Geometric Factors and the Unity of Physical Law: How Identical Substrate Relationships Generate Diverse Formulas

J. Rogers, SE Ohio, 22 Jun 2025, 1403

Abstract

The Formula Forge framework reveals that apparently distinct physical laws often emerge from identical dimensional substrate relationships, differing only in geometric factors that encode specific physical constraints. Using escape velocity and orbital velocity as a case study, we demonstrate how the same fundamental energy scaling M/r generates two "different" formulas through distinct geometric embeddings. This analysis illuminates the deep structure of physical law derivation and explains why geometric factors must be incorporated into initial postulates rather than dimensional projections.

Introduction

Classical physics presents escape velocity v_e = √(2GM/r) and orbital velocity v_o = √(GM/r) as distinct formulas governing different physical phenomena. However, dimensional analysis reveals these as manifestations of the same underlying substrate relationship, differing only in geometric factors that encode the specific physical constraints of each scenario.

This paper examines how geometric factors function within the Formula Forge framework, demonstrating that they represent essential physics embedded in the initial postulate rather than mere mathematical corrections to dimensional analysis.

All Planck units are non reduced form.  1/2pi has nothing to do with unit scaling, in fact hbar is not only a profound misunderstanding but an active category error that conflates real geometry with unit scaling artifacts.

The Substrate Relationship

Both escape and orbital velocities emerge from the same fundamental energy scaling:

Substrate relationship: Kinetic energy scale ~ Gravitational potential energy scale

In dimensionless form: (v²/c²) ~ (GM/rc²)

This gives the base dimensional structure: v ~ √(GM/r)

The crucial insight is that this substrate relationship is identical for both scenarios. The physics at the deepest level concerns the balance between kinetic and gravitational energy scales.

Geometric Constraint Encoding

The geometric factors (√2 vs 1) arise from different physical constraints applied to the same substrate relationship:

Escape Velocity: √2 Factor

• Physical constraint: Total energy = 0
• Mathematical expression: ½mv² - GMm/r = 0
• Geometric factor: √2 emerges from the factor of ½ in kinetic energy
• Postulate: v/v_P ~ √2 × √(M/m_P) / √(r/l_P)

Orbital Velocity: Unity Factor

• Physical constraint: Centripetal force = Gravitational force
• Mathematical expression: mv²/r = GMm/r²
• Geometric factor: 1 (no additional factors)
• Postulate: v/v_P ~ 1 × √(M/m_P) / √(r/l_P)

The Postulate Structure

The key insight is that geometric factors must be incorporated into the initial postulate, not added as corrections to dimensional analysis:

Postulate = Geometric Factor × Substrate Relationship

This structure has profound implications:

1. Geometric factors encode physics: They represent the essential physical constraints that distinguish different scenarios
2. Substrate relationships are universal: The same dimensional scaling applies across multiple physical situations
3. Postulates are complete: They contain all necessary information for exact formula derivation

Why Geometric Factors Cannot Be Dimensional Projections

Geometric factors cannot emerge from pure dimensional analysis because:

1. Dimensional analysis is constraint-agnostic: It reveals scaling relationships but cannot distinguish between different physical scenarios that share the same dimensional structure

2. Geometric factors are dimensionless: They represent pure numbers arising from the geometric embedding of physical constraints, not unit conversions

3. Physical insight is required: Identifying the correct geometric factor requires understanding the specific physics of each scenario

4. Uniqueness of solutions: Without geometric factors, dimensional analysis would predict the same formula for all scenarios sharing the same substrate relationship

The Two-Level Structure of Physical Laws

This analysis reveals physical laws have a two-level structure:

Level 1: Substrate Relationships (Universal)

• Dimensional scaling between physical quantities
• Shared across multiple physical scenarios
• Revealed by dimensional analysis
• Example: v ~ √(GM/r) for all velocity-mass-radius relationships

Level 2: Geometric Embeddings (Scenario-Specific)

• Numerical factors encoding physical constraints
• Unique to specific physical situations
• Require physical insight to identify
• Example: √2 for escape, 1 for orbit

Implications for the Formula Forge Framework

This structure explains several key features of the Formula Forge:

1. Why postulates work: They combine universal substrate relationships with scenario-specific geometric factors

2. Where physics insight is needed: In identifying the correct geometric factors for each physical situation

3. Why the method is systematic: Once the postulate is correctly formulated, dimensional analysis mechanically produces the exact formula

4. How apparent complexity arises: A small number of substrate relationships, combined with various geometric embeddings, generates the apparent diversity of physical laws

Detailed Analysis: Four Related Gravitational Formulas

To fully understand how geometric constraints shape postulates, we examine four related gravitational formulas in detail:

1. Escape Velocity: v = √(2GM/r)

Physical Constraint: Object must have exactly zero total energy at infinite distance.

Energy Analysis:

• Total energy: E ~ KE + PE ~ ½mv² - Mm/r
• Escape condition: E = 0 (zero energy at infinity)
• Therefore: ½mv²  ~ Mm/r
• Solving: v² ~ 2M/r → v ~ √(2M/r)

Geometric Origin of √2: The factor of ½ in kinetic energy (½mv²) creates the factor of 2 under the square root when solving for velocity.

Formula Forge Postulate:

v/v_P = √2 × √(M/m_P) × √(l_P/r)

Constraint Encoding: The √2 factor directly encodes the "zero total energy" constraint. This is not a correction but the essential physics of escape - the object must have exactly enough kinetic energy to overcome the gravitational potential well.

2. Orbital Velocity: v = √(GM/r)

Physical Constraint: Centripetal force must exactly equal gravitational force for stable circular motion.

Force Analysis:

• Gravitational force: F_g ~ Mm/r²
• Required centripetal force: F_c = mv²/r
• Orbital condition: F_c ~ F_g
• Therefore: mv²/r ~ Mm/r²
• Solving: v² ~ M/r → v = √(M/r)
  
• Create the postulate v/V_P =  √((M/m_P)/(r/l_P))
• Solve for v in SI

Geometric Origin of Unity Factor: The direct force balance (F_c = F_g) involves no additional geometric factors - the centripetal acceleration v²/r directly balances the gravitational acceleration GM/r².

Formula Forge Postulate:

v/v_P = 1 × √(M/m_P) × √(l_P/r)

Constraint Encoding: The unity factor encodes "perfect circular force balance" - no excess or deficit of centripetal force. This represents the precise balance required for stable orbital motion.

3. Parabolic Velocity: v = √(2GM/r)

Physical Constraint: Object follows a parabolic trajectory with total energy exactly zero.

Energy Analysis:

• Parabolic orbit: eccentricity e = 1
• Total energy: E ~ -M/2a (where a is semi-major axis)
• For parabolic orbit: a → ∞, so E → 0
• At periapsis: E ~ ½mv² - Mm/r = 0
• Therefore: ½mv² ~ Mm/r
• Solving: v² ~ 2M/r → v ~ √(2M/r)

Geometric Origin of √2: Identical to escape velocity - the parabolic trajectory is the limiting case where the object just barely escapes on a curved path rather than radial escape.

Formula Forge Postulate:

v/v_P = √2 × √(M/m_P) × √(l_P/r)

Constraint Encoding: The √2 factor encodes "zero total energy trajectory" - the same constraint as escape velocity but applied to curved rather than radial motion. This reveals that escape velocity and parabolic velocity are the same physical phenomenon viewed in different coordinate systems.

4. Surface Gravity: g = GM/r²

Physical Constraint: Acceleration experienced by a test mass at rest on the surface.

Force Analysis:

• Gravitational force on test mass m: F ~ Mm/r²
• Acceleration: a ~ F/m ~ M/r²
• No motion involved - pure acceleration measurement

Dimensional Structure: Note that this has different dimensional structure than the velocities:

• Velocities: [LT⁻¹] ~ √([M][L³T⁻²]/[L]) = √([ML²T⁻²])
• Acceleration: [LT⁻²] ~ ([M][L³T⁻²]/[L²]) = [ML T⁻²]/[L²] = [LT⁻²]

Formula Forge Postulate:

g/a_P = (M/m_P) × (l_P/r)²

Where a_P is Planck acceleration.

Constraint Encoding: The constraint here is "stationary test mass" - no kinetic energy involved, only the direct measurement of gravitational field strength. The r² dependence comes from the inverse square law of gravitational field strength, not from energy balance.

How Constraints Shape Postulates

Energy-Based Constraints (Escape, Parabolic)

• Physical principle: Total energy = 0
• Mathematical consequence: Kinetic energy = Potential energy
• Geometric factor: √2 (from ½ in kinetic energy formula)
• Postulate form: v/v_P = √2 × √(M/m_P) × √(l_P/r)

Force-Based Constraints (Orbital)

• Physical principle: Centripetal force = Gravitational force
• Mathematical consequence: Direct force balance
• Geometric factor: 1 (no additional factors in force balance)
• Postulate form: v/v_P = 1 × √(M/m_P) × √(l_P/r)

Field-Based Constraints (Surface Gravity)

• Physical principle: Direct measurement of gravitational field
• Mathematical consequence: Inverse square law
• Geometric factor: Built into dimensional structure (r⁻²)
• Postulate form: g/a_P = (M/m_P) × (l_P/r)²

The Pattern Recognition

The Formula Forge succeeds because it systematically encodes these constraint types:

1. Identify the physical constraint (energy balance, force balance, field measurement)
2. Determine the mathematical consequence (factors of 2, unity, inverse powers)
3. Encode in the postulate (geometric factors, dimensional structure)
4. Apply dimensional analysis (mechanical derivation of exact formula)

This reveals why apparently "different" physical laws are actually the same substrate relationship viewed through different constraint systems. The constraints don't change the underlying physics (GM/r scaling) but determine how that physics manifests in the final formula.

Philosophical Implications

This analysis supports the broader Formula Forge thesis that physical laws are coordinate artifacts rather than fundamental truths:

1. Apparent diversity from underlying unity: What appear as different laws are projections of the same substrate relationship through different coordinate systems

2. Observer dependence: The choice of physical constraints (escape vs orbit) determines which geometric factor appears in the final formula

3. Coordinate system artifacts: The complexity of physics partially stems from the combinatorial explosion of ways to embed simple relationships in various geometric constraints

Conclusion

The identical substrate relationships underlying escape and orbital velocity demonstrate that geometric factors represent essential physics embedded in initial postulates, not corrections to dimensional analysis. This structure explains how the Formula Forge can systematically derive exact physical laws: by correctly identifying both the universal substrate relationship and the scenario-specific geometric embedding.

This analysis suggests that the apparent complexity of physics may arise largely from the combinatorial ways simple substrate relationships can be embedded in different geometric constraint systems. The Formula Forge framework provides a systematic method for decomposing this complexity into its fundamental components: universal dimensional scalings and scenario-specific geometric factors.

The deeper implication is that physical "laws" may be better understood as a catalog of the various ways fundamental energy and length scales can be geometrically embedded in different physical constraint systems, rather than as independent discoveries about nature's fundamental structure.

Appendix: Relativistic Escape Velocity

Physical Constraint: Object must have exactly zero total energy at infinite distance (same as classical)

Relativistic Energy Analysis:

• Total energy: E = KE + PE = (γ - 1)mc² - GMm/r
• Where γ = 1/√(1 - v²/c²) is the Lorentz factor
• Escape condition: E = 0 at infinity
• Therefore: (γ - 1)mc² = GMm/r

Solving for velocity:

    (γ - 1)c² = GM/r
    γ - 1 = GM/(rc²)
    γ = 1 + GM/(rc²)
    1/√(1 - v²/c²) = 1 + GM/(rc²)

Taking the reciprocal and squaring:

    1 - v²/c² = 1/[1 + GM/(rc²)]²
    v²/c² = 1 - 1/[1 + GM/(rc²)]²

For weak field approximation (GM/rc² << 1):

    1/[1 + GM/(rc²)]² ≈ 1 - 2GM/(rc²)
    v²/c² ≈ 2GM/(rc²)
    v² ≈ 2GM/r

This recovers the classical result! But for strong fields, we get the full relativistic correction.

The beautiful insight: The same geometric constraint (zero total energy) works perfectly - we just replace the classical kinetic energy with relativistic kinetic energy. The constraint remains identical; only the energy expression changes.

Dimensionless form: v ~ √(2M/r) with relativistic corrections encoded in the energy relationship, showing that even relativistic effects emerge from the same underlying proportional constraints

Let me show the Formula Forge postulate format for relativistic escape velocity:

Relativistic Escape Velocity Derivation

Step 1: Pure geometric constraint analysis (no constants)

    Relativistic KE: E_k ~ (γ - 1)mc²
    Gravitational PE: E_p ~ -Mm/r  
    Escape condition: E_k ~ E_p
    Therefore: (γ - 1)mc² ~ Mm/r

Step 2: Express as dimensionless postulate

For relativistic escape, the constraint becomes:

    (γ - 1) ~ M/(rc²/m)

In Planck units where c = 1, m_P = 1, this becomes:

    (γ - 1)/1 ~ (M/m_P)/(r/l_P × c²/c²)

The Formula Forge Postulate:

    γ - 1 = (M/m_P) × (l_P/r) × (c²/c²)

Which simplifies to:

    γ - 1 = (M/m_P) × (l_P/r)

Step 3: Solve for v in SI units

Substituting back:

    1/√(1 - v²/c²) - 1 = GM/(rc²)

Solving for v gives the full relativistic escape velocity formula.

The key insight: The postulate structure remains identical - we just encode the relativistic energy constraint (γ - 1 instead of ½mv²) in the dimensionless form. The same geometric constraint (zero total energy) projects through relativistic energy expressions instead of classical ones.

The relativistic correction emerges automatically from the constraint, not as an added complexity!