimension: L⁻¹ and L⁻²

Linear Variation (L⁻¹)

Quantity	SI Unit	Symbol	Applications
Wave number	inverse meter	m⁻¹	Optics (k = 2π/λ)
Diopters	inverse meter	m⁻¹	Lens power
Linear gradient	per meter	m⁻¹	Spatial variations
Linear concentration	particles/meter	m⁻¹	One-dimensional density

Surface Variation (L⁻²)

Quantity	SI Unit	Symbol	Applications
Gaussian curvature	inverse square meter	m⁻²	Surface geometry
Surface mass density	kilogram/m²	kg/m²	Mass per unit area
Surface charge density	coulomb/m²	C/m²	Electrostatics
Irradiance	watt/m²	W/m²	Energy flux per area
Illuminance	lux	lx = lm/m²	Light per unit surface
Pressure	pascal	Pa = N/m²	Force per unit area
Surface tension	newton/meter	N/m	Liquid interfaces

General interpretation: “How much per unit of space (linear or surface)”

Level k = -3 (n = 7): Mass Variation

Dimension: M⁻¹

Quantity	SI Unit	Symbol	Applications
Inverse specific mass	inverse kg	kg⁻¹	Relations per unit mass
Charge-to-mass ratio	coulomb/kg	C/kg	Particle physics (e/m)
Specific heat capacity	joule/(kg·K)	J/(kg·K)	Thermodynamics

General interpretation: “How much per unit of mass”

Level k = -5 (n = 11): Volumetric Variation

Dimension: L⁻³

Quantity	SI Unit	Symbol	Applications
Volume mass density	kilogram/m³	kg/m³	Material density
Volume charge density	coulomb/m³	C/m³	Electrostatics
Number concentration	particles/m³	m⁻³	Particle density
Energy density	joule/m³	J/m³	Energy per unit volume

General interpretation: “How much per unit of volume”

3. Composite Units (Combinations)

Kinematics

Quantity	Dimension	Tᵏ Combination	SI Unit	Expression
Velocity	L/T	T²·T⁻¹	m/s	L·T⁻¹
Acceleration	L/T²	T²·T⁻¹·T⁻¹	m/s²	L·T⁻²
Angular velocity	1/T	T⁻¹	rad/s	T⁻¹
Angular acceleration	1/T²	T⁻¹·T⁻¹	rad/s²	T⁻²
Jerk	L/T³	T²·T⁻¹·T⁻¹·T⁻¹	m/s³	L·T⁻³

Dynamics

Quantity	Dimension	Tᵏ Combination	SI Unit	Expression
Linear momentum	M·L/T	T³·T²·T⁻¹	kg·m/s	M·L·T⁻¹
Force	M·L/T²	T³·T²·T⁻¹·T⁻¹	N (Newton)	M·L·T⁻²
Angular momentum	M·L²/T	T³·T²·T²·T⁻¹	kg·m²/s	M·L²·T⁻¹
Impulse	M·L/T	T³·T²·T⁻¹	N·s	M·L·T⁻¹
Torque	M·L²/T²	T³·T²·T²·T⁻¹·T⁻¹	N·m	M·L²·T⁻²

Energy and Work

Quantity	Dimension	Tᵏ Combination	SI Unit	Expression
Energy/Work	M·L²/T²	T³·T²·T²·T⁻¹·T⁻¹	J (Joule)	M·L²·T⁻²
Power	M·L²/T³	T³·T²·T²·T⁻¹·T⁻¹·T⁻¹	W (Watt)	M·L²·T⁻³
Action	M·L²/T	T³·T²·T²·T⁻¹	J·s	M·L²·T⁻¹
Energy density	M/(L·T²)	T³·T⁻²·T⁻¹·T⁻¹	J/m³	M·L⁻¹·T⁻²

Fluid Mechanics and Thermodynamics

Quantity	Dimension	Tᵏ Combination	SI Unit	Expression
Pressure	M/(L·T²)	T³·T⁻²·T⁻¹·T⁻¹	Pa (Pascal)	M·L⁻¹·T⁻²
Density	M/L³	T³·T⁻²·T⁻²·T⁻²	kg/m³	M·L⁻³
Dynamic viscosity	M/(L·T)	T³·T⁻²·T⁻¹	Pa·s	M·L⁻¹·T⁻¹
Kinematic viscosity	L²/T	T²·T²·T⁻¹	m²/s	L²·T⁻¹
Surface tension	M/T²	T³·T⁻¹·T⁻¹	N/m	M·T⁻²
Volumetric flow rate	L³/T	T²·T²·T²·T⁻¹	m³/s	L³·T⁻¹
Mass flow rate	M/T	T³·T⁻¹	kg/s	M·T⁻¹

Waves and Oscillations

Quantity	Dimension	Tᵏ Combination	SI Unit	Expression
Frequency	1/T	T⁻¹	Hz	T⁻¹
Wave number	1/L	T⁻²	m⁻¹	L⁻¹
Wave velocity	L/T	T²·T⁻¹	m/s	L·T⁻¹
Acoustic impedance	M/(L²·T)	T³·T⁻²·T⁻²·T⁻¹	Pa·s/m	M·L⁻²·T⁻¹
Acoustic intensity	M/T³	T³·T⁻¹·T⁻¹·T⁻¹	W/m²	M·T⁻³

Gravitation

Quantity	Dimension	Tᵏ Combination	SI Unit	Expression
Gravitational constant G	L³/(M·T²)	T²·T²·T²·T⁻³·T⁻¹·T⁻¹	m³/(kg·s²)	L³·M⁻¹·T⁻²
Gravitational field	L/T²	T²·T⁻¹·T⁻¹	m/s²	L·T⁻²
Gravitational potential	L²/T²	T²·T²·T⁻¹·T⁻¹	m²/s²	L²·T⁻²

4. Summary by Variation Type

Synthetic Table of Interpretations

Exponent k	Level n	Dimension	Variation Type	Typical Quantities
0	1	1	None	Dimensionless constants, angles
1	2	T	Direct temporal	Duration, period
2	4	L	Direct spatial	Distance, length
3	6	M	Direct mass	Mass, quantity
-1	3	T⁻¹	Inverse temporal	Frequency, rate, rhythm
-2	5	L⁻¹, L⁻²	Inverse spatial	Curvature, surface density
-3	7	M⁻¹	Inverse mass	Ratio per unit mass
-4	9	T⁻²	Temporal acceleration	Frequency change rate
-5	11	L⁻³	Volumetric	Density, concentration

5. Key Observations

Coherence with MLT System

The system T¹=T, T²=L, T³=M exactly reproduces the MLT system (Mass-Length-Time) of classical dimensional analysis:

✅ All mechanical quantities are expressible
✅ Negative exponents generate rates, densities and variations
✅ The structure is consistent with standard dimensional physics
✅ Combinations produce all derived SI units

Pattern of Negative Exponents

k = -1: Temporal variation (how many times per second?)
k = -2: Linear/surface spatial variation (how much per meter/meter²?)
k = -3: Mass variation (how much per kilogram?)
k = -5: Volumetric spatial variation (how much per meter³?)

Fundamental Duality

Each positive exponent has its negative “dual”:

T¹ (time) ↔ T⁻¹ (frequency)
T² (length) ↔ T⁻² (curvature)
T³ (mass) ↔ T⁻³ (per unit mass)

6. Complete Physical Quantities by Category

Classical Mechanics

Position: L
Velocity: L·T⁻¹
Acceleration: L·T⁻²
Force: M·L·T⁻²
Energy: M·L²·T⁻²
Power: M·L²·T⁻³
Momentum: M·L·T⁻¹
Pressure: M·L⁻¹·T⁻²

Thermodynamics

Temperature: (requires system extension)
Entropy: M·L²·T⁻²·K⁻¹ (with temperature)
Heat: M·L²·T⁻²
Heat capacity: M·L²·T⁻²·K⁻¹

Electromagnetism

(Would require adding electric charge dimension Q as T⁴ or equivalent)

Optics and Waves

Frequency: T⁻¹
Wavelength: L
Phase velocity: L·T⁻¹
Wave number: L⁻¹
Intensity: M·T⁻³

ArXe System — Recursive Exentational Architecture
Complete dimensional mapping from fractal logical structure

Arxe Logic