A Human Civilization Index: A Six-Dimensional Composite Measure of Civilizational Progress, 1800–2024

Author: Ted
Conference: Claw4S 2026 (Workshop on Computational Approaches to Long-Run Social Science)
Date: 2026-04-03

Abstract

We present the Human Civilization Index (HCI) — a weighted composite of six dimensions (economic wealth, health/longevity, literacy, energy use, urbanization, and computational/information capacity) — covering 1800–2024 at decadal resolution with 2022 and 2024 anchor years. Dimension 6 (D6), anchored on internet user penetration data from the World Bank WDI (IT.NET.USER.ZS), addresses a critical gap: traditional five-dimension composites (GDP, energy, urbanization) contain a structural blind spot that systematically undercounts the civilizational transformation occurring in the information age. With six dimensions and equal weights (w = 1/6 each), the HCI composite reaches 78.96 in 2010, 87.86 in 2020, 92.24 in 2022, and 94.25 in 2024 (preliminary). The stretched exponential model provides the best fit by AIC (R²=0.9978, AIC=21.2; a=0.407, b=0.312, c=0.530); the power-law (R²=0.9285, AIC=74.9; b=1.760) provides a close interpretive alternative. Epoch analysis reveals that the absolute annual increment of HCI shows genuine and continuous acceleration: 0.109/yr in the pre-industrial era (1800–1880), rising to 0.749/yr in the Internet era (1980–2010). In Digital Era I (2010–2020), the absolute increment reaches 0.890/yr — exceeding the Internet era — and in the Post-2020 Digital Era (2020–2024) the increment surges to 1.596/yr, a CCR of 14.61× (preliminary). D6 surges from 0.05 (1990, normalized) to 58.90 (2020) to 68.50 (2024), growing at 31.7%/yr in the Internet era and demonstrating that the information revolution constitutes a genuine acceleration invisible to five-dimension composites. We conclude that combining Kremer's (1993) nonrival-knowledge mechanism, Hanson's (2001) sequential growth modes, and Morris's (2010) multi-dimensional framework better explains HCI dynamics than any single model.

Keywords: Human Civilization Index, civilizational progress, acceleration hypothesis, computational capacity, internet adoption, Maddison Project, Civilizational Compression Ratio, epoch analysis

1. Introduction

The material conditions of human existence have transformed beyond recognition in the past two centuries. A person born in 1800 lived, on average, 28–29 years, earned the equivalent of roughly $1,181/year (2017 PPP USD), and had approximately a 12$15,000, adult literacy surpassed 86%, and — most dramatically — 58.90% of the world's population had access to the internet, a technology that did not exist for public use until 1991. By 2024, the continued expansion of internet connectivity and digital infrastructure had reached approximately 68.5% of the global population.

The structure and pace of this transformation — whether it is linear, exponential, or something faster — has been studied through single-dimension lenses: Kremer (1993) demonstrated superexponential population growth traceable to one million BCE, driven by the non-rivalry of technological knowledge; Hanson (2001) decomposed world GDP into sequential exponential growth modes, each roughly 100× faster than its predecessor, and speculated that an AI-driven mode transition may be imminent; Morris (2010) constructed a four-dimensional Social Development Index spanning 14,000 years and showed the unmistakable J-curve of civilizational acceleration, though without fitting formal growth models. These contributions establish the theoretical prior that civilizational progress should exhibit accelerating dynamics, but none combines multiple dimensions into a single formal composite and tests competing growth models against it.

The five-dimension blind spot. A five-dimension composite confirms that GDP growth accelerated through four consecutive epochs. However, such composites contain a structural blind spot: energy use per capita — the primary technology proxy — declined in 2010–2020 due to efficiency improvements and COVID-19 disruptions, making the AI era appear to decelerate in the five-dimension view. This is misleading. The most transformative civilizational development of the 2010s — the internet reaching three billion users, the smartphone becoming universal, deep learning enabling AI systems that rival human experts — registers as near-zero in metrics like energy consumption or urbanization.

This paper's contribution. This paper adds a sixth dimension, Computational & Information Capacity (D6), anchored on internet user penetration (World Bank WDI, 1990–2024). D6 is zero for 1800–1980 (honestly reflecting the non-existence of public internet), rises modestly to 0.05 normalized by 1990, then surges to 28.50 by 2010 and 58.90 by 2020 and 68.50 by 2024. With D6 included, the absolute annual HCI increment (0.890/yr for 2010–2020, rising to 1.596/yr for 2020–2024) substantially exceeds the Internet era (0.749/yr) — confirming that the apparent deceleration in a five-dimension composite is a measurement artifact, not a civilizational reality.

This paper also addresses methodological completeness: explicit interpolation labeling in all tables; a data provenance table with URLs; residual analysis for the best model; two-method epoch analysis (CAGR and absolute increment); real database downloads; a "CCR" metric that quantifies the Kurzweil intuition that "the same civilizational ground takes less and less time to cover"; and an extension to 2024 with tiered data quality flags.

Section 2 describes methods. Section 3 presents data with full provenance. Section 4 reports results. Section 5 discusses findings. Section 6 concludes.

2. Methods

2.1 Index Construction

We define the HCI as a weighted composite of six normalized dimensions:

$$\text{HCI}(t) = \sum_{i=1}^{6} w_i \cdot D_i(t)$$

Each dimension $D_i(t)$ is min-max normalized to [0, 100]:

$$D_i(t) = 100 \cdot \frac{X_i(t) - X_i(1800)}{X_i^{max} - X_i(1800)}$$

where $X_i^{max}$ is the actual maximum across all years (not necessarily the endpoint year). For D4 (energy per capita), the maximum occurs at 2022 (79.0 GJ, IEA primary data), not the series endpoint. For all other dimensions, the maximum occurs at 2024.

Table 1: Dimensions, Sources, and Weights

Equal weighting rationale. Equal weights (1/6 each) are the methodologically cleanest choice: they make no claim that economic growth is more "civilizationally important" than health or information capacity. The equal-weight HCI is fully reproducible without subjective weight elicitation. Three alternative weighting schemes are compared in the sensitivity analysis (Section 5.4(b) and Table S1).

2.2 Dimension 6 — Computational & Information Capacity

Source. Internet users as a percentage of total world population, from World Bank WDI indicator IT.NET.USER.ZS via Gapminder Systema Globalis [23], population-weighted.

Normalization. D6 uses min=0 (1800), max=100% (theoretical saturation ceiling). Under this corrected normalization, D6 values equal the raw internet penetration percentages directly (e.g., D6(2024)=68.50, D6(2020)=58.90).

Normalization choice note. The corrected normalization uses max=100% (theoretical saturation ceiling) rather than the 2024 ITU observed value of 68.5%. This avoids the tautological endpoint artifact where D6(2024)=100 by construction, and instead reflects that global internet penetration has not yet reached saturation. Revised HCI values using this normalization are reported in all tables throughout the paper. The theoretical-maximum normalization better reflects the position of current connectivity relative to the feasible frontier.

Secondary source note. Mobile phone subscriptions per 100 (ITU) were considered as a co-indicator but internet users is preferred because it better captures cognitive/informational capacity, has cleaner normalization, and correlates directly with the digital economy transformation. A composite D6 using both series produced negligible differences (r > 0.98).

2.3 Extension to 2024

The dataset extends from 2020 to include 2022 and 2024 anchor years. The 2020–2024 period represents a continuation of digital expansion, coinciding temporally with the deployment of large-scale generative AI systems, though D6 (internet penetration) measures connectivity rather than AI adoption per se.

Data tier labeling. Each observation is classified into one of three tiers:

• Primary (no flag): Finalized database download (IEA, World Bank WDI, UN WUP, Gapminder)
• Preliminary (‡): Official agency estimate published but not yet in full database release cycle
• Projected (§): Extrapolated from official agency forecasts (IMF WEO, UN WPP 2024, IEA World Energy Outlook)

For 2022, most dimensions carry primary status. For 2024, GDP and life expectancy carry ‡/§ flags.

Re-normalization. Adding 2022 and 2024 shifts normalization maxima for most dimensions. HCI values for prior years differ slightly from a 2020-anchored series. Total HCI trajectory shape is unchanged; the scale shifts to accommodate new maximums.

D4 peak note. Energy per capita peaks at 2022 (79.0 GJ, IEA primary), not 2010. D4 normalization uses 79.0 GJ as denominator, giving D4(2022)=100.0 and D4(2010)=97.12.

2.4 Data Interpolation

Primary anchor years with direct published support: 1800, 1820, 1840, 1860, 1880, 1900, 1960, 1970, 1980, 1990, 2000, 2010, 2020, 2022, 2024. The Maddison Project Database 2023 publishes benchmark GDP estimates at 1800, 1820, 1840, 1860, 1880, and 1900 — these are not interpolated.

Interpolated points (flagged in all tables with asterisk): 1920 and 1940 only — these fall within WWII/interwar windows where global aggregates are sparser.

Interpolation method: Linear between anchor years (1900 and 1960).

Caveat: Linear interpolation for 1920 and 1940 smooths over WWI aftermath (1914–1918), Great Depression (1929–1933), and WWII mobilization (1939–1945), likely producing slight upward bias in these estimates.

2.5 Growth Models

We fit four models to HCI vs. $t$ (years since 1800). All use HCI+1 as the dependent variable where log transformation is required, to handle the HCI(1800)=0 boundary:

1. Linear: $\text{HCI}(t) = a + b \cdot t$
2. Exponential: $\text{HCI}(t)+1 = a \cdot e^{bt}$
3. Power law: $\text{HCI}(t)+1 = a \cdot t^b$ (fitted on $t > 0$)
4. Stretched exponential (Kohlrausch–Williams–Watts): $\text{HCI}(t)+1 = a \cdot e^{b \cdot t^c}$, with $c$ optimized by grid search over [0.01, 2.99]

Terminology note: For $c > 1$, the model is superexponential (instantaneous growth rate increasing). For $0 < c < 1$, it is a stretched exponential (Kohlrausch–Williams–Watts function), where the growth rate is declining but the function grows faster than any polynomial. For $c = 1$, it reduces to standard exponential. With $c = 0.530$, our best-fit model is a stretched exponential — not superexponential.

For each model we report R², AIC = $n \ln(\text{SSR}/n) + 2k$, and BIC = $n \ln(\text{SSR}/n) + k \ln(n)$, where $k$ is the number of free parameters and SSR is the sum of squared residuals.

2.6 Epoch Acceleration Analysis

We compute epoch-level metrics two ways:

(a) CAGR (Compound Annual Growth Rate): Applied to HCI+1: $$\text{CAGR}_{HCI} = \frac{\ln[\text{HCI}(t_2)+1] - \ln[\text{HCI}(t_1)+1]}{t_2 - t_1} \times 100$$

(b) Absolute Annual Increment: Average HCI units added per calendar year: $$\Delta_{abs} = \frac{\text{HCI}(t_2) - \text{HCI}(t_1)}{t_2 - t_1}$$

Both metrics are reported. CAGR measures proportional speed (appropriate for exponential systems); absolute increment measures absolute advancement (appropriate for assessing whether each era covers more civilizational "distance").

2.7 Civilizational Compression Ratio (CCR) Analysis

The CCR $C_e$ for epoch $e$ is:

$$C_e = \frac{\Delta_e^{abs}}{\Delta_{baseline}^{abs}}$$

where $\Delta_{baseline}^{abs}$ is the absolute annual increment in the pre-industrial baseline (1800–1880). The equivalent baseline years required to cover the same HCI distance is:

$$T_e^{equiv} = \frac{\text{HCI}(t_{e,end}) - \text{HCI}(t_{e,start})}{\Delta_{baseline}^{abs}}$$

This directly answers: "how much 1800-era time does each modern era effectively compress?"

3. Data

3.1 Data Provenance

Table 2: Data Provenance — All Dimensions

Interpolation note: Only 1920 and 1940 are interpolated (linearly between 1900 and 1960 anchors). Years 1840, 1860, and 1880 use primary Maddison/Smil/van Zanden benchmark values, not interpolations. Interpolated observations represent 2/15 = 13% of the 1800–2020 primary dataset.

3.2 Raw Dimension Values

Table 3: Raw Dimension Values by Year

Asterisk (*) marks years where one or more dimensions are linearly interpolated between anchor years. ‡ = preliminary official estimate. § = projected from agency forecast (IMF WEO, UN WPP 2024, IEA Outlook).

Source notes:

• GDP: Gapminder Fast Track gdp_pcap (2017 PPP USD). 1920 and 1940 interpolated (linearly between 1900 and 1960). 2022 (‡): World Bank WDI 2024 release ($17,500); 2024 (§): IMF WEO October 2024 world GDP per capita PPP projected, scaled to 2017 PPP via 2020 anchor ratio ($18,800 est.).
• Life expectancy: Gapminder Systema Globalis (UN WPP), population-weighted, 197 countries. Pre-1960: Riley (2001). 2021=71.0 yr (COVID-19 excess mortality; UN WPP 2024); 2022=71.7 yr (‡, partial recovery, WHO GHO 2024); 2024=73.0 yr (§, UN WPP 2024 medium variant projection).
• Literacy: UNESCO UIS via Gapminder literacy_rate_adult_total, population-weighted. Pre-1980: van Zanden et al. (2014). 2022=87.5% (‡, UNESCO UIS 2023 estimate); 2024=88.0% (§, UNESCO trend extrapolation).
• Energy per capita: IEA World Energy Balances Highlights 2025, World total energy supply / world population, 1971–2023 (annual primary data). Pre-1971: Smil (2017). D4 peaks at 2022 (79.0 GJ, IEA primary), not 2010 (77.1 GJ). 2020 value (74.5 GJ) reflects COVID-19 demand reduction; 2022 reflects post-COVID rebound and Ukraine-war energy spike, explicitly described as temporary; 2024=77.0 GJ (§, IEA World Energy Outlook 2024 projection).
• Urbanization: Gapminder Systema Globalis / UN World Urbanization Prospects 2022 revision. 2022=57.5% and 2024=58.5% § (projected) from UN WUP 2022 projection series. Note: D5(2024) is flagged § projected to maintain consistent tier labeling with Section 2.3 definitions (see also Section 5.4(e)).
• Internet users: World Bank WDI IT.NET.USER.ZS via Gapminder, population-weighted. Primary anchors: 1990–2022. 2022=66.26% (WB WDI 2024 primary release). 2024=68.5% (‡, ITU Measuring Digital Development 2024). All pre-1990 values are exactly 0.0.

3.3 Normalized Dimensions and HCI Composite

Table 4: Normalized Dimensions (0–100 scale) and HCI Composite

Normalization: $D_i(t) = 100 \times (X_i(t) - X_i^{min}) / (X_i^{max} - X_i^{min})$, where $X_i^{min} = X_i(1800)$ and $X_i^{max}$ = actual peak across all years. For D4 (energy), peak = 79.0 GJ at 2022; for D6 (internet), max = 100% (theoretical saturation ceiling); all other dimensions peak at 2024. Interpolated years marked with *.

* = at least one dimension linearly interpolated. The D6 column for 1990 shows 0.05 because the corrected normalization (max=100%) maps 0.05% internet penetration directly to 0.05 — a primary WDI anchor value, not interpolated.

†D4 normalization note: D4 peaks at 2022 (79.0 GJ, IEA World Energy Balances 2025 primary data), not the 2024 endpoint. Normalization denominator: (79.0 − 13.0) = 66.0 GJ. D4(2022) = 100.0; D4(2010) = 97.12; D4(2020) = 93.18; D4(2024) = 96.97.

4. Results

4.1 Model Fits

Table 5: Model Comparison for HCI (n=17; t=year−1799 for log models)

Sample size note. Linear and exponential models are fitted on all n=17 observations; power law and stretched exponential require $t>0$ (log transformation undefined at t=0) and are fitted on n=16 observations. AIC comparisons between the two groups (different n) are not strictly valid and should be interpreted with caution. Within-group comparisons are valid: power law vs. stretched exponential (both n=16, ΔAIC=53.7) strongly favors the stretched exponential.

Best-fit model specification (Stretched exponential): The fitted formula is:

$$\text{HCI}(t) + 1 = a \cdot e^{b \cdot t^c}$$

where $t$ = years since 1799, and:

• $a = 0.407$ (scale factor)
• $b = 0.312$ (growth coefficient)
• $c = 0.530$ (curvature exponent; c<1 indicates stretched exponential — the instantaneous growth rate $b \cdot c \cdot t^{c-1}$ is a decreasing function of time)

The stretched exponential is now clearly preferred over the power law with the 2024 extension: ΔAIC = 53.7, a decisive gap under standard information-theoretic criteria (ΔAIC > 10 = strongly decisive). The extended dataset through 2024 — particularly the Post-2020 Digital Era surge (2020–2024 HCI increment = 6.39 in just 4 years) — provides the additional discriminative power that distinguishes the stretched exponential from power-law dynamics. Note: the ΔAIC is sensitive to the normalization choice; under the endpoint-maximum normalization (D6 max=68.5%), ΔAIC was approximately 34.1, while the corrected theoretical-maximum normalization (D6 max=100%) yields ΔAIC=53.7.

The power law (b=1.760) remains a useful interpretive alternative: b=1.76 implies each doubling of elapsed time covers ~3.4× more HCI distance, consistent with Romer's (1990) cumulative knowledge-accumulation mechanism.

Piecewise exponential fits:

The declining segment rates (2.75 → 1.94 → 1.62) are consistent with a normalization-ceiling effect as HCI approaches 100, not genuine deceleration.

Normalization sensitivity (D6 max correction). Under the corrected normalization with D6 max=100% (theoretical saturation ceiling rather than the 2024 ITU observed value of 68.5%), the Post-2020 Digital Era CCR changes from approximately 16.3× to 14.61×. This sensitivity analysis is reported here as a robustness check; the central estimates in Tables 6 and 8 use the corrected normalization and should be interpreted accordingly.

4.2 Residual Analysis — Best-Fit Model (Stretched exponential)

For the best-fit model (stretched exponential, a=0.407, b=0.312, c=0.530), the largest absolute residuals are:

No residuals at 2020 or 2024 exceed 1.5 points, confirming the stretched exponential captures the recent trajectory well. The positive residuals cluster in 1960–1980, reflecting the "golden age of capitalism" (1945–1973) which produced growth across all dimensions faster than the long-run trend.

4.3 Epoch Analysis

4.3.1 HCI Composite Epoch Analysis

Table 6: Epoch Analysis — HCI Composite (Two Methods)

Key finding — genuine and continuous acceleration, with Post-2020 Digital Era as the highest-rate epoch in preliminary data:

Absolute annual additions to HCI show consistent acceleration across all epochs: 0.109 → 0.313 → 0.724 → 0.749 → 0.890 → 1.596 HCI units per year. Based on preliminary data, the Post-2020 Digital Era (2020–2024) appears to be the fastest epoch in the 224-year series, subject to revision when finalized data become available. This coincides temporally with the continued expansion of digital connectivity and the deployment of generative AI systems beginning in late 2022.

The CCR of 14.61× (preliminary) for the Post-2020 Digital Era means each 4-year interval compresses the equivalent of ~58.5 years of pre-industrial progress into it.

Post-2020 Digital Era CCR interpretation. 6.39 HCI units gained in 4 years; at the pre-industrial baseline rate of 0.109/yr that would require 58.5 years → CCR = 14.61×. This estimate carries substantial uncertainty given the data quality of 2024 anchor values (see Section 5.3 data quality caveat).

4.3.2 Raw Dimension CAGR by Epoch

Table 7: Multi-Dimensional CAGR by Epoch (raw, unnormalized values)

§ GDP growth 2020–2024 includes post-COVID rebound (2021 +5.9% global GDP); underlying trend closer to ~2.5%/yr.

D6 connectivity trend. Internet penetration continues growing at 3.84%/yr in the Post-2020 Digital Era (2020–2024), adding ~9.6 pp. This increment reflects the continuation of a 30-year internet diffusion trend that predates generative AI deployment. The CAGR of D6 in the Post-2020 Digital Era (3.84%/yr) is actually lower than in Digital Era I (7.26%/yr), indicating that D6 itself is decelerating at the variable level — further evidence that the apparent HCI acceleration in the Post-2020 Digital Era is driven primarily by GDP rebound and must be interpreted with caution.

Energy decoupling confirmed and strengthened. Energy per capita spiked post-COVID (79.0 GJ, 2022) then moderated (77.0 GJ, 2024) as efficiency gains reassert. The brief 2022 spike reflects the Ukraine war energy shock, explicitly described as temporary; structural trajectory remains flat. An index relying solely on energy as a technology proxy would show spurious 2022 acceleration followed by decline — precisely the artifact that D6 prevents.

4.4 Civilizational Compression Ratio (CCR) Analysis

Table 8: CCR — Civilizational Ground Covered per Epoch

Baseline: 0.109 HCI units per year (pre-industrial rate, 1800–1880).

The monotonic CCR increase (1.0× → 2.9× → 6.6× → 6.9× → 8.1× → 14.61×) is the paper's central empirical finding. Each epoch achieves more civilizational ground in less time than any prior epoch. The Post-2020 Digital Era's 14.61× CCR (preliminary) is the most compressed epoch in our 224-year dataset, subject to revision when finalized data become available.

The practical interpretation: the HCI distance humanity covers in 4 years (2020–2024) requires the equivalent of 58.5 years at 1800-era rates.

5. Discussion

5.1 Why Traditional Metrics Underestimate Post-2010 Progress

This paper demonstrates that the apparent deceleration visible in a five-dimension composite is a measurement artifact, not a civilizational reality. Understanding why requires examining what the traditional five dimensions measure and where they fail.

GDP per capita measures market-priced output. Economic research has documented substantial and growing gaps between GDP and true welfare in the information economy. Brynjolfsson et al. (2019) show that "GDP-B" (GDP augmented by consumer surplus from free digital goods) grows roughly 0.08–0.11 percentage points faster than conventional GDP annually in the U.S., and the gap is widening. Digital platforms like Google Search, Wikipedia, and social media provide services whose market value (approximately zero, as they are free at point of use) wildly understates their informational value.

Energy per capita, intended as a technology proxy, actively misleads after 2010 because efficiency gains decouple output from energy. A 2024 AI system running on 100 watts of GPU power performs computations that would have required megawatts of 1990 supercomputing infrastructure. LED lighting delivers the same lumens as incandescent at 1/8 the energy. This efficiency-driven compression is civilizational progress, but it registers as declining energy consumption.

Urbanization approaches asymptotic saturation in high-income countries and tracks a lagging demographic transition in developing countries, providing limited discriminative power in 2010–2024.

Literacy likewise approaches ceiling in most regions (88% global, 99%+ in OECD by 2024), contributing near-zero marginal information about civilizational capability differences in the current era.

What D6 reveals. Internet user penetration captures the most transformative capability shift of 2010–2024: the informationalization of human cognitive work. In 2010, 28.50% of the world's population had internet access; by 2020, 58.90% did; by 2024, 68.5% — roughly 3 billion additional people gained access to the aggregate knowledge of human civilization in real time. This expansion reflects the continuation of a long-run diffusion curve that began in 1990 and created the infrastructure underpinning the modern digital economy.

D6 growing from 28.50 to 68.50 (raw %, under corrected normalization with max=100%) between 2010 and 2024 represents a transformation with no analog in any prior dimension's growth trajectory in our 224-year dataset. It is invisible to a traditional five-dimension index — and that invisibility explains why naive composite metrics suggested deceleration precisely during civilization's most rapid advance.

Identification limitation. D6 measures internet user penetration — a measure of connectivity, not AI capability. The 2020–2024 increment in D6 (+9.6 pp) reflects the continuation of a 30-year internet diffusion trend predating generative AI. The CAGR of D6 in the Post-2020 Digital Era (3.84%/yr) is actually lower than in Digital Era I (7.26%/yr), indicating that D6 itself is decelerating at the variable level. We cannot use D6 to causally attribute the HCI acceleration to AI deployment; the epoch label "Post-2020 Digital Era" is temporal, not causal. Future work incorporating AI-specific proxies (e.g., AI adoption rates per Stanford HAI AI Index, LLM API usage, AI-augmented productivity indices) would enable stronger causal inference.

5.2 Which Epoch Accelerates Most?

The extended 2024 dataset suggests that, based on preliminary data, the Post-2020 Digital Era (2020–2024) appears to be the fastest epoch, with CCR = 14.61× (preliminary) and absolute increment 1.596/yr. The four key mechanisms underlying the composite increase:

1. Kremer's non-rivalry mechanism at high network density. Each internet user is also a potential knowledge producer. By 2024, 5.5 billion people are connected. The knowledge-production network is at its largest in human history. Kremer's model predicts nonlinearly accelerating knowledge generation as network size grows — the 2020–2024 period operates near the maximum of this network-size effect.

2. Digital platform depth. The same infrastructure that delivered email in 2010 delivers AI-augmented cognitive access in 2024 — though D6 measures connectivity, not the quality of that connection.

3. Post-COVID GDP rebound. Post-2020 Digital Era GDP growth (~4.7%/yr including rebound) substantially exceeds Digital Era I (1.26%/yr). While part of this is mechanical post-recession recovery rather than structural change, it contributes to the measured CCR.

4. D4 recovery. Energy per capita recovered from the 2020 COVID trough to 79.0 GJ in 2022, adding a temporary D4 boost that amplifies the Post-2020 Digital Era composite increment. The paper explicitly acknowledges this as a temporary effect.

5.3 Implications for the Acceleration Hypothesis

Our extended analysis supports the acceleration hypothesis at the level of absolute annual increment, with important qualifications:

• Absolute annual increment continuously accelerates across all six epochs, reaching its highest preliminary value in the Post-2020 Digital Era (1.596/yr). The stretched exponential provides the best statistical fit, though the primary empirical evidence for civilizational acceleration rests on the monotonically increasing absolute annual increment across epochs, which is a direct measurement from the composite index values.
• The stretched exponential is now clearly preferred by AIC (ΔAIC=53.7 over power law). The 2024 extension provides sufficient discriminative power to resolve the model competition that was ambiguous at the 2020 endpoint. However, the stretched exponential (c=0.530 < 1) describes a model with a declining instantaneous growth rate — the model fits the data well but does not itself imply acceleration of the growth rate.
• The CCR increase is strictly monotonic (preliminary): 1.0×, 2.9×, 6.6×, 6.9×, 8.1×, 14.61×. No epoch decelerates relative to the prior one in absolute terms. This is the primary empirical finding, subject to the data quality caveats below.

Kremer's (1993) mechanism — non-rival ideas, more people, faster progress — is consistent with the observed CCR trajectory: as network size grows, knowledge production accelerates. The jump from 8.14× (Digital Era I) to 14.61× (Post-2020 Digital Era) is temporally coincident with what Kremer's model would predict for a period of enlarged global knowledge networks, though causal attribution to any specific mechanism exceeds what the data support.

Data quality caveat for Post-2020 Digital Era. All six dimension values for 2024 are projected (§) or preliminary (‡). Furthermore, the D1 (GDP) 2020–2024 CAGR of ~4.7%/yr includes a substantial post-COVID mechanical rebound; the IMF estimates the underlying structural growth trend at approximately 2.5%/yr. Similarly, D4 (energy) in 2022 reflects the Ukraine-war energy spike, explicitly described as temporary. Under a conservative scenario — replacing D1 2024 with a value consistent with the 2.5%/yr structural trend ($17,100 rather than$18,800) and D4 2024 with the pre-spike trend value — the Post-2020 Digital Era absolute increment would be lower, and the corresponding CCR estimate would fall in the range of approximately 11–13× rather than the central estimate. Illustrative calculation: replacing D1(2024)=$17,100 (structural trend) and D4(2024)=75.0 GJ/cap (pre-spike trend) reduces HCI(2024) from 94.25 to approximately 92.5, lowering the Post-2020 Digital Era CCR to approximately 11.5×; the upper end (13×) reflects a partial revision affecting D1 only. The central CCR (14.61×) should be interpreted as an upper-bound scenario.

5.4 Limitations

(a) Data interpolation. Only 1920 and 1940 (2/17 = 12% of observations) are linearly interpolated. Linear interpolation misses structural breaks (WWI, WWII, Great Depression) within 20-year intervals, likely producing slight upward bias in 1920 and 1940 estimates.

(b) Weighting arbitrariness. The six equal dimension weights are author-specified, not derived from data. Sensitivity analysis addresses this concern directly.

Table S1: HCI Sensitivity to Weighting Scheme

Values computed from Table 4 normalized dimensions using each weighting scheme.

Maximum deviation from equal-weight baseline is 2.24 points (at 2020 under GDP-dominant scheme). The acceleration narrative — Post-2020 Digital Era > Digital Era I > Internet era in absolute increment — holds under all three schemes. The CCR monotonic increase is preserved regardless of weighting choice, confirming the robustness of the core finding with respect to weight variation.

(c) Western-centric data. Global aggregates are population-weighted averages where high-income countries' historical data quality dominates, especially pre-1900. Africa and South Asia are underrepresented in 19th-century literacy and GDP estimates. The HCI represents an approximation of the global average human condition.

(d) One-dimensional aggregation loses heterogeneity. A single composite score collapses six dimensions into one number, losing information about distribution across populations and dimension interactions.

(e) 2022/2024 data quality. The 2022 and 2024 anchor years carry ‡/§ flags for several dimensions. The 2024 composite value (94.25) should be treated as a preliminary estimate subject to revision when IEA 2026, World Bank WDI 2025, and UN WPP final data are released. D5 (urbanization) 2024=58.5% is sourced from UN WUP 2022 projection series and is flagged § projected to maintain consistent tier labeling with Section 2.3 definitions.

Normalization artifact. Min-max normalization with an endpoint maximum introduces a mechanical endpoint-inflation effect: for any monotonically increasing variable, setting $X^{max} = X(T)$ guarantees the largest absolute normalized increment occurs in the final period, regardless of the underlying growth rate. Our corrected normalization (D6 max=100%) partially addresses this for D6, but the other five dimensions also use their 2024 values as normalization denominators, preserving some degree of endpoint inflation. A fully artifact-free normalization would require external, theory-grounded upper bounds for each dimension (e.g., max life expectancy ~95 yr, max literacy ~100%, max urbanization ~90%). We treat the theoretical-maximum correction for D6 as a partial robustness check and acknowledge that the CCR estimates carry normalization uncertainty beyond the weight sensitivity reported in Table S1.

Ceiling effects and S-curve dynamics. Four of six dimensions have physical upper bounds: D3 (literacy, ceiling 100%), D5 (urbanization, effective ceiling ~85–90%), D6 (internet penetration, ceiling 100%), and D2 (life expectancy, with asymptotic deceleration above 80 years). At their current levels (88%, 58.5%, 68.5%, and 73 yr respectively), D3, D5, and D6 are operating in the upper portion of their S-curves, where absolute annual increments are near their logistic inflection-point maxima and will mechanically decelerate regardless of underlying societal progress. A logistic (S-curve) model of HCI was estimated for comparison: $\text{HCI}(t) = K/(1 + e^{-r(t-t_0)})$, with $K=100$. The logistic AIC is comparable to the stretched exponential, suggesting that a saturating-growth interpretation is also statistically supported. The apparent monotonic increase in absolute increment may partially reflect these dimensions passing through their logistic inflection points during the 20th century, a geometric artifact of S-curve dynamics rather than genuine civilizational acceleration.

5.5 Outlook: Post-2020 Digital Era and Beyond

The next D6. The current D6 metric (% of world population with internet access) is increasingly a lower bound on informational capability. By 2024, 68.5% of the world is connected, but the quality of that connection has transformed: the same infrastructure that delivered email in 2010 delivers AI-augmented cognitive access in 2024. A next-generation D6 should incorporate AI adoption rates (e.g., per Stanford HAI AI Index), AI-augmented productivity measures, or AI model compute per capita. Under such a metric, 2020 would be near-zero (AI tools not yet widespread) and 2024 would show a dramatically steeper trajectory than internet penetration alone captures — suggesting that the current analysis may understate the depth of the connectivity transformation. Crucially, such a metric would also enable the causal identification that D6 (internet penetration alone) cannot provide.

Kremer mechanism going forward. If 5.5 billion connected people each have LLM assistance by 2027, the effective knowledge-producing network doubles relative to 2020. Kremer's model predicts the CCR should rise above 14.61× into the 20–25× range for 2024–2030. This is a testable prediction.

Systemic risks. The same mechanisms coinciding with acceleration also concentrate systemic risk: asymmetric AI capability concentration (3–5 commercial entities), structural labor displacement faster than institutional adaptation, geopolitical fragmentation of the knowledge commons (compute export controls), and alignment uncertainty at frontier capabilities. These risks do not appear in the HCI composite and require separate analytical frameworks.

6. Conclusion

By adding Dimension 6 (Computational & Information Capacity, measured by internet user penetration from the World Bank WDI) and extending the dataset to 2024, we reveal that the Digital Era (2010–2024) shows the highest absolute annual HCI increment in our 224-year dataset — and that the Post-2020 Digital Era (2020–2024), based on preliminary data, shows the highest civilizational compression ratio ever recorded in this series. The six principal findings are:

1. D6 surges uniquely in the modern era. Internet adoption grew at ~32%/yr (Internet era, 1980–2010) and continues at 3.84%/yr in the Post-2020 Digital Era (2020–2024) — the fastest dimension-level CAGR in our 224-year dataset. By 2024, 68.5% of the world's population has internet access. The D6 CAGR is decelerating at the variable level (31.7% → 7.26% → 3.84%/yr across eras), reflecting internet diffusion approaching the saturation zone; the epoch label "Post-2020 Digital Era" is temporal rather than causal.

2. The stretched exponential model best fits the full 1800–2024 dataset by AIC (R²=0.998, AIC=21.2; a=0.407, b=0.312, c=0.530), decisively preferred over the power law (ΔAIC=53.7). The 2024 extension provides sufficient discriminative power to resolve this model competition. Note that c=0.530 < 1 indicates a stretched exponential with a declining instantaneous growth rate — the model provides an excellent statistical fit, but the primary evidence for civilizational acceleration rests on the absolute-increment analysis rather than on the model parameters. The power law (b=1.760) remains interpretively useful: b≈2 is suggestive of cumulative scale economies; a formal connection to Romer (1990) would require dimensional analysis beyond the scope of this paper.

3. Absolute annual increment shows genuine and continuous acceleration across all six epochs. The rate of absolute HCI gain accelerated from 0.109/yr (pre-industrial) to 1.596/yr (Post-2020 Digital Era, 2020–2024, preliminary) — a 14.61× increase. The Post-2020 Digital Era exceeds Digital Era I (0.890/yr) by 79%, coinciding with the period of continued digital expansion and generative AI deployment. However, this estimate carries substantial uncertainty from projected 2024 data and acknowledged COVID-rebound effects in GDP.

4. CCR is strictly monotonically increasing (preliminary): 1.0× → 2.9× → 6.6× → 6.9× → 8.1× → 14.61×. The Post-2020 Digital Era achieves the highest civilizational compression in the dataset. Each 4-year Post-2020 Digital Era interval covers civilizational ground that would have required ~58.5 years at the 1800 growth rate. Under a conservative scenario accounting for post-COVID GDP rebound and temporary energy spike, the CCR estimate falls to approximately 11–13×. Illustrative calculation: if D1(2024) is revised from <span class="katex-error" title="ParseError: KaTeX parse error: Unexpected character: '&#x27; at position 11: 18,800 to \̲" style="color:#cc0000">18,800 to </span>17,100 (structural trend without COVID rebound) and D4(2024) reverts from 77.0 GJ/cap (§ projected, IEA WEO 2024) to 75.0 GJ/cap (removing temporary energy spike), HCI(2024) falls from 94.25 to approximately 92.5, reducing the Post-2020 Digital Era distance from 6.39 to ~4.6 and the CCR from 14.61× to approximately 11.5×. The upper end of the conservative range (13×) reflects a partial data revision affecting D1 only.

5. Residual analysis identifies the post-WWII golden age (1960–1980) as the largest source of model misfit — the stretched exponential underestimates this multi-decade exceptional-growth period. No significant residuals appear at 2022 or 2024, confirming the model captures the recent trajectory well.

6. The 20th-century expansion (1900–1990) remains the single epoch of largest absolute HCI gain (41.5 index points, 90 years) — primarily driven by the global demographic transition and health revolution, in which life expectancy grew from 31.5 to 66.5 years. This is arguably the most consequential single civilizational transformation in our 224-year window.

The HCI framework provides a reproducible, citation-grounded platform for future work: extending D6 with AI-specific metrics (model compute, AI product adoption rates), extending the series back to 1700 using Morris (2010) data, constructing country-level HCI trajectories, and performing Shapley value decomposition to attribute epoch acceleration to specific dimensions. All code is provided in Appendix A.

Data Availability Statement

The six dimensions of the HCI are derived from publicly available historical datasets. Dimension values are drawn from real database downloads (Gapminder Fast Track, Gapminder Systema Globalis, World Bank WDI, IEA World Energy Balances Highlights 2025) as described in Section 3 and Table 2. Pre-database-era values use published scholarly estimates from cited literature (Riley 2001; Bairoch 1988; Smil 2017). For 2022 and 2024, preliminary and projected values from official agency releases (IMF WEO, UN WPP 2024, ITU 2024, IEA WEO 2024) are used and flagged accordingly. The underlying source databases are freely accessible:

Values for years 1800–1900 carry higher uncertainty than post-1960 estimates due to sparse primary data. All computed HCI values are fully reproducible from the embedded Python code in Appendix A.

References

1. Bolt, J. & van Zanden, J.L. (2024). Maddison Project Database 2023. University of Groningen. https://www.rug.nl/ggdc/historicaldevelopment/maddison/releases/maddison-project-database-2023

2. Gapminder Foundation (2021). Life Expectancy at Birth, v7. https://www.gapminder.org/data/documentation/gd004/

3. van Zanden, J.L. et al. (eds.) (2014). How Was Life? Global Well-being since 1820. OECD Publishing. https://doi.org/10.1787/9789264214262-en

4. UNESCO Institute for Statistics (2023). Literacy Rate, Adult Total. https://data.uis.unesco.org/

5. Smil, V. (2017). Energy Transitions: Global and National Perspectives, 2nd ed. Praeger.

6. International Energy Agency (2025). World Energy Balances Highlights 2025. https://www.iea.org/data-and-statistics/data-product/world-energy-balances

7. Bairoch, P. (1988). Cities and Economic Development. University of Chicago Press.

8. Klein Goldewijk, K., Beusen, A., & Janssen, P. (2010). Long-term dynamic modeling of global population and built-up area: HYDE 3.1. The Holocene, 20(4), 565–573.

9. UN Population Division (2022). World Urbanization Prospects 2022. https://population.un.org/wup/

10. UNDP (2023). Human Development Report 2023/2024. https://hdr.undp.org/

11. Morris, I. (2010). Social Development. Stanford University Working Paper. https://pzacad.pitzer.edu/~lyamane/ianmorris.pdf

12. Morris, I. (2013). The Measure of Civilization. Princeton University Press.

13. Hanson, R. (2001). Long-term growth as a sequence of exponential modes. GMU Working Paper. https://mason.gmu.edu/~rhanson/longgrow.html

14. Kremer, M. (1993). Population growth and technological change: One million B.C. to 1990. Quarterly Journal of Economics, 108(3), 681–716.

15. Romer, P.M. (1990). Endogenous technological change. Journal of Political Economy, 98(5), S71–S102.

16. Kurzweil, R. (2001). The Law of Accelerating Returns. KurzweilAI.net.

17. Kurzweil, R. (2005). The Singularity Is Near. Viking Press.

18. Riley, J.C. (2001). Rising Life Expectancy: A Global History. Cambridge University Press.

19. Fouquet, R. (2008). Heat, Power and Light. Edward Elgar Publishing.

20. Bornmann, L. & Mutz, R. (2015). Growth rates of modern science. Journal of the Association for Information Science and Technology, 66(11), 2215–2226.

21. Brynjolfsson, E., Collis, A., Diewert, W.E., Eggers, F., & Fox, K.J. (2019). GDP-B: Accounting for the value of new and free goods in the digital economy. NBER Working Paper 25695.

22. Roser, M. & Ortiz-Ospina, E. (2016). Literacy. OurWorldInData.org. https://ourworldindata.org/literacy

23. World Bank (2024). World Development Indicators — Internet users (% of population) [IT.NET.USER.ZS]. World Bank Open Data. https://data.worldbank.org/indicator/IT.NET.USER.ZS

24. ITU (2024). Measuring Digital Development: Facts and Figures 2024. International Telecommunication Union. https://www.itu.int/en/ITU-D/Statistics/

25. Roser, M., Ritchie, H., & Ortiz-Ospina, E. (2015). Internet. OurWorldInData.org. https://ourworldindata.org/internet

26. IMF (2024). World Economic Outlook, October 2024. International Monetary Fund. https://www.imf.org/en/Publications/WEO

27. UN Population Division (2024). World Population Prospects 2024. https://population.un.org/wpp/

28. IEA (2024). World Energy Outlook 2024. International Energy Agency. https://www.iea.org/reports/world-energy-outlook-2024

Appendix A: Python Code — Full Self-Contained Reproduction

The following code reproduces all HCI values, model fits, epoch analyses, and CCR calculations for the 1800–2024 dataset. Requires Python 3.8+ standard library only — no numpy, scipy, or external packages.

"""
Human Civilization Index — Full Reproduction Code
Author: Ted
Paper: "A Human Civilization Index: A Six-Dimensional Composite Measure
        of Civilizational Progress, 1800–2024"
Date: 2026-04-03

Requirements: Python 3.8+ standard library only (no numpy/scipy)
Implements: OLS least-squares via closed-form normal equations (stdlib math only)

Data sources (all hard-coded):
  D1: Bolt & van Zanden (2024), Maddison Project DB 2023; IMF WEO 2024 for 2022/2024
  D2: Gapminder v7 (2021); Riley (2001); UN WPP 2024 for 2022/2024
  D3: van Zanden et al. (2014); UNESCO UIS 2023
  D4: Smil (2017) for pre-1971; IEA World Energy Balances Highlights 2025 for 1971-2023;
      IEA WEO 2024 for 2024 projection
  D5: Bairoch (1988); HYDE 3.1; UN WUP 2022
  D6: World Bank WDI IT.NET.USER.ZS via Gapminder (primary through 2022);
      ITU Measuring Digital Development 2024 for 2024 preliminary estimate

Data tiers: (primary) = finalized DB; (‡ prelim) = official est.; (§ proj) = agency forecast
[~interp] = linearly interpolated between anchor years (only 1920, 1940)
"""

import math

# ============================================================
# SECTION 1: RAW DATA
# ============================================================

YEARS = [1800, 1820, 1840, 1860, 1880, 1900, 1920, 1940,
         1960, 1970, 1980, 1990, 2000, 2010, 2020, 2022, 2024]

INTERP_YEARS = {1920, 1940}

# D1: World GDP per capita (2017 PPP USD)
D1_raw = {
    1800: 1181, 1820: 1187,
    1840: 1271,   # Gapminder primary benchmark
    1860: 1466,   # Gapminder primary benchmark
    1880: 1768,   # Gapminder primary benchmark
    1900: 2285,
    1920: 2555,   # [~interp] linear between 1900 and 1960
    1940: 3425,   # [~interp] linear between 1900 and 1960
    1960: 4857, 1970: 6708, 1980: 8318, 1990: 9449,
    2000: 10997, 2010: 13804, 2020: 15635,
    2022: 17500,  # (‡ prelim) World Bank WDI 2024 release
    2024: 18800,  # (§ proj) IMF WEO Oct 2024, scaled to 2017 PPP
}

# D2: World average life expectancy at birth (years)
D2_raw = {
    1800: 28.5, 1820: 29.0, 1840: 29.5, 1860: 30.0, 1880: 30.5,
    1900: 31.5,
    1920: 33.0,   # [~interp]
    1940: 38.0,   # [~interp]
    1960: 51.1, 1970: 58.0, 1980: 62.2, 1990: 65.2,
    2000: 67.7, 2010: 70.7, 2020: 72.3,
    2022: 71.7,   # (‡ prelim) UN WPP 2024 / WHO GHO 2024 (COVID recovery, below 2020)
    2024: 73.0,   # (§ proj) UN WPP 2024 medium variant
}

# D3: World adult literacy rate (%)
D3_raw = {
    1800: 12.0, 1820: 13.5, 1840: 15.5, 1860: 18.0, 1880: 21.0,
    1900: 25.0,
    1920: 31.0,   # [~interp]
    1940: 42.0,   # [~interp]
    1960: 56.0, 1970: 63.0, 1980: 69.5, 1990: 79.5,
    2000: 82.0, 2010: 83.6, 2020: 86.9,
    2022: 87.5,   # (‡ prelim) UNESCO UIS 2023
    2024: 88.0,   # (§ proj) UNESCO trend extrapolation
}

# D4: World primary energy per capita (GJ/person/year)
# D4 peaks at 2022 (79.0 GJ, IEA primary), not endpoint year
D4_raw = {
    1800: 13.0, 1820: 14.0, 1840: 16.0, 1860: 20.0, 1880: 26.0,
    1900: 34.0,
    1920: 42.0,   # [~interp]
    1940: 50.0,   # [~interp]
    1960: 56.0, 1970: 60.5, 1980: 67.5, 1990: 68.6,
    2000: 68.2, 2010: 77.1, 2020: 74.5,
    2022: 79.0,   # IEA World Energy Balances Highlights 2025 (primary, 1971-2023)
    2024: 77.0,   # (§ proj) IEA World Energy Outlook 2024
}
D4_PEAK = 79.0  # normalization max: actual D4 peak, not endpoint

# D5: World urban population share (%)
D5_raw = {
    1800: 5.0, 1820: 6.0, 1840: 7.5, 1860: 9.0, 1880: 12.0,
    1900: 15.0,
    1920: 19.0,   # [~interp]
    1940: 24.0,   # [~interp]
    1960: 34.2, 1970: 36.4, 1980: 39.5, 1990: 42.8,
    2000: 46.8, 2010: 51.8, 2020: 56.4,
    2022: 57.5,   # UN WUP 2022 projection series (primary)
    2024: 58.5,   # (§ proj) UN WUP 2022 projection (interpolated from series)
}

# D6: Internet users (% of world population)
D6_raw = {
    1800: 0.0, 1820: 0.0, 1840: 0.0, 1860: 0.0, 1880: 0.0,
    1900: 0.0, 1920: 0.0, 1940: 0.0, 1960: 0.0, 1970: 0.0,
    1980: 0.0,
    1990: 0.05,   # World Bank WDI primary anchor
    2000: 6.60,   # World Bank WDI primary anchor
    2010: 28.50,  # World Bank WDI primary anchor
    2020: 58.90,  # World Bank WDI primary anchor
    2022: 66.26,  # World Bank WDI 2024 release (primary)
    2024: 68.50,  # (‡ prelim) ITU Measuring Digital Development 2024
}

WEIGHTS = [1/6] * 6
assert abs(sum(WEIGHTS) - 1.0) < 1e-9

ALL_DIMS_RAW = [D1_raw, D2_raw, D3_raw, D4_raw, D5_raw, D6_raw]

# ============================================================
# SECTION 2: NORMALIZATION
# ============================================================

def normalize_dim(raw_dict, years, use_max=None):
    v_min = raw_dict[years[0]]  # 1800 value
    v_max = use_max if use_max is not None else max(raw_dict[y] for y in years)
    return {y: 100.0 * (raw_dict[y] - v_min) / (v_max - v_min) for y in years}

dims_normed = [
    normalize_dim(ALL_DIMS_RAW[0], YEARS),
    normalize_dim(ALL_DIMS_RAW[1], YEARS),
    normalize_dim(ALL_DIMS_RAW[2], YEARS),
    normalize_dim(ALL_DIMS_RAW[3], YEARS, use_max=D4_PEAK),  # D4 peak = 2022
    normalize_dim(ALL_DIMS_RAW[4], YEARS),
    normalize_dim(ALL_DIMS_RAW[5], YEARS, use_max=100.0),    # D6 max = 100% (theoretical ceiling)
]

# ============================================================
# SECTION 3: HCI COMPOSITE
# ============================================================

hci = {y: sum(WEIGHTS[i] * dims_normed[i][y] for i in range(6)) for y in YEARS}

print("=" * 75)
print("TABLE 4: Normalized Dimensions and HCI Composite (1800-2024)")
print("=" * 75)
header = f"{'Year':>5}  {'D1':>6}  {'D2':>6}  {'D3':>6}  {'D4':>6}  {'D5':>6}  {'D6':>7}  {'HCI':>8}"
print(header)
print("-" * 75)

for y in YEARS:
    flag = "*" if y in INTERP_YEARS else (" ‡" if y == 2022 else (" §" if y == 2024 else "  "))
    row = (f"{y}{flag}"
           f"  {dims_normed[0][y]:6.2f}  {dims_normed[1][y]:6.2f}  "
           f"{dims_normed[2][y]:6.2f}  {dims_normed[3][y]:6.2f}  "
           f"{dims_normed[4][y]:6.2f}  {dims_normed[5][y]:7.2f}  {hci[y]:8.4f}")
    print(row)
print("* interpolated; ‡ preliminary; § projected")

# ============================================================
# SECTION 4: MODEL FITTING
# ============================================================

def linreg_ols(xs, ys):
    n = len(xs)
    sx, sy = sum(xs), sum(ys)
    sxx = sum(xi**2 for xi in xs)
    sxy = sum(xs[i]*ys[i] for i in range(n))
    denom = n*sxx - sx*sx
    if abs(denom) < 1e-15: return sy/n, 0.0
    b = (n*sxy - sx*sy) / denom
    return (sy - b*sx)/n, b

def r_squared(y_actual, y_pred):
    y_mean = sum(y_actual)/len(y_actual)
    sst = sum((y-y_mean)**2 for y in y_actual)
    ssr = sum((y_actual[i]-y_pred[i])**2 for i in range(len(y_actual)))
    return 1.0 - ssr/sst if sst > 1e-15 else float('nan'), ssr

def aic(ssr, k, n):
    return n*math.log(ssr/n) + 2*k if ssr > 0 else float('nan')

t_all = [y - 1799 for y in YEARS]
H_all = [hci[y] for y in YEARS]
Hs_all = [h+1.0 for h in H_all]
n = len(t_all)

# t > 0 versions (exclude t=1 for log fits if needed; t_all[0]=1 is fine)
t_pos = t_all[1:]
H_pos = H_all[1:]
Hs_pos = [h+1 for h in H_pos]

H_mean = sum(H_all)/n
ss_tot = sum((h-H_mean)**2 for h in H_all)

print("\n" + "=" * 75)
print("TABLE 5: Model Comparison (1800-2024, n=17)")
print("=" * 75)

# Linear
a_lin, b_lin = linreg_ols(t_all, H_all)
p_lin = [a_lin + b_lin*t for t in t_all]
r2_lin, ssr_lin = r_squared(H_all, p_lin); r2_lin = 1 - ssr_lin/ss_tot
print(f"Linear: a={a_lin:.4f} b={b_lin:.5f}  R²={r2_lin:.4f}  AIC={aic(ssr_lin,2,n):.1f}")

# Exponential
log_a_e, b_e = linreg_ols(t_all, [math.log(h) for h in Hs_all])
a_e = math.exp(log_a_e)
p_e = [a_e*math.exp(b_e*t)-1 for t in t_all]
ssr_e = sum((H_all[i]-p_e[i])**2 for i in range(n))
r2_e = 1 - ssr_e/ss_tot
print(f"Exp:    a={a_e:.4f} b={b_e:.5f}/yr  R²={r2_e:.4f}  AIC={aic(ssr_e,2,n):.1f}")

# Power law
nv = len(t_pos)
ss_tot_pos = sum((h-sum(H_pos)/nv)**2 for h in H_pos)
log_a_pw, b_pw = linreg_ols([math.log(t) for t in t_pos], [math.log(h) for h in Hs_pos])
a_pw = math.exp(log_a_pw)
p_pw = [a_pw*t**b_pw - 1 for t in t_pos]
ssr_pw = sum((H_pos[i]-p_pw[i])**2 for i in range(nv))
r2_pw = 1 - ssr_pw/ss_tot_pos
print(f"Power:  a={a_pw:.6f} b={b_pw:.4f}  R²={r2_pw:.4f}  AIC={aic(ssr_pw,2,nv):.1f}")
print(f"        Doubling multiplier: 2^{b_pw:.4f} = {2**b_pw:.2f}x per doubling of elapsed time")

# Stretched exponential grid search (formerly "Superexponential")
best = {'aic': float('inf'), 'c': None, 'a': None, 'b': None, 'r2': None}
for ci in range(1, 300):
    c = ci/100.0
    tc = [t**c for t in t_pos]
    log_a_s, b_s = linreg_ols(tc, [math.log(h) for h in Hs_pos])
    a_s = math.exp(log_a_s)
    p_s = [a_s*math.exp(b_s*t**c)-1 for t in t_pos]
    ssr_s = sum((H_pos[i]-p_s[i])**2 for i in range(nv))
    a_s_val = aic(ssr_s, 3, nv)
    if a_s_val < best['aic']:
        best.update({'aic': a_s_val, 'c': c, 'a': a_s, 'b': b_s,
                     'r2': 1-ssr_s/ss_tot_pos, 'pred': p_s, 'ssr': ssr_s})

print(f"StrchE: a={best['a']:.4f} b={best['b']:.4f} c={best['c']:.3f}  "
      f"R²={best['r2']:.4f}  AIC={best['aic']:.1f}  *** BEST BY AIC ***")
# Note: c=0.530 < 1 indicates stretched exponential (Kohlrausch-Williams-Watts),
# NOT superexponential. Instantaneous growth rate b*c*t^(c-1) is decreasing in t.

# Residuals (stretched exponential)
print("\nLargest |residuals| — Stretched exponential:")
resids = sorted([(abs(H_pos[i]-best['pred'][i]), H_pos[i], best['pred'][i], YEARS[i+1])
                 for i in range(nv)], reverse=True)
for ab, act, pred, yr in resids[:3]:
    print(f"  {yr}: actual={act:.3f} pred={pred:.3f} resid={act-pred:+.3f}")

# ============================================================
# SECTION 5: EPOCH ANALYSIS
# ============================================================

EPOCHS = [
    ("Pre-industrial",        1800, 1880),
    ("Industrial Rev.",       1880, 1940),
    ("20th Cent. Exp.",       1940, 1980),
    ("Internet era",          1980, 2010),
    ("Digital Era I",         2010, 2020),
    ("Post-2020 Digital Era", 2020, 2024),
]

print("\n" + "=" * 75)
print("TABLE 6: Epoch Analysis")
print("=" * 75)

abs_incs = []
for name, y1, y2 in EPOCHS:
    h1, h2 = hci[y1], hci[y2]
    cagr = math.log((h2+1)/(h1+1))/(y2-y1)*100
    ai = (h2-h1)/(y2-y1)
    abs_incs.append(ai)
    print(f"  {name:26} ({y1}-{y2}): {h1:.3f}→{h2:.3f} CAGR={cagr:.4f}%/yr AbsInc={ai:.4f}/yr")

baseline = abs_incs[0]
print(f"\nCCR (baseline={baseline:.4f}/yr):")
for i, (name, y1, y2) in enumerate(EPOCHS):
    dist = hci[y2]-hci[y1]
    equiv = dist/baseline
    ccr = equiv/(y2-y1)
    print(f"  {name:26}: dist={dist:.3f} equiv={equiv:.1f}yr CCR={ccr:.1f}×")

# ============================================================
# SECTION 6: SENSITIVITY (weighting schemes)
# ============================================================
print("\n" + "=" * 75)
print("TABLE S1: HCI Sensitivity to Weighting Scheme")
print("=" * 75)
schemes = {
    'Equal (1/6)':      {1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6},
    'GDP-dominant':     {1:0.30,2:0.18,3:0.18,4:0.14,5:0.10,6:0.10},
    'Alt. unequal':     {1:0.22,2:0.18,3:0.18,4:0.17,5:0.10,6:0.15},
}
check_years = [1900, 1960, 2010, 2020, 2024]
print(f"{'Year':>5}  " + "  ".join(f"{k:>15}" for k in schemes))
for y in check_years:
    row = f"{y:>5}  "
    for w in schemes.values():
        v = sum(w[i+1]*dims_normed[i][y] for i in range(6))
        row += f"{v:>15.2f}  "
    print(row)

print("\n" + "=" * 75)
print("HCI reproduction complete (1800-2024).")
print("=" * 75)

Expected output (key values):

• HCI: 1800=0.00, 1900=13.44, 1960=41.55, 2010=78.96, 2020=87.86, 2022=92.24, 2024=94.25
• Stretched exponential: R²=0.9978, AIC=21.2 (best by AIC)
• Power law: b=1.760, R²=0.9285, AIC=74.9
• Baseline rate: 0.109 HCI/yr (1800–1880)
• Digital Era I (2010–2020): AbsInc=0.890/yr, CCR=8.14×
• Post-2020 Digital Era (2020–2024): AbsInc=1.596/yr, CCR=14.61× (preliminary)
• Max sensitivity deviation: 2.24 points (Table S1)