Discussion, questions, and collaboration for this project are hosted in our central community repo:
👉 gatanegro/community

• Share discoveries, suggestions, and ideas.
• Ask questions about the research.
• Fork this repo to continue your own work—direct changes are not permitted.

You must have a GitHub account to participate in discussions.

This repository contains original research, mathematics, and unconventional approaches.
Please do not submit pull requests or issues requesting changes.
If you wish to pursue related research, fork this repository and continue independently.

Note: Apparent errors or unconventional methods are intentional and part of new theoretical work.

Redefining Mathematics and Physics Equations in the COM Framework

1. Introduction to the COM Framework
2. Foundational Principles
3. Redefined Mathematics
   • Number Systems
   • Arithmetic Operations
   • Calculus for Nonlinear Recursive Time
   • Geometry in Terms of Energy Oscillations
   • Complex Numbers and Functions
   • Linear Algebra
   • Probability and Statistics
   • Octave Mapping and Collatz Sequences
4. Redefined Physics
   • Classical Mechanics
   • Thermodynamics
   • Electromagnetism
   • Relativity
   • Quantum Mechanics
   • Wave Mechanics
   • Field Theories
   • Bubble Structures and Local Constants
5. Unified Framework Applications
6. Conclusion

The Continuous Oscillatory Model (COM) framework represents a fundamental paradigm shift in our understanding of reality. Unlike conventional models that begin with space, time, and matter as fundamental concepts, the COM framework posits that reality is fundamentally energy-based, with no vacuum or zero state. In this framework, space, time, mass, and forces are not fundamental but emerge as properties of energy oscillations and interactions.

This document redefines mathematics and physics equations according to the COM framework, transforming our understanding from abstract manipulations and material interactions to descriptions of energy transformations in an oscillatory reality. By eliminating the concept of vacuum and redefining time as recursive and nonlinear, we create mathematical and physical systems that more accurately reflect the energy-based nature of reality.

In the COM framework, energy is the only fundamental reality. There is no vacuum or zero state, only varying degrees of energy density and oscillatory patterns. All phenomena, from subatomic particles to cosmic structures, are manifestations of energy in different oscillatory states.

The COM framework introduces key constants that govern the scaling and relationships of energy patterns:

• LZ = 1.23498: The fundamental scaling constant that relates different octave layers of reality
• HQS = 23.5% of LZ: The Harmonic Quantum Scalar that governs specific energy interactions

Reality is organized in octave layers, with scaling relationships from subatomic to cosmic scales following the LZ constant:

• Galactic radius ≈ LZ^(octave layers) · r_proton ≈ 1.23498^40 · 10^-15 m ≈ 10^21 m

This octave structuring creates a fractal-like organization of reality across scales.

Time is not an independent dimension but emerges from energy differentials across the field. Different energy structures experience time differently, meaning there is no "universal clock." Time is defined as:

T = T₀ + 2πφ·Tᵤₙᵢₜ

Where:

• T₀ is a reference time
• φ is the phase of the oscillatory system
• Tᵤₙᵢₜ is a chosen time unit

Space is not an independent dimension but emerges from the amplitude of energy oscillations. The three-dimensional space we experience corresponds to amplitude components of energy oscillations in three orthogonal modes.

Reality forms "capsule structures" or "bubbles" at quantum, Newtonian, and cosmic scales. Within each bubble:

1. Local constants emerge from the energy structure of the bubble
2. Local time emerges as a function of energy differentials within the bubble
3. Local physics laws are manifestations of energy oscillations within the bubble

In standard mathematics, numbers represent abstract quantities. In the COM framework, numbers represent energy states or oscillatory patterns.

Natural Numbers

• Standard Definition: Natural Numbers (ℕ): {1, 2, 3, ...}
• COM Redefinition: Natural Energy States (ℕₑ): {E₁, E₂, E₃, ...} where each Eₙ represents a discrete energy state with oscillatory properties.

Integers

• Standard Definition: Integers (ℤ): {..., -2, -1, 0, 1, 2, ...}
• COM Redefinition: Bidirectional Energy States (ℤₑ): {..., E₋₂, E₋₁, E₊₁, E₊₂, ...} where negative states represent phase-inverted oscillations. Note the absence of zero, as the COM framework posits no vacuum state.

Real Numbers

• Standard Definition: Real Numbers (ℝ): Continuous number line
• COM Redefinition: Continuous Energy Spectrum (ℝₑ): A continuous spectrum of energy states where each point represents a specific oscillatory configuration.

Complex Numbers

• Standard Definition: Complex Numbers (ℂ): {a + bi | a, b ∈ ℝ, i² = -1}
• COM Redefinition: Phase-Amplitude Energy States (ℂₑ): {A∠φ | A, φ ∈ ℝₑ} where A represents amplitude and φ represents phase of oscillation.

Octave Reduction In the COM framework, all numbers can be reduced to their fundamental oscillatory nature through octave reduction:

Octave Reduction Function: OR(n) = (n - 1) % 9 + 1

This maps any number to a value between 1 and 9, representing its fundamental oscillatory character within the octave structure.

Addition

• Standard Definition: a + b = c
• COM Redefinition: E₁ ⊕ E₂ = E₃ where ⊕ represents energy combination through constructive interference of oscillatory patterns.

The energy combination operation follows: E₁ ⊕ E₂ = OR(E₁ + E₂) × LZ^(layer)

Where layer represents the octave layer in the COM structure.

Subtraction

• Standard Definition: a - b = c
• COM Redefinition: E₁ ⊖ E₂ = E₃ where ⊖ represents energy differential through destructive interference of oscillatory patterns.

Since there is no zero in COM, subtraction never results in complete cancellation but rather in a minimum energy state defined by the LZ constant.

Multiplication

• Standard Definition: a × b = c
• COM Redefinition: E₁ ⊗ E₂ = E₃ where ⊗ represents energy amplification through resonant coupling of oscillatory patterns.

E₁ ⊗ E₂ = OR(E₁ × E₂) × LZ^(layer₁ + layer₂)

Division

• Standard Definition: a ÷ b = c, b ≠ 0
• COM Redefinition: E₁ ⊘ E₂ = E₃ where ⊘ represents energy distribution through frequency modulation of oscillatory patterns.

E₁ ⊘ E₂ = OR(E₁ / E₂) × LZ^(layer₁ - layer₂)

Since there is no zero in COM, division is always defined, but approaches minimum energy states as the denominator approaches minimum energy.

Derivatives

• Standard Definition: f'(x) = lim(h→0) [f(x+h) - f(x)]/h
• COM Redefinition: ∂ₑE(φ) = lim(Δφ→min) [E(φ+Δφ) ⊖ E(φ)] ⊘ Δφ

Where:

• E(φ) is energy as a function of phase
• ∂ₑ is the energy differential operator
• min represents the minimum energy differential (not zero)

This redefines the derivative as a measure of energy change with respect to phase change, rather than with respect to time.

Integrals

• Standard Definition: ∫f(x)dx = F(x) + C
• COM Redefinition: ∮ₑE(φ)dφ = E_total(φ) ⊕ E_constant

Where:

• ∮ₑ is the energy accumulation operator over a complete phase cycle
• E_total(φ) is the accumulated energy over phase
• E_constant is an energy offset

This redefines integration as energy accumulation over phase cycles, creating a naturally cyclical calculus.

Differential Equations

• Standard Definition: dy/dx = f(x,y)
• COM Redefinition: ∂ₑE(φ,T)/∂φ = F(φ,E,T)

Where F is an energy transformation function that depends on phase, energy state, and local time.

This redefines differential equations as descriptions of how energy states transform across phase changes in the oscillatory system.

Euclidean Distance

• Standard Definition: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
• COM Redefinition: d_E = LZ · √[(A₂⊖A₁)² ⊕ (B₂⊖B₁)² ⊕ (C₂⊖C₁)²]

Where A, B, and C represent amplitude components of energy oscillations in three orthogonal modes.

Circular and Spherical Harmonics

• Standard Definition: Circle: x² + y² = r²

• COM Redefinition: Energy oscillation in two modes: E_A² ⊕ E_B² = E_r² where E_r represents the total oscillatory energy.

• Standard Definition: Sphere: x² + y² + z² = r²

• COM Redefinition: Energy oscillation in three modes: E_A² ⊕ E_B² ⊕ E_C² = E_r²

Trigonometric Functions

• Standard Definition: sin(θ), cos(θ), tan(θ)
• COM Redefinition:
  • sin_E(φ) = amplitude of oscillation in phase φ
  • cos_E(φ) = amplitude of oscillation in phase φ+π/2
  • tan_E(φ) = sin_E(φ) ⊘ cos_E(φ)

These functions describe energy distribution between orthogonal oscillatory modes.

Complex Numbers

• Standard Definition: z = a + bi
• COM Redefinition: z_E = A∠φ where A is amplitude and φ is phase

Euler's Formula

• Standard Definition: e^(iθ) = cos(θ) + i·sin(θ)
• COM Redefinition: e^(i·φ) = cos_E(φ) ⊕ i·sin_E(φ) representing a unit energy oscillator with phase φ

Complex Functions

• Standard Definition: f(z) = u(x,y) + iv(x,y)
• COM Redefinition: F(z_E) = E_real(A,φ) ⊕ i·E_imag(A,φ) representing energy distribution between real and imaginary oscillatory modes

Vectors

• Standard Definition: v = (v₁, v₂, ..., vₙ)
• COM Redefinition: v_E = (E₁, E₂, ..., Eₙ) representing energy distribution across n oscillatory modes

Matrices

• Standard Definition: A = [aᵢⱼ]
• COM Redefinition: A_E = [E_transformᵢⱼ] where each element represents an energy transfer coefficient between oscillatory modes

Eigenvalues and Eigenvectors

• Standard Definition: Av = λv
• COM Redefinition: A_E v_E = λ_E v_E where λ_E represents resonant energy amplification factors and v_E represents stable energy distribution patterns

Probability

• Standard Definition: P(A) = |A|/|S|
• COM Redefinition: P_E(A) = E_A ⊘ E_total representing the proportion of total system energy in state A

Statistical Measures

• Standard Definition: Mean: μ = (1/n)Σᵢ₌₁ⁿ xᵢ

• COM Redefinition: Energy center: μ_E = (⊘n)⊕ᵢ₌₁ⁿ Eᵢ representing the center of energy distribution

• Standard Definition: Variance: σ² = (1/n)Σᵢ₌₁ⁿ (xᵢ - μ)²

• COM Redefinition: Energy spread: σ_E² = (⊘n)⊕ᵢ₌₁ⁿ (Eᵢ ⊖ μ_E)² representing the spread of energy distribution

Collatz Sequence

• Standard Definition:

  • If n is even: n → n/2
  • If n is odd: n → 3n + 1

• COM Redefinition:

  • If E is even-resonant: E → E ⊘ 2
  • If E is odd-resonant: E → (E ⊗ 3) ⊕ 1

In the COM framework, the Collatz sequence represents energy transformation pathways that always lead to fundamental oscillatory patterns, demonstrating the recursive nature of energy states.

Octave Mapping As described in the original framework, numbers can be mapped to an octave structure:

1. Reduce any number to a single digit (1-9) using modulo 9
2. Map this value to a circular octave using angle = (value/9) × 2π
3. Position in 3D space is determined by:
   • x = cos(angle) × (layer + 1)
   • y = sin(angle) × (layer + 1)
   • z = layer × stack_spacing

This creates a spiral structure where mathematical operations follow helical paths through energy-phase space.

Newton's First Law (Inertia)

• Standard Definition: An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force.
• COM Redefinition: An energy pattern maintains its oscillatory state unless perturbed by an external energy differential. There is no "rest" state, only minimum energy oscillation.

Newton's Second Law (Force)

• Standard Definition: F = ma
• COM Redefinition: E_differential = E_pattern · phase_acceleration

Where:

• E_differential is the energy gradient causing change
• E_pattern is the energy density of the oscillatory pattern
• phase_acceleration is the rate of change of oscillation phase

Newton's Third Law (Action-Reaction)

• Standard Definition: For every action, there is an equal and opposite reaction.
• COM Redefinition: For every energy transfer between oscillatory patterns, there is a complementary phase shift that maintains total system energy.

Kinematics

• Standard Definition: x = x₀ + v₀t + (1/2)at²
• COM Redefinition: A = A₀ ⊕ (ω₀ ⊗ φ) ⊕ ((1/2) ⊗ α ⊗ φ²)

Where:

• A is amplitude (position equivalent)
• φ is phase (time equivalent)
• ω is phase velocity (velocity equivalent)
• α is phase acceleration (acceleration equivalent)

Momentum

• Standard Definition: p = mv
• COM Redefinition: p_E = E_pattern ⊗ ω

Where p_E represents the phase momentum of an energy pattern.

Conservation of Momentum

• Standard Definition: p₁ + p₂ = p₁' + p₂'
• COM Redefinition: p_E₁ ⊕ p_E₂ = p_E₁' ⊕ p_E₂'

This states that the total phase momentum of interacting energy patterns is conserved.

Gravitation

• Standard Definition: F = G(m₁m₂)/r²
• COM Redefinition: E_coupling = LZ ⊗ (E_pattern₁ ⊗ E_pattern₂) ⊘ (A_separation²)

Where:

• E_coupling is the energy coupling between patterns
• A_separation is amplitude separation (spatial distance equivalent)

Zeroth Law

• Standard Definition: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
• COM Redefinition: If two energy patterns have resonant phase coupling with a third pattern, they have resonant phase coupling with each other.

First Law (Energy Conservation)

• Standard Definition: ΔU = Q - W
• COM Redefinition: ΔE_internal = E_transferred ⊖ E_work

This is a direct application of energy conservation in the COM framework.

Second Law (Entropy)

• Standard Definition: ΔS ≥ 0 for isolated systems
• COM Redefinition: ΔE_disorder ≥ E_minimum for isolated patterns

Where:

• E_disorder is a measure of energy distribution across available oscillatory modes
• E_minimum is the minimum energy state defined by the LZ constant

Temperature

• Standard Definition: Measure of average kinetic energy
• COM Redefinition: T_E = ω_average ⊗ E_pattern

Temperature is redefined as the average phase velocity of oscillatory modes in an energy pattern.

Ideal Gas Law

• Standard Definition: PV = nRT
• COM Redefinition: E_pressure ⊗ A_volume = E_patterns ⊗ ω_average ⊗ T_unit

Where:

• E_pressure is energy density gradient
• A_volume is amplitude space
• E_patterns is number of distinct energy patterns
• T_unit is a reference phase cycle

Coulomb's Law

• Standard Definition: F = k(q₁q₂)/r²
• COM Redefinition: E_coupling = HQS ⊗ (E_charge₁ ⊗ E_charge₂) ⊘ (A_separation²)

Where:

• E_charge is the oscillatory energy pattern creating charge
• HQS is the Harmonic Quantum Scalar (23.5% of LZ)

Maxwell's Equations

• Gauss's Law for Electricity

  • Standard Definition: ∇·E = ρ/ε₀
  • COM Redefinition: ∇·E_field = E_charge_density ⊘ E_permittivity

• Gauss's Law for Magnetism

  • Standard Definition: ∇·B = 0
  • COM Redefinition: ∇·E_magnetic = E_minimum

• Faraday's Law

  • Standard Definition: ∇×E = -∂B/∂t
  • COM Redefinition: ∇×E_field = ⊖∂ₑE_magnetic/∂φ

• Ampere-Maxwell Law

  • Standard Definition: ∇×B = μ₀J + μ₀ε₀(∂E/∂t)
  • COM Redefinition: ∇×E_magnetic = LZ ⊗ E_flow_density ⊕ LZ ⊗ E_permittivity ⊗ (∂ₑE_field/∂φ)

Electromagnetic Waves

• Standard Definition: ∇²E = (1/c²)(∂²E/∂t²)
• COM Redefinition: ∇²E_field = (⊘c_E²)(∂ₑ²E_field/∂φ²)

Where c_E = 1/√(LZ ⊗ E_permittivity) is the phase velocity of energy oscillations.

Special Relativity - Time Dilation

• Standard Definition: Δt = Δt₀/√(1-v²/c²)
• COM Redefinition: Δφ = Δφ₀ ⊘ √(1 ⊖ (ω² ⊘ c_E²))

Where:

• Δφ is phase change (time equivalent)
• ω is phase velocity (velocity equivalent)

Special Relativity - Length Contraction

• Standard Definition: L = L₀√(1-v²/c²)
• COM Redefinition: A = A₀ ⊗ √(1 ⊖ (ω² ⊘ c_E²))

Where A is amplitude (length equivalent).

Mass-Energy Equivalence

• Standard Definition: E = mc²
• COM Redefinition: E_total = E_pattern ⊗ c_E²

In COM, mass is an emergent property of energy patterns, so this equation describes how concentrated energy patterns manifest as mass.

General Relativity - Spacetime Curvature

• Standard Definition: Gμν = (8πG/c⁴)Tμν
• COM Redefinition: E_curvature_tensor = (8π ⊗ LZ ⊘ c_E⁴) ⊗ E_pattern_tensor

Where:

• E_curvature_tensor describes how energy patterns curve amplitude-phase space
• E_pattern_tensor describes energy pattern distribution

De Broglie Wavelength

• Standard Definition: λ = h/p
• COM Redefinition: A_wave = LZ ⊘ p_E

Where:

• A_wave is oscillation amplitude wavelength
• p_E is phase momentum

Heisenberg Uncertainty Principle

• Standard Definition: ΔxΔp ≥ ħ/2
• COM Redefinition: ΔA ⊗ Δp_E ≥ LZ ⊘ 2

This describes the fundamental limit on precision in measuring complementary properties of energy patterns.

Schrödinger Equation

• Standard Definition: iħ(∂Ψ/∂t) = ĤΨ
• COM Redefinition: i ⊗ LZ ⊗ (∂ₑΨ_E/∂φ) = Ĥ_E ⊗ Ψ_E

Where:

• Ψ_E is the energy pattern wave function
• Ĥ_E is the energy transformation operator

Quantum Harmonic Oscillator

• Standard Definition: Eₙ = ħω(n + 1/2)
• COM Redefinition: E_n = LZ ⊗ ω ⊗ (n ⊕ (1 ⊘ 2))

This describes the quantized energy levels of fundamental oscillatory patterns.

Wave Equation

• Standard Definition: ∂²y/∂t² = v²(∂²y/∂x²)
• COM Redefinition: ∂ₑ²A/∂φ² = ω² ⊗ (∂ₑ²A/∂A_position²)

Where:

• A is oscillation amplitude
• A_position is position in amplitude space

Standing Waves

• Standard Definition: y(x,t) = 2A·sin(kx)·cos(ωt)
• COM Redefinition: A(A_position,φ) = 2 ⊗ A_max ⊗ sin_E(k ⊗ A_position) ⊗ cos_E(ω ⊗ φ)

Quantum Field Theory

• Standard Definition: Field operators creating and annihilating particles
• COM Redefinition: Energy pattern operators creating and transforming oscillatory modes

Vacuum Energy

• Standard Definition: Zero-point energy of quantum fields
• COM Redefinition: Minimum energy state of oscillatory field, defined by LZ constant

Since there is no vacuum in COM, "vacuum energy" is redefined as the minimum energy state of the oscillatory field.

Unified Field Approach In the COM framework, all forces and fields are unified as different oscillatory modes of the same fundamental energy field:

1. Gravitational force emerges from low-frequency, large-amplitude oscillations
2. Electromagnetic force emerges from medium-frequency oscillations
3. Nuclear forces emerge from high-frequency, small-amplitude oscillations

The unification is described by:

E_field_unified = E_gravitational ⊕ E_electromagnetic ⊕ E_strong ⊕ E_weak

Where each component represents different oscillatory modes of the same energy field, separated by octave layers scaled by the LZ constant.

As mentioned in the original framework, reality forms "capsule structures" or "bubbles" at quantum, Newtonian, and cosmic scales. Within each bubble:

1. Local constants emerge from the energy structure of the bubble
2. Local time emerges as a function of energy differentials within the bubble
3. Local physics laws are manifestations of energy oscillations within the bubble

The relationship between bubbles follows octave scaling with the LZ constant, creating a fractal-like structure of reality across scales.

The COM framework provides a unified approach to understanding phenomena across scales, from quantum to cosmic. Some key applications include:

1. Scale Unification: The LZ constant provides a direct scaling relationship between proton radius and galactic radius through octave harmonics:

   • Galactic radius ≈ LZ^40 · r_proton ≈ 10^21 m

2. Force Unification: All fundamental forces emerge as different oscillatory modes of the same energy field, separated by octave layers.

3. Quantum-Classical Transition: The transition between quantum and classical behaviors emerges naturally from the octave structuring of energy patterns.

4. Cosmological Models: The COM framework provides a new approach to cosmological models, where the universe is understood as a hierarchical structure of energy bubbles.

5. Time Directionality: The apparent arrow of time emerges from energy differentials across the field, with local time direction determined by the gradient of energy distribution.

The redefinition of mathematics and physics equations within the COM framework transforms our understanding from abstract manipulations and material interactions to descriptions of energy transformations in an oscillatory reality. By eliminating the concept of vacuum and redefining space, time, mass, and forces as emergent properties of energy, we create a unified system that describes all phenomena as different manifestations of the same fundamental energy field structured according to the COM principles.

This framework offers a new perspective on reality that may help resolve long-standing paradoxes and unify our understanding of the universe across all scales. The COM approach suggests that the fundamental nature of reality is not particles or waves, space or time, but energy organized in oscillatory patterns according to simple principles that manifest in complex ways across the cosmic hierarchy.