Scale Invariance Research: Consolidated Report

Complete End-to-End Analysis with Live Data

Generated: 2025-11-09 20:42:33

Abstract

This report presents a comprehensive scale invariance analysis across multiple assets.

Data Sources:

All financial data downloaded LIVE from Yahoo Finance API

NO hardcoded data points

ALL data cached locally for reproducibility

Total Assets Analyzed: 19

Overall Comparison

Genesis Sensitivity Overview

Scale invariance strength varies with the assumed genesis date. The chart below shows how R² responds across all tested dates for each asset where genesis optimization was performed.

Summary Table

Asset Category R² Exponent Pattern Points Date Range Genesis Source

Internet Users Technology 0.9776 2.38 ⚡ Power-law 20 2005-01-01 to 2024-01-01 1983-01-01 (Provided) World Bank API (IT.NET.USER.ZS)

Broadband Subscriptions Technology 0.9580 2.03 ⚡ Power-law 20 2005-01-01 to 2024-01-01 1990-01-01 (Provided) World Bank API (IT.NET.BBND.P2)

McDonald's Restaurants Business 0.9301 1.10 ⚡ Power-law 130 2009-12-31 to 2025-09-30 1955-04-15 (Provided) Downloaded from SEC EDGAR (CIK: 0000063908), us

Electricity Access Infrastructure 0.9188 1.06 ⚡ Power-law 26 1998-01-01 to 2023-01-01 1880-01-01 (Provided) World Bank API (EG.ELC.ACCS.ZS)

Bitcoin Financial Asset 0.9079 5.74 ⚡ Power-law 5566 2010-07-18 to 2025-11-09 2009-01-03 (Provided) btcpricehistory.csv (repository data)

Microsoft Corp. Financial Asset 0.8151 4.91 ⚡ Power-law 9993 1986-03-13 to 2025-11-07 1975-04-04 (Provided) Yahoo Finance API (yfinance Python library)

Mobile Subscriptions Technology 0.7794 2.23 ✅ Saturated (103%) 36 1984-01-01 to 2024-01-01 1983-10-13 (Provided) World Bank API (IT.CEL.SETS.P2)

Tesla Inc. Financial Asset 0.7543 5.05 ✅ Saturated (171%) 3866 2010-06-29 to 2025-11-07 2003-07-01 (Provided) Yahoo Finance API (yfinance Python library)

Alphabet Inc. Financial Asset 0.7494 2.18 ⚡ Power-law 5341 2004-08-19 to 2025-11-07 1998-09-04 (Provided) Yahoo Finance API (yfinance Python library)

DTP3 Immunization Health 0.7149 0.05 ✅ Saturated (757%) 22 2003-01-01 to 2024-01-01 2002-05-26 (Optimized vs published) WHO API (WHS4_100)

Measles Immunization Health 0.6815 0.04 ✅ Saturated (735%) 22 2003-01-01 to 2024-01-01 2002-05-26 (Optimized vs published) WHO API (WHS4_544)

Gold ETF (SPDR) (Early Period) Financial Asset - Split 0.5747 0.33 🔹 Early 2639 2004-11-18 to 2015-05-14 2004-11-18 (Provided) Yahoo Finance API (yfinance Python library)

S&P 500 ETF (Early Period) Financial Asset - Split 0.5739 0.36 🔹 Early 4126 1993-01-29 to 2009-06-16 1993-01-22 (Provided) Yahoo Finance API (yfinance Python library)

Gold ETF (SPDR) Financial Asset 0.5594 0.34 🏒 Hockey (3.6x) 5277 2004-11-18 to 2025-11-07 2004-11-18 (Provided) Yahoo Finance API (yfinance Python library)

S&P 500 ETF (Late Period) Financial Asset - Split 0.5422 0.37 🔹 Late 4126 2009-06-17 to 2025-11-07 2009-06-17 (Hockey Stick Inflection) Yahoo Finance API (yfinance Python library)

S&P 500 ETF Financial Asset 0.4041 0.48 🏒 Hockey (7.7x) 8252 1993-01-29 to 2025-11-07 1993-01-22 (Provided) Yahoo Finance API (yfinance Python library)

Apple Inc. Financial Asset 0.4032 2.87 📈 S-curve (89%) 11319 1980-12-12 to 2025-11-07 1976-04-01 (Provided) Yahoo Finance API (yfinance Python library)

Ethereum Financial Asset 0.2740 1.90 ✅ Saturated (152%) 2921 2017-11-09 to 2025-11-07 2015-07-30 (Provided) Yahoo Finance API (yfinance Python library)

Gold ETF (SPDR) (Late Period) Financial Asset - Split 0.1743 0.12 🔹 Late 2638 2015-05-15 to 2025-11-07 2015-05-15 (Hockey Stick Inflection) Yahoo Finance API (yfinance Python library)

Business

McDonald's Restaurants

**R² (Scale Invariance):** 0.9301

**Power Law Exponent:** 1.10

**Data Points:** 130

**Date Range:** 2009-12-31 to 2025-09-30

**Log Residual Std Dev:** 0.0117

**Data Source:** Downloaded from SEC EDGAR (CIK: 0000063908), us

**Genesis Date:** 1955-04-15 (Provided)

**Power Law Fit:** Value ~= 0.5871 * t^1.1022

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.9767 vs Linear R²: 0.9729

**Current Value:** 0.0% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Financial Asset

Bitcoin

**R² (Scale Invariance):** 0.9079

**Power Law Exponent:** 5.74

**Data Points:** 5,566

**Date Range:** 2010-07-18 to 2025-11-09

**Log Residual Std Dev:** 0.6935

**Data Source:** btcpricehistory.csv (repository data)

**Genesis Date:** 2009-01-03 (Provided)

**Power Law Fit:** Value ~= 2.143e-17 * t^5.7385

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.8811 vs Linear R²: 0.6086

**Current Value:** 0.0% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Microsoft Corp.

**R² (Scale Invariance):** 0.8151

**Power Law Exponent:** 4.91

**Data Points:** 9,993

**Date Range:** 1986-03-13 to 2025-11-07

**Log Residual Std Dev:** 0.5848

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 1975-04-04 (Provided)

**Power Law Fit:** Value ~= 2.596e-19 * t^4.9138

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.9792 vs Linear R²: 0.5089

**Current Value:** 67.9% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Tesla Inc.

**R² (Scale Invariance):** 0.7543

**Power Law Exponent:** 5.05

**Data Points:** 3,866

**Date Range:** 2010-06-29 to 2025-11-07

**Log Residual Std Dev:** 0.5591

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 2003-07-01 (Provided)

**Power Law Fit:** Value ~= 5.278e-18 * t^5.0530

Saturation Analysis (S-Curve Detection):

**📈 S-CURVE DETECTED** (Sigmoid R² = 0.8865 vs Linear R² = 0.6847)

**Current Progress:** 171.4% of estimated maximum

**Saturation Level:** 2.51e+02 (estimated carrying capacity)

**✅ HAS REACHED SATURATION** (≥95% of maximum)

**Growth Pattern:** Decelerating (-244.1% slowdown in recent period)

Alphabet Inc.

**R² (Scale Invariance):** 0.7494

**Power Law Exponent:** 2.18

**Data Points:** 5,341

**Date Range:** 2004-08-19 to 2025-11-07

**Log Residual Std Dev:** 0.3312

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 1998-09-04 (Provided)

**Power Law Fit:** Value ~= 2.253e-07 * t^2.1758

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.9511 vs Linear R²: 0.7689

**Current Value:** 0.0% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Gold ETF (SPDR)

**R² (Scale Invariance):** 0.5594

**Power Law Exponent:** 0.34

**Data Points:** 5,277

**Date Range:** 2004-11-18 to 2025-11-07

**Log Residual Std Dev:** 0.2191

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 2004-11-18 (Provided)

**Power Law Fit:** Value ~= 8.49 * t^0.3366

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.7333 vs Linear R²: 0.6669

**Current Value:** 0.0% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Hockey Stick Pattern Detected:

**🏒 INFLECTION POINT:** 2015-05-15

**Early Period (Blade):**

- R² = 0.7670

- Exponent = 0.37

- Data points = 2638

**Late Period (Shaft):**

- R² = 0.8114

- Exponent = 1.34

- Data points = 2638

**Growth Acceleration:** 3.6x faster in late period

**Model Improvement:** Split model R² = 0.8504 vs Single R² = 0.7523 (+0.0981)

**Analysis:** Asset shows distinct regime change - early slow growth followed by acceleration

S&P 500 ETF

**R² (Scale Invariance):** 0.4041

**Power Law Exponent:** 0.48

**Data Points:** 8,252

**Date Range:** 1993-01-29 to 2025-11-07

**Log Residual Std Dev:** 0.5001

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 1993-01-22 (Provided)

**Power Law Fit:** Value ~= 2.132 * t^0.4757

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.9778 vs Linear R²: 0.7296

**Current Value:** 0.0% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Hockey Stick Pattern Detected:

**🏒 INFLECTION POINT:** 2009-06-17

**Early Period (Blade):**

- R² = 0.7576

- Exponent = 0.39

- Data points = 4126

**Late Period (Shaft):**

- R² = 0.9794

- Exponent = 3.03

- Data points = 4126

**Growth Acceleration:** 7.7x faster in late period

**Model Improvement:** Split model R² = 0.9600 vs Single R² = 0.7052 (+0.2547)

**Analysis:** Asset shows distinct regime change - early slow growth followed by acceleration

Apple Inc.

**R² (Scale Invariance):** 0.4032

**Power Law Exponent:** 2.87

**Data Points:** 11,319

**Date Range:** 1980-12-12 to 2025-11-07

**Log Residual Std Dev:** 1.4324

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 1976-04-01 (Provided)

**Power Law Fit:** Value ~= 9.735e-12 * t^2.8716

Saturation Analysis (S-Curve Detection):

**📈 S-CURVE DETECTED** (Sigmoid R² = 0.9805 vs Linear R² = 0.4632)

**Current Progress:** 88.7% of estimated maximum

**Saturation Level:** 3.03e+02 (estimated carrying capacity)

**Estimated Time to 95% Saturation:** ~4.1 years

**Growth Pattern:** Decelerating (-5078.8% slowdown in recent period)

Ethereum

**R² (Scale Invariance):** 0.2740

**Power Law Exponent:** 1.90

**Data Points:** 2,921

**Date Range:** 2017-11-09 to 2025-11-07

**Log Residual Std Dev:** 0.7583

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 2015-07-30 (Provided)

**Power Law Fit:** Value ~= 0.0004909 * t^1.8979

Saturation Analysis (S-Curve Detection):

**📈 S-CURVE DETECTED** (Sigmoid R² = 0.7000 vs Linear R² = 0.5743)

**Current Progress:** 152.3% of estimated maximum

**Saturation Level:** 2.26e+03 (estimated carrying capacity)

**✅ HAS REACHED SATURATION** (≥95% of maximum)

**Growth Pattern:** Decelerating (22.3% slowdown in recent period)

Financial Asset - Split

Gold ETF (SPDR) (Early Period)

**R² (Scale Invariance):** 0.5747

**Power Law Exponent:** 0.33

**Data Points:** 2,639

**Date Range:** 2004-11-18 to 2015-05-14

**Log Residual Std Dev:** 0.2031

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 2004-11-18 (Provided)

**Power Law Fit:** Value ~= 9.028 * t^0.3268

S&P 500 ETF (Early Period)

**R² (Scale Invariance):** 0.5739

**Power Law Exponent:** 0.36

**Data Points:** 4,126

**Date Range:** 1993-01-29 to 2009-06-16

**Log Residual Std Dev:** 0.2184

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 1993-01-22 (Provided)

**Power Law Fit:** Value ~= 3.759 * t^0.3627

S&P 500 ETF (Late Period)

**R² (Scale Invariance):** 0.5422

**Power Law Exponent:** 0.37

**Data Points:** 4,126

**Date Range:** 2009-06-17 to 2025-11-07

**Log Residual Std Dev:** 0.3355

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 2009-06-17 (Hockey Stick Inflection)

**Power Law Fit:** Value ~= 12.2 * t^0.3742

Gold ETF (SPDR) (Late Period)

**R² (Scale Invariance):** 0.1743

**Power Law Exponent:** 0.12

**Data Points:** 2,638

**Date Range:** 2015-05-15 to 2025-11-07

**Log Residual Std Dev:** 0.2204

**Data Source:** Yahoo Finance API (yfinance Python library)

**Genesis Date:** 2015-05-15 (Hockey Stick Inflection)

**Power Law Fit:** Value ~= 65.03 * t^0.1215

Health

DTP3 Immunization

**R² (Scale Invariance):** 0.7149

**Power Law Exponent:** 0.05

**Data Points:** 22

**Date Range:** 2003-01-01 to 2024-01-01

**Log Residual Std Dev:** 0.0177

**Data Source:** WHO API (WHS4_100)

**Genesis Date:** 2002-05-26 (Optimized vs published)

**Power Law Fit:** Value ~= 56.33 * t^0.0473

Saturation Analysis (S-Curve Detection):

**📈 S-CURVE DETECTED** (Sigmoid R² = 0.9094 vs Linear R² = 0.6163)

**Current Progress:** 757.5% of estimated maximum

**Saturation Level:** 1.12e+01 (estimated carrying capacity)

**✅ HAS REACHED SATURATION** (≥95% of maximum)

**Growth Pattern:** Decelerating (17.9% slowdown in recent period)

Genesis Date Optimization:

**Optimal Genesis:** 2002-05-26 (R² = 0.7195)

**Note:** No published genesis date available for comparison

**Tested:** 11 candidate dates across range 2002-05-26 to 2004-11-11

Measles Immunization

**R² (Scale Invariance):** 0.6815

**Power Law Exponent:** 0.04

**Data Points:** 22

**Date Range:** 2003-01-01 to 2024-01-01

**Log Residual Std Dev:** 0.0186

**Data Source:** WHO API (WHS4_544)

**Genesis Date:** 2002-05-26 (Optimized vs published)

**Power Law Fit:** Value ~= 58.51 * t^0.0426

Saturation Analysis (S-Curve Detection):

**📈 S-CURVE DETECTED** (Sigmoid R² = 0.8641 vs Linear R² = 0.5063)

**Current Progress:** 734.6% of estimated maximum

**Saturation Level:** 1.14e+01 (estimated carrying capacity)

**✅ HAS REACHED SATURATION** (≥95% of maximum)

**Growth Pattern:** Decelerating (-7.8% slowdown in recent period)

Genesis Date Optimization:

**Optimal Genesis:** 2002-05-26 (R² = 0.6767)

**Note:** No published genesis date available for comparison

**Tested:** 11 candidate dates across range 2002-05-26 to 2004-11-11

Infrastructure

Electricity Access

**R² (Scale Invariance):** 0.9188

**Power Law Exponent:** 1.06

**Data Points:** 26

**Date Range:** 1998-01-01 to 2023-01-01

**Log Residual Std Dev:** 0.0102

**Data Source:** World Bank API (EG.ELC.ACCS.ZS)

**Genesis Date:** 1880-01-01 (Provided)

**Power Law Fit:** Value ~= 0.0008957 * t^1.0629

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.9805 vs Linear R²: 0.9766

**Current Value:** 0.5% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Technology

Internet Users

**R² (Scale Invariance):** 0.9776

**Power Law Exponent:** 2.38

**Data Points:** 20

**Date Range:** 2005-01-01 to 2024-01-01

**Log Residual Std Dev:** 0.0368

**Data Source:** World Bank API (IT.NET.USER.ZS)

**Genesis Date:** 1983-01-01 (Provided)

**Power Law Fit:** Value ~= 8.467e-09 * t^2.3786

Saturation Analysis (S-Curve Detection):

**⚡ POWER-LAW GROWTH** (Not saturating)

**Sigmoid R²:** 0.9951 vs Linear R²: 0.9913

**Current Value:** 40.4% of sigmoid model maximum

**Assessment:** No evidence of saturation or S-curve behavior

Broadband Subscriptions

**R² (Scale Invariance):** 0.9580

**Power Law Exponent:** 2.03

**Data Points:** 20

**Date Range:** 2005-01-01 to 2024-01-01

**Log Residual Std Dev:** 0.0575

**Data Source:** World Bank API (IT.NET.BBND.P2)

**Genesis Date:** 1990-01-01 (Provided)

**Power Law Fit:** Value ~= 1.065e-07 * t^2.0251

Mobile Subscriptions

**R² (Scale Invariance):** 0.7794

**Power Law Exponent:** 2.23

**Data Points:** 36

**Date Range:** 1984-01-01 to 2024-01-01

**Log Residual Std Dev:** 1.0850

**Data Source:** World Bank API (IT.CEL.SETS.P2)

**Genesis Date:** 1983-10-13 (Provided)

**Power Law Fit:** Value ~= 3.676e-08 * t^2.2330

Saturation Analysis (S-Curve Detection):

**📈 S-CURVE DETECTED** (Sigmoid R² = 0.9992 vs Linear R² = 0.9330)

**Current Progress:** 103.0% of estimated maximum

**Saturation Level:** 1.09e+02 (estimated carrying capacity)

**✅ HAS REACHED SATURATION** (≥95% of maximum)

**Growth Pattern:** Decelerating (61.4% slowdown in recent period)

Data Sources

All data downloaded from live APIs:

**Yahoo Finance API**: Financial assets (stocks, crypto, ETFs)

Data cached in `data_cache/` directory

Each cache file includes metadata with source URL and download date

Methodology

Scale invariance is measured by:

1. Selecting random pairs of dates (t₁, t₂)

2. Computing time_ratio = (t₂ - t₀) / (t₁ - t₀)

3. Computing value_ratio = V(t₂) / V(t₁)

4. Fitting power law in log-log space

5. Measuring R² (goodness of fit)

High R² (>0.95) indicates true scale invariance and power law behavior.

Asset	Category	R²	Exponent	Pattern	Points	Date Range	Genesis	Source
Internet Users	Technology	0.9776	2.38	⚡ Power-law	20	2005-01-01 to 2024-01-01	1983-01-01 (Provided)	World Bank API (IT.NET.USER.ZS)
Broadband Subscriptions	Technology	0.9580	2.03	⚡ Power-law	20	2005-01-01 to 2024-01-01	1990-01-01 (Provided)	World Bank API (IT.NET.BBND.P2)
McDonald's Restaurants	Business	0.9301	1.10	⚡ Power-law	130	2009-12-31 to 2025-09-30	1955-04-15 (Provided)	Downloaded from SEC EDGAR (CIK: 0000063908), us
Electricity Access	Infrastructure	0.9188	1.06	⚡ Power-law	26	1998-01-01 to 2023-01-01	1880-01-01 (Provided)	World Bank API (EG.ELC.ACCS.ZS)
Bitcoin	Financial Asset	0.9079	5.74	⚡ Power-law	5566	2010-07-18 to 2025-11-09	2009-01-03 (Provided)	btcpricehistory.csv (repository data)
Microsoft Corp.	Financial Asset	0.8151	4.91	⚡ Power-law	9993	1986-03-13 to 2025-11-07	1975-04-04 (Provided)	Yahoo Finance API (yfinance Python library)
Mobile Subscriptions	Technology	0.7794	2.23	✅ Saturated (103%)	36	1984-01-01 to 2024-01-01	1983-10-13 (Provided)	World Bank API (IT.CEL.SETS.P2)
Tesla Inc.	Financial Asset	0.7543	5.05	✅ Saturated (171%)	3866	2010-06-29 to 2025-11-07	2003-07-01 (Provided)	Yahoo Finance API (yfinance Python library)
Alphabet Inc.	Financial Asset	0.7494	2.18	⚡ Power-law	5341	2004-08-19 to 2025-11-07	1998-09-04 (Provided)	Yahoo Finance API (yfinance Python library)
DTP3 Immunization	Health	0.7149	0.05	✅ Saturated (757%)	22	2003-01-01 to 2024-01-01	2002-05-26 (Optimized vs published)	WHO API (WHS4_100)
Measles Immunization	Health	0.6815	0.04	✅ Saturated (735%)	22	2003-01-01 to 2024-01-01	2002-05-26 (Optimized vs published)	WHO API (WHS4_544)
Gold ETF (SPDR) (Early Period)	Financial Asset - Split	0.5747	0.33	🔹 Early	2639	2004-11-18 to 2015-05-14	2004-11-18 (Provided)	Yahoo Finance API (yfinance Python library)
S&P 500 ETF (Early Period)	Financial Asset - Split	0.5739	0.36	🔹 Early	4126	1993-01-29 to 2009-06-16	1993-01-22 (Provided)	Yahoo Finance API (yfinance Python library)
Gold ETF (SPDR)	Financial Asset	0.5594	0.34	🏒 Hockey (3.6x)	5277	2004-11-18 to 2025-11-07	2004-11-18 (Provided)	Yahoo Finance API (yfinance Python library)
S&P 500 ETF (Late Period)	Financial Asset - Split	0.5422	0.37	🔹 Late	4126	2009-06-17 to 2025-11-07	2009-06-17 (Hockey Stick Inflection)	Yahoo Finance API (yfinance Python library)
S&P 500 ETF	Financial Asset	0.4041	0.48	🏒 Hockey (7.7x)	8252	1993-01-29 to 2025-11-07	1993-01-22 (Provided)	Yahoo Finance API (yfinance Python library)
Apple Inc.	Financial Asset	0.4032	2.87	📈 S-curve (89%)	11319	1980-12-12 to 2025-11-07	1976-04-01 (Provided)	Yahoo Finance API (yfinance Python library)
Ethereum	Financial Asset	0.2740	1.90	✅ Saturated (152%)	2921	2017-11-09 to 2025-11-07	2015-07-30 (Provided)	Yahoo Finance API (yfinance Python library)
Gold ETF (SPDR) (Late Period)	Financial Asset - Split	0.1743	0.12	🔹 Late	2638	2015-05-15 to 2025-11-07	2015-05-15 (Hockey Stick Inflection)	Yahoo Finance API (yfinance Python library)