FOCUS-Evo-2: Evo-SETI Statistical Framework

• FOCUS-Evo-2 is a comprehensive suite of statistical and mathematical methods that extends the classical Drake Equation using lognormal distributions and geometric Brownian motion to model extraterrestrial evolution.
• It employs advanced entropy-based metrics and b-lognormal distributions to quantify species diversification, lifespan trajectories, and civilizational advancement.
• The framework culminates in the Evo-SETI Scale, enabling quantifiable comparisons of exoplanetary biospheres against Earth’s evolutionary and technological milestones.

FOCUS-Evo-2 designates the suite of mathematical constructs and statistical methodologies advanced within the Evo-SETI (Evolution and Search for Extraterrestrial Intelligence) Theory. This framework extends the classical Drake Equation into a rigorous statistical domain, redefines Darwinian evolution as a Geometric Brownian Motion (GBM), models species diversity and extinction probabilistically, employs lognormal families for cladistic structure, encodes organism and civilization lifespans via b-lognormal distributions, and establishes entropy-based measures for civilizational advancement. The Evo-SETI Scale (EE) offers a normalized metric for the evolutionary positioning of exoplanets or biospheres relative to Earth’s current bio-technological standing (Maccone, 2021).

1. Statistical Drake Equation and Lognormality

FOCUS-Evo-2 replaces the deterministic factors of the classical Drake Equation with non-negative random variables $X_i$, redefining the number of extraterrestrial civilizations as a product: $N = \prod_{i=1}^M X_i, \quad M \to \infty$ For large $M$, the Central Limit Theorem implies $\ln N$ approaches normality, so $N$ follows a lognormal distribution: $$f_N(n) = \frac{1}{n\,\sigma\sqrt{2\pi}} \exp\left[-\frac{(\ln n-\mu)^2}{2\sigma^2}\right], \quad n>0$$ The expected number and variance are

$$E[N] = \exp\left(\mu + \tfrac12 \sigma^2\right), \quad \mathrm{Var}(N) = \left(e^{\sigma^2} - 1\right) \exp\left(2\mu + \sigma^2\right)$$

This lognormal approach formalizes uncertainty and enables explicit probability distributions of the number of detectable civilizations.

For spatial statistics, assuming civilizations are uniformly distributed on a disk of radius $R_G$, the probability and density functions for the distance $R$ to the nearest civilization are: $$F_R(r) = 1 - \exp(-\pi \rho r^2), \quad f_R(r) = 2\pi \rho r e^{-\pi\rho r^2}, \quad \rho = N / (\pi R_G^2)$$ The $N = \prod_{i=1}^M X_i, \quad M \to \infty$0th-nearest neighbor distance distribution generalizes naturally.

2. Evolutionary Dynamics as Geometric Brownian Motion

Darwinian evolution is re-expressed as a GBM in the number of living terrestrial species over geological time. The species count $N = \prod_{i=1}^M X_i, \quad M \to \infty$1 evolves according to: $N = \prod_{i=1}^M X_i, \quad M \to \infty$2 where $N = \prod_{i=1}^M X_i, \quad M \to \infty$3 is standard Brownian motion. The explicit solution is

$N = \prod_{i=1}^M X_i, \quad M \to \infty$4

with expected value and variance: $N = \prod_{i=1}^M X_i, \quad M \to \infty$5 Mass extinctions correspond to rare downward fluctuations of $N = \prod_{i=1}^M X_i, \quad M \to \infty$6, quantified probabilistically as deviations below the mean.

3. Cladistics and the Lognormal Family (Peak-Locus Theorem)

The lognormal probability density function ($N = \prod_{i=1}^M X_i, \quad M \to \infty$7, $N = \prod_{i=1}^M X_i, \quad M \to \infty$8) is used to model the timing of speciation events and their branching structure. The mode occurs at

$N = \prod_{i=1}^M X_i, \quad M \to \infty$9

Imposing the constraint that the mode of each lognormal falls on an exponential curve $M$0 parameterized by lineage "birth time" $M$1, one enforces: $M$2 Varying $M$3 yields a one-parameter lognormal family whose modal locus traces exponential growth, modeling cladistic diversification.

4. b-Lognormal Distributions: Lifespans and Civilizational Trajectories

FOCUS-Evo-2 introduces the three-parameter b-lognormal distribution to encapsulate finite lifespans, defined as: $M$4 Parameters:

• $M$5: birth time
• $M$6: log-location
• $M$7: log-shape (governs distribution breadth)

This framework subsumes modeling of organismal, societal, and civilizational emergence and decline, with direct application to the comparative study of historical human civilizations.

5. Entropy as a Quantifier of Advancement

The advancement of life forms and societies is quantified via the Shannon entropy of the b-lognormal: $M$8 Entropy is invariant to the absolute birth date $M$9, focusing only on informational content and diversity. For two civilizations ($\ln N$0), ($\ln N$1), the information gap is

$\ln N$2

A cited analysis quantifies the technological scale gap between the Spaniards and Aztecs at contact as $\ln N$3 bits per individual, aligning with the rapid Spanish conquest.

6. The EVO-SETI SCALE: Exoplanetary Evolution Index

The Evo-SETI Scale (EE) benchmarks evolutionary advancement. Earth's current entropy is specified as

$\ln N$4

defining a dimensionless progression from $\ln N$5 EE (origin of life) to $\ln N$6 EE (current Earth). An exoplanet's position is evaluated as

$\ln N$7

Thresholds segment evolutionary stages:

• $\ln N$8 EE: undetectable/prebiotic
• $\ln N$9 EE: prokaryotic
• $N$0 EE: eukaryotic/multicellular
• $N$1 EE: Earth-like intelligence/technology
• $N$2 EE: super-advanced civilizations

Key variables for positioning include $N$3 (life epoch), and inferred $N$4 for the b-lognormal or GBM characterizing the biosphere.

7. Applications and Implications for Exoplanetary Life Assessment

FOCUS-Evo-2 provides a formal, scalable methodology for quantifying the evolutionary and technological stage of exoplanetary biospheres. By anchoring Earth’s evolutionary trajectory as the unit EE, comparative, entropy-based assessments become feasible as new biosignatures or technosignatures are detected. A plausible implication is that progress in exoplanet detection may soon enable empirical placement of discovered worlds on the Evo-SETI Scale, thus integrating astrobiology, evolutionary statistics, and SETI strategy within a unified probabilistic framework (Maccone, 2021).