ArXe Theory proposes a fundamental correspondence between logical structures and the dimensional architecture of physics. At its core, it suggests that each level of logical complexity maps directly to a specific physical dimension.

The Key Concept

Each number of exentation (n) represents a level in a recursive logical hierarchy. Starting from an initial point (n = 1), each new level is built by systematically applying logical operations to the previous one, generating an infinite ladder of increasing complexity.

The Dimensional Connection

Through a precise mathematical formula, each of these logical levels (n) is transformed into a dimensional exponent (k). This exponent defines fundamental temporal dimensions of the form T^(k,) where:

• T⁰ represents the dimensionless (the origin point)
• T¹ corresponds to Time
• T² corresponds to Length (space)
• T³ corresponds to Mass

Conversion formula:

[ e(n) = (-1)^(n) cdot lfloor n/2 rfloor, quad n > 1 ]
[ e(1) = 0 ]

This simple expression generates the sequence:
0, 1, −1, 2, −2, 3, −3, 4, −4…

Remarkable Feature

Positive exponents (1, 2, 3…) correspond to the “direct” fundamental dimensions (time, length, mass), while negative exponents (−1, −2, −3…) generate their “variations” (frequency, curvature, density).

Deeper Implication

The ArXe framework suggests that the dimensional structure of physics is not arbitrary but emerges naturally from the architecture of logical recursion.

Physical Units System by Exentation Exponent

Fundamental Assignment

System basis:

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

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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

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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”

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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)”

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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”

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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”

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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

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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

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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)

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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

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