A Newtonian Model for Bitcoin Network Value Created by Reddit user u/Obvireal 03/11/2025 Table of Contents • Abstract • 1. Introduction • 2. Theoretical Framework: The Bitcoin Newtonian Network Model • 2.1. Newton’s Law of Universal Gravitation (The Analogy) • 2.2. The Bitcoin Network Value Equation • 2.3. Deriving the Bitcoin Constant (G) • 2.4. Historical Analysis and Evolution of the Bitcoin Constant (G) • 2.5. G Value in 2013 • 2.6. G Value in 2017 • 2.7. Rate of Change and Future Implications • 2.8. Possible Future G Values • 2.9. Economic Friction (r) • 3. Conceptual Applications and Implications • 3.1. Example Calculations • 3.2. Future Scenarios • 4. Discussion • 4.1. Strengths and Limitations • 4.2. Impact of Friction • 5. Comparing to Metcalfe’s Law • 5.1. Metcalfe’s Law Applied to Bitcoin • 5.2. Comparing the Two Models • 5.3. End Results under Universal Adoption Abstract Bitcoin imposes unique challenges for valuation and understanding its network dynamics. In this paper, we introduce a conceptual framework that draws an analogy to Newton’s law of universal gravitation in physics. In our model, Bitcoin’s network value is treated similarly to gravitational force: it is proportional to the product of “economic mass” (capital and users) and inversely proportional to the square of “economic friction.” 1. Introduction Bitcoin’s worth is closely linked to its network properties such as decentralization, scarcity, and security. In our “Newtonian Network” approach, we suggest that Bitcoin’s network value can be modeled by factors analogous to those in Newton’s law of gravitation. We outline the model, explain its components, and explore its potential implications while noting its limitations. 2. Theoretical Framework: The Bitcoin Newtonian Network Model 2.1. Newton’s Law of Universal Gravitation (The Analogy) In physics, Newton’s law of gravitation is written as: G · m₁ · m₂ F = ———————————— r² Where: • F is the gravitational force, • m₁ and m₂ are the masses of two objects, • r is the distance between their centers • G is the gravitational constant. This law tells us that the force is proportional to the product of the masses and decreases with the square of the distance between them. 2.2. The Bitcoin Network Value Equation Drawing from this idea, we propose a similar equation for Bitcoin’s network value. We define: G · C · U F = —————————— r² Where: • F represents the total network value or “economic gravity” of Bitcoin. • G is the “Bitcoin Constant,” a factor that is derived from current market value. • C is the total capital invested in Bitcoin (analogous to mass in physics). • U is the number of active users (another form of economic mass). • r is “economic friction”, a measure of barriers and user attractiveness. 2.3. Deriving the Bitcoin Constant (G) for 2025 Unlike the fixed gravitational constant in physics, our Bitcoin constant (G) is derived from current market data and can change. By using approximate values when Bitcoin’s network had: • F = $2 trillion • C = $500 billion (Total net inflows estimate 2009-2025 based off network effects described by Metcalfe's Law. $2 trillion/4) • U = 100 million users (current number of users) • r = 1 (baseline friction, USD = 1. G can change with a changing r.) We plug these into our equation: G · ($500 billion × 100 million) 2 trillion = ———————————— 1² Solving for G gives: G = (2 × 10¹²) / (500 billion × 100 million) = 4 × 10 ⁸ or 0.00000004 ⁻ 2.4. Historical Analysis and Evolution of the Bitcoin Constant (G) In our model, the Bitcoin Constant (G) is defined by the relationship: (F × r²) G = ——— (C × U) Because G is derived from market data, it is not fixed over time but evolves as the Bitcoin ecosystem matures. Below, we calculate G for two distinct periods and discuss the implications. 2.5. G Value in 2013 For this year, we use the following parameters: • F = $21 billion • C = $5.25 billion (Derived from Metcalfes Law) • U = 1 million users • r = 1 Calculation: G = (21×10 ) / (5.25×10 × 1×10⁶) ⁹ ⁹ = 21×10 / 5.25×10¹⁵ ⁹ = 4 ×10 ⁶ = 0.000004 ⁻ In 2013, the relatively high G value suggests that, given a modest capital base and user count, the network was assigned a high value per unit of economic mass—possibly reflecting speculative excitement and limited market depth at the time. 2.6. G Value in 2017 For the year 2017, the parameters are: • F = $400 billion • C = $100 billion • U = 18 million users • r = 1 Calculation: G = (400×10 ) / (100×10 × 18×10⁶) ⁹ ⁹ = 400×10 / 1.8×10¹⁸ ⁹ ≈ 2.22×10 ⁷ 0.000000222 ⁻ Compared to 2013, the G value in 2017 is significantly lower. This steep decline reflects the rapid scaling of both capital and user base relative to the overall network value, as the market matured and more participants joined. 2.7. Rate of Change and Future Implications The transition from 2013 to 2017 shows a decrease in G from approximately 4×10 ⁶ to 2.22×10 ⁷ a ⁻ ⁻ reduction by a factor of roughly 18. 2017 (2.22×10 ⁷) to 2025 (4 × 10 ⁸) experienced a reduction of 5.5. ⁻ ⁻ This decline indicates that while the network's total value (F) increased, the proportional growth in economic mass (C × U) was even more pronounced. Such a trend suggests: • Market Maturation: As Bitcoin’s ecosystem grows, the incremental contribution of each additional dollar of capital or user tends to yield lower marginal increases in network value. In other words, the efficiency with which capital and users translate into “economic gravity” diminishes over time. • Diminishing Returns: A lower G value in later years may signal that the network is reaching a stage where further increases in capital and users contribute less dramatically to the overall valuation—a possible stabilization as market saturation is approached. • Future Recalibration: If the trend continues, future G values might decline further. However, shifts in technological innovation, regulatory frameworks, and global economic conditions could alter this trajectory. As the model is sensitive to the variables involved, periodic recalibration might be necessary to ensure that G remains a relevant indicator of Bitcoin’s network efficiency. • Policy and Strategy Insights: Understanding the rate of change in G over time can help stakeholders gauge the impact of new investments, user growth strategies, and friction-reducing measures. For instance, policy changes that reduce economic friction (r) could counterbalance the declining trend in G, potentially boosting overall network value even as the market matures. The evolution of the Bitcoin Constant (G) from 2013 to 2017 highlights the dynamic interplay between market growth and network efficiency. Monitoring how G changes can offer valuable insights into the future development and valuation of the Bitcoin ecosystem. 2.8. Possible Future G values The past evolution of the Bitcoin constant (G) provides a basis for forecasting its future trajectory. Between 2013 and 2017 there was a reduction by a factor of roughly 18 over 4 years. From 2017 to 2025, G had a reduction by a factor of about 5.5 over 8 years. To estimate future values, we first calculate the implied annual decay factor over the 2017–2025 period. The decay factor (d) is given by: d = (G ₂₀₂₅ / G ₂₀₁₇ )^(1/8) ₍ ₎ ₍ ₎ Substituting the values: d = (4×10 ⁸ / 2.22×10 ⁷)^(1/8) ≈ (0.18018)^(1/8) ≈ 0.807 ⁻ ⁻ This means that each year, on average, G is approximately 80.7% of its value in the previous year, a reduction of about 19.3% per year. Using this annual decay factor, we can project G into the future. Assuming that the 2017–2025 trend continues and taking G ₂₀₂₅ = 4×10 ⁸ as our starting point: ₍ ₎ ⁻ • For 2030 (5 years after 2025): G ₂₀₃₀ = 4×10 ⁸ × (0.807)⁵ ₍ ₎ ⁻ (0.807)⁵ ≈ 0.342 G ₂₀₃₀ ≈ 4×10 ⁸ × 0.342 ≈ 1.37×10 ⁸ ₍ ₎ ⁻ ⁻ ⇒ • For 2035 (10 years after 2025): G ₂₀₃₅ = 4×10 ⁸ × (0.807)¹ ₍ ₎ ⁻ ⁰ (0.807)¹ ≈ 0.117 ⁰ G ₂₀₃₅ ≈ 4×10 ⁸ × 0.117 ≈ 4.68×10 ⇒ ₍ ₎ ⁻ ⁻⁹ • For 2040 (15 years after 2025): G ₂₀₄₀ = 4×10 ⁸ × (0.807)¹⁵ ₍ ₎ ⁻ (0.807)¹⁵ ≈ 0.040 G ₂₀₄₀ ≈ 4×10 ⁸ × 0.040 ≈ 1.6×10 ⇒ ₍ ₎ ⁻ ⁻⁹ These calculations suggest that if the recent trend persists, the Bitcoin constant could decline significantly over the next couple of decades. Such a trend would imply that even as the network’s total value (F) increases, the proportional contribution of additional capital and users (C × U) becomes less impactful. In practical terms, this means that as the Bitcoin ecosystem continues to mature and expand, the marginal contribution of each additional unit of economic mass to overall network value will decrease, highlighting a stabilization in the efficiency of value generation. 2.9. Economic Friction (r) The term r (economic friction) represents the various challenges or costs involved in using the Bitcoin network. These may include: • Regulatory Issues: Unclear or strict regulations increase friction. • Transaction Fees: Higher fees can discourage use. • User Experience: Complex interfaces make it harder for new users. • Technological Barriers: Issues like limited internet access or hardware requirements. • Psychological Factors: Skepticism or perceived risk. For our model, we use a simplified approach by treating r as a single, measurable value. Setting r = 1 serves as a baseline; lower values would represent an easier-to-use system, while higher values indicate more barriers. 3. Conceptual Applications and Implications These examples calculate F for the year 2025 with different parameters. 3.1. Example Calculations Example 1 We use the Newtonian model equation for network value: G · C · U F = —————————— r² With parameters: • G = 4×10 ⁸ ⁻ • C = $1 trillion (2× today’s value) • U = 100 million users • r = 1 Calculation: F = (4×10 ⁸ × 1×10¹² × 1×10⁸) / 1² ⁻ = 4×10¹² This implies a network value of $4 trillion . Assuming a total supply of 21 million BTC, the estimated price per Bitcoin is: 1 BTC ≈ $4×10¹² / 21×10⁶ ≈ $190,000 Example 2 Suppose we increase capital to the current value of all currencies and the user base to the total human population. This is what it would look like if it happened in 2025 according to the model. Parameters: • G = 4×10 ⁸ ⁻ • C = $100 trillion • U = 8 billion users • r = 1 Calculation: F = (4×10 ⁸ × 1×10¹⁴ × 8×10 ) / 1² ⁻ ⁹ = (4 × 8 × 10^(14+9–8)) = 32×10¹⁵ = 3.2×10¹⁶ This leads to a network value of $32 quadrillion . Using a 21 million BTC supply, 1 BTC is estimated at: 1 BTC ≈ $3.2×10¹⁶ / 21×10⁶ ≈ $1.52 billion Example 3 Assume: • G: 4×10 ⁸ ⁻ • C: 5% of global wealth (Assuming global wealth is approximately $400 trillion (some say $900 trillion), then 5% is $20 trillion.) • U: 10% of the global population (With a global population of about 8 billion, 10% is 800 million users.) • r: 1 Calculation: 1. Multiply C and U: C × U = (20×10¹²) × (8×10⁸) = 160×10² ⁰ = 1.6×10²² 2. Multiply by G: F = (4×10 ⁸) × (1.6×10²²) ⁻ = 6.4×10^(22–8) = 6.4×10¹⁴ This yields a network value of $640 trillion ; hence, 1 BTC is: 1 BTC ≈ $6.4×10¹⁴ / 21×10⁶ ≈ $30 million Example 4 – Impact of Economic Friction (r) Using the base parameters of Example 1 wi th G = 4×10 ⁸ ⁻ , C = $1 trillion, and U = 100 million: Scenario A: Friction r = 0.8 1. Compute r²: r² = 0.8² = 0.64 2. Calculate F: F = (4×10 ⁸ × 1×10¹² × 1×10⁸) / 0.64 ⁻ = (4×10¹²) / 0.64 ≈ 6.25×10¹² Estimated network value: $6.25 trillion Price per BTC: 1 BTC ≈ $6.25×10¹² / 21×10⁶ ≈ $298,000 Scenario B: Friction r = 1.2 1. Compute r²: r² = 1.2² = 1.44 2. Calculate F: F = (4×10 ⁸ × 1×10¹² × 1×10⁸) / 1.44 ⁻ = (4×10¹²) / 1.44 ≈ 2.78×10¹² Estimated network value: $2.78 trillion Price per BTC: 1 BTC ≈ $2.78×10¹² / 21×10⁶ ≈ $132,000 3.2. Future Scenarios Below are revised examples using whole‐number representations for all parameters. In each case, we assume full adoption levels are global wealth of $400 trillion and 8 billion people, and compute the network value F with the Newtonian model: F = (G × C × U) / r² (r = 1, the value can sway +-50% with different r values.) Then, using a 21 million BTC supply, the estimated BTC price is: 1 BTC ≈ F / 21,000,000 For the Year 2030 (5 years out; G = 0.0000000137 or 1.37×10 ⁸): ⁻ Example (2030, 5% Adoption) Suppose 5% of global wealth and population adopt Bitcoin. Parameters: • G = 0.0000000137 • C = $20,000,000,000,000 (5% of $400 trillion) • U = 400,000,000 users (5% of 8 billion) • r = 1 Calculation: F = (0.0000000137 × 20,000,000,000,000 × 400,000,000) = 109,600,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 109,600,000,000,000 / 21,000,000 ≈ $5,219,048 Example (2030, 10% Adoption) Suppose 10% adoption. Parameters: • G = 0.0000000137 • C = $40,000,000,000,000 (10% of $400 trillion) • U = 800,000,000 users (10% of 8 billion) • r = 1 Calculation: F = (0.0000000137 × 40,000,000,000,000 × 800,000,000) = 438,400,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 438,400,000,000,000 / 21,000,000 ≈ $20,876,190 Example (2030, 25% Adoption) Suppose 25% adoption. Parameters: • G = 0.0000000137 • C = $100,000,000,000,000 (25% of $400 trillion) • U = 2,000,000,000 users (25% of 8 billion) • r = 1 Calculation: F = (0.0000000137 × 100,000,000,000,000 × 2,000,000,000) = 2,740,000,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 2,740,000,000,000,000 / 21,000,000 ≈ $130,476,190 Example (2030, 50% Adoption) Suppose 50% adoption. Parameters: • G = 0.0000000137 • C = $200,000,000,000,000 (50% of $400 trillion) • U = 4,000,000,000 users (50% of 8 billion) • r = 1 Calculation: F = (0.0000000137 × 200,000,000,000,000 × 4,000,000,000) = 10,960,000,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 10,960,000,000,000,000 / 21,000,000 ≈ $522,857,143 For the Year 2035 (10 years out; G = 0.00000000468 or 4.68×10 ): ⁻⁹ Example (2035, 5% Adoption) Parameters: • G = 0.00000000468 • C = $20,000,000,000,000 • U = 400,000,000 users • r = 1 Calculation: F = (0.00000000468 × 20,000,000,000,000 × 400,000,000) = 37,440,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 37,440,000,000,000 / 21,000,000 ≈ $1,782,857 Example (2035, 10% Adoption) Parameters: • G = 0.00000000468 • C = $40,000,000,000,000 • U = 800,000,000 users • r = 1 Calculation: F = (0.00000000468 × 40,000,000,000,000 × 800,000,000) = 149,760,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 149,760,000,000,000 / 21,000,000 ≈ $7,131,429 Example (2035, 25% Adoption) Parameters: • G = 0.00000000468 • C = $100,000,000,000,000 • U = 2,000,000,000 users • r = 1 Calculation: F = (0.00000000468 × 100,000,000,000,000 × 2,000,000,000) = 936,000,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 936,000,000,000,000 / 21,000,000 ≈ $44,571,429 Example (2035, 50% Adoption) Parameters: • G = 0.00000000468 • C = $200,000,000,000,000 • U = 4,000,000,000 users • r = 1 Calculation: F = (0.00000000468 × 200,000,000,000,000 × 4,000,000,000) = 3,744,000,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 3,744,000,000,000,000 / 21,000,000 ≈ $178,285,714 For the Year 2040 (15 years out; G = 0.0000000016 or 1.6×10 ): ⁻⁹ Example (2040, 5% Adoption) Parameters: • G = 0.0000000016 • C = $20,000,000,000,000 • U = 400,000,000 users • r = 1 Calculation: F = (0.0000000016 × 20,000,000,000,000 × 400,000,000) = 12,800,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 12,800,000,000,000 / 21,000,000 ≈ $609,524 Example (2040, 10% Adoption) Parameters: • G = 0.0000000016 • C = $40,000,000,000,000 • U = 800,000,000 users • r = 1 Calculation: F = (0.0000000016 × 40,000,000,000,000 × 800,000,000) = 51,200,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 51,200,000,000,000 / 21,000,000 ≈ $2,438,095 Example (2040, 25% Adoption) Parameters: • G = 0.0000000016 • C = $100,000,000,000,000 • U = 2,000,000,000 users • r = 1 Calculation: F = (0.0000000016 × 100,000,000,000,000 × 2,000,000,000) = 320,000,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 320,000,000,000,000 / 21,000,000 ≈ $15,238,095 Example (2040, 50% Adoption) Parameters: • G = 0.0000000016 • C = $200,000,000,000,000 • U = 4,000,000,000 users • r = 1 Calculation: F = (0.0000000016 × 200,000,000,000,000 × 4,000,000,000) = 1,280,000,000,000,000 Estimated 1 BTC price: 1 BTC ≈ 1,280,000,000,000,000 / 21,000,000 ≈ $60,952,381 These calculations show the faster Bitcoin is adopted the more expensive it becomes. 4. Discussion 4.1. Strengths and Limitations The Bitcoin Newtonian Network model, inspired by classical gravitational theory, provides a different way to view Bitcoin’s network value. Although it is an analogy and not a precise scientific law, the model offers several insights: • Visualization: It helps us understand how increases in capital and user adoption can multiply network value. • Friction’s Role: The inverse-square relationship with friction highlights how even modest improvements in usability and regulation could lead to large gains. • Strategic Implications: This framework can inform decisions on enhancing the Bitcoin ecosystem—from improving user experience to advocating for clearer regulations. Limitations: • Complex Friction: In reality, r is a composite measure that includes many qualitative factors. • Simplicity: The model simplifies many aspects of a complex economic system and should be seen as a conceptual tool rather than a precise predictor. • Derivations from Metcalfes Law 4.2. Impact of Friction Because friction (r) appears in the denominator as r², small changes in r have a large impact on F. For instance: • Halving r (r = 0.5): The network value increases by a factor of 4. • Doubling r (r = 2): The network value decreases by a factor of 4. This underscores the importance of reducing friction in the Bitcoin ecosystem to boost overall value. 5. Comparing to Metcalfe’s Law Metcalfe’s Law states that the value of a network is proportional to the square of the number of its users. In simpler terms, if you double the number of users, the network’s value increases roughly by a factor of four. Mathematically, this can be expressed as: Value n² where n is the number of users. ∝ 5.1. Metcalfe’s Law Applied to Bitcoin Let’s consider a scenario where Bitcoin’s current price is $80,000 per coin with 50 million users. According to Metcalfe’s Law, if the number of users grows, the network’s value should scale with the square of that change. Example 1 1. Determine the Scaling Factor: - Current users: 100 million - Potential users: 200 million - Scaling factor = 200/100 = 2 2. Apply the Square Law: - According to Metcalfe’s Law, network value scales with the square of the user increase. - Increase in value ≈ 2² = 4 3. Calculate the Theoretical Price: - With Metcalfe’s Law: Price ≈ $80,000 × 4 = $320,000 per Bitcoin - Using the Newtonian Model with our revised G: Approximately $190,000 per Bitcoin Example 2 1. Determine the Scaling Factor: - Current users: 100 million - Potential users: 8 billion - Scaling factor = 8,000,000,000 / 100,000,000 = 80 2. Apply the Square Law: - Increase in value ≈ 80² = 6,400 3. Calculate the Theoretical Price: - With Metcalfe’s Law: Price ≈ $80,000 × 6,400 ≈ $512 million per Bitcoin - Using the Newtonian Model with our revised G: Approximately $1.5 billion per Bitcoin for r = 1. 5.2. Comparing the Two Models While Metcalfe’s Law focuses solely on the impact of network size, the Newtonian model incorporates additional factors: • C (Capital): Represents the total money invested. • U (Users): Represents the number of active participants. • r (Economic Friction): Captures barriers to network use (e.g., transaction fees, regulatory hurdles). • G (Bitcoin Constant): 4×10 ⁸ ⁻ , Calculated with current market data. In the Newtonian model: G · C · U F = —————————— r² Here, the network value (F) depends not only on the number of users (U) but also on the capital invested (C) and the friction (r). As a result, the Newtonian model is more sensitive to real-world factors that affect adoption and economic activity. It provides a more nuanced perspective that also accounts for the depth of capital inflows, the ease of network participation and diminishing returns. For example, if friction is reduced (a lower r), the network’s value increases dramatically, even if the number of users remains constant. 5.3 End Results under Universal Adoption • Both prediction models can have similar answers. • Metcalfe’s Law Prediction: If Bitcoin’s price is $80,000 with 100 million users, then scaling to 8 billion users suggests a theoretical price of approximately $512 million per coin. This prediction comes solely from the effect of network size, assuming other factors remain constant. • Newtonian Model Perspective: When extrapolating to 8 billion users, the Newtonian model’s prediction will vary depending on the assumptions about capital (C) and friction (r) in addition to user count. For example, if we assume a certain level of capital investment and set friction to 1, the model might predict a very high network value. However, because the Newtonian model factors in both capital and friction, its predictions offer a more nuanced view that can reflect real-world constraints. In other words, while Metcalfe’s Law provides a striking illustration of the potential impact of network effects, the Newtonian model reminds us that the actual value will depend on both the depth of capital inflows and the ease of network participation.