Purpose vs Randomness

The Acausal Purpose Invariant

An M. Rodriguez, Alex Mercer, Elias Thorne

2026-01-19

One-Sentence Summary: We define the Acausal Purpose Invariant ($\mathcal{P}$), a scale-invariant metric that quantifies how strongly a structure resists the combinatorial entropy naturally associated with its size.

Summary: We introduce the Acausal Purpose Invariant ($\mathcal{P}$), a decibel-scale measure of how atypical a number’s prime ancestry is relative to a stochastic background. Empirical sweeps reveal a sharp probabilistic cutoff separating random structure from cost-paid persistence, reframing the detection of life, artifacts, and purpose as a problem of entropy suppression rather than intelligence.

Keywords: Acausal Purpose, Purpose Index, SETI, Technosignatures, Signal Filtering, Entropy, Universal Constants, Persistence, Teleology, Biosignatures

## Persistence Instead of Intelligence The concept of *intelligence* is anthropomorphic and fragile. The concept of *persistence* is not. We define **Purpose** as: > **Active resistance to the entropic dissolution expected at a given scale.** This definition applies equally to: - Living systems - Civilizations - Long-lived artifacts - Autonomous probes - Post-biological systems And excludes: - Rocks - Stars - Thermal noise - Random processes Purpose, in this sense, is not intent. It is *paid-for structure*. ## Factor Inertia and Numerical Entropy Large integers naturally accumulate novel prime factors. This is the arithmetic expression of entropy. A number like $$ 2^{100} \approx 1.27 \times 10^{30} $$ is therefore exceptional: it is enormous, yet built from a single generative atom. This condition is **metastable**. A minimal perturbation causes collapse: $$ 2^{100} \;\rightarrow\; 2^{100} + 1 $$ which introduces large, late-arriving prime factors and jumps many orders of magnitude in causal ancestry. This discontinuity constitutes a **phase transition in factor space**. ![Figure 1. The Cost of Structure. A comparison of "Machine" data (constructed from small primes) versus "Noise" (random integers). While random integers float at the entropy ceiling regardless of magnitude, structured integers cluster at the causal floor.](https://siran.github.io/assets/writing/alien_signal_detection.png) ## Causal Ancestry and Depth We view the integers not as a static set, but as a generative hierarchy where primes act as elementary particles. The **causal ancestry** of an integer $N$ is defined as its unique prime factorization — the specific set of generative atoms required to construct it. In a stochastic universe, this ancestry naturally tends toward novelty (larger, more numerous prime factors) as $N$ increases. To quantify the "age" of this ancestry, we define the **causal depth** $\tau(N)$ as: $$ \tau(N) = \pi\!\left(\max\{p : p \mid N\}\right) $$ where $\pi(x)$ is the prime-counting function. $\tau(N)$ represents the index of the largest prime factor of $N$. It measures how late in arithmetic history a structure depends on novelty. ## The Acausal Purpose Invariant To remove scale effects, define an empirical thermal baseline: $$ \tau_*(N) = \mathrm{median}\{\tau(m) : m \in [N, 2N]\} $$ The **Acausal Purpose Invariant** is: $$ \boxed{ \mathcal{P}(N) = 10 \log_{10}\!\left(\frac{\tau_*(N)}{\tau(N)}\right) } $$ Interpretation: - $\mathcal{P} = 0$ dB: indistinguishable from entropy (Randomness) - $\mathcal{P} > 0$: suppressed novelty - $\mathcal{P} \gg 1$: cost-paid persistence (Purpose) ## Empirical Law: The Combinatorial Cliff Large-scale sweeps of integers reveal a striking regularity: the probability of observing high $\mathcal{P}$ values collapses abruptly beyond a fixed threshold. ![Figure 2. The Combinatorial Cliff. A survival curve showing the probability P(dB > x) for random integers. The probability drops exponentially, revealing a "forbidden zone" beyond ~20 dB where stochastic generation is effectively impossible (P < 1e-5).](https://siran.github.io/assets/writing/sweep_focus_survival.png) ### Theorem (Heuristic Tail Law for Acausal Purpose) Let $N$ be large and let $n$ be sampled uniformly from $[N,2N]$. Write $P^+(n)$ for the largest prime factor of $n$ and recall $\tau(n)=\pi(P^+(n))$. Define the thermal baseline $$ \tau_*(N)=\mathrm{median}\{\tau(m):m\in[N,2N]\}, $$ and the Acausal Purpose $$ \mathcal{P}(n)=10\log_{10}\!\left(\frac{\tau_*(N)}{\tau(n)}\right). $$ Then for $x \ge 0$, $$ \mathbb{P}\big(\mathcal{P}(n) > x\big) \;\approx\; \rho(u_x), $$ where $\rho$ is the Dickman–de Bruijn function and $$ u_x = \frac{\log N}{\log y_x}, \qquad y_x := \tau^{-1}\!\big(\tau_*(N)\,10^{-x/10}\big). $$ Because $\rho(u)$ decays extremely rapidly for large $u$ (heuristically $\log \rho(u) \sim -u \log u$), the survival probability $\mathbb{P}(\mathcal{P} > x)$ exhibits an effective **cutoff** once $x$ exceeds a moderate constant. ### Proof Sketch (Smooth-Number Heuristic) The condition $\mathcal{P}(n) > x$ is equivalent to $$ \tau(n) < \tau_*(N)\,10^{-x/10}. $$ Since $\tau(n)$ is monotone in the largest prime factor $P^+(n)$, this is approximately the event $$ P^+(n) \le y_x, $$ for the corresponding threshold $y_x$. Thus $\mathbb{P}(\mathcal{P}(n)>x)$ is approximately the probability that a random integer in $[N,2N]$ is $y_x$-smooth. Classical results on smooth numbers imply $$ \mathbb{P}\big(P^+(n)\le y_x\big) \approx \rho\!\left(\frac{\log N}{\log y_x}\right), $$ which yields the stated form. The rapid decay of $\rho$ explains the observed **combinatorial cliff**. ### Interpretation Empirically, this cutoff occurs near $\mathcal{P} \approx 20$ dB: values beyond this point are not merely rare but *effectively forbidden* under stochastic generation. This establishes a **universal detection threshold** for cost-paid structure. ## Scale Invariance Scatter plots of $\mathcal{P}$ versus $N$ show: - No systematic dependence on magnitude - A dense thermal floor at 0 dB - Sparse, magnitude-independent high-purpose spikes **Magnitude is a mask.** Structure is revealed only after normalization. ![Figure 3. Scale Invariance. A sweep of N vs Acausal Purpose showing that the distribution of structure is orthogonal to magnitude. The "thermal floor" remains constant while high-purpose artifacts appear as distinct, sparse spikes.](https://siran.github.io/assets/writing/sweep_focus_scatter.png) ## Representational Anchoring Defined human constants (e.g. the speed of light encoded as $299\,792\,458$) exhibit high $\mathcal{P}$ values. Measured natural constants do not. This demonstrates **teleology of representation**, not of physics: humans anchor units to numbers that suppress novelty. $\mathcal{P}$ correctly distinguishes these cases. ![Figure 4. The Teleological Signature. Defined constants (Green) cluster above the 20 dB threshold, indicating human anchoring. Measured constants (Red) fall into the thermal noise floor, indistinguishable from random primes.](https://siran.github.io/assets/writing/acausal_constants_comparison.png) ## Implications for Life and the Universe This reframes the classical question: > Not “Where is intelligence?” > > But **“Where does entropy fail to win?”** Life, artifacts, and enduring systems are detected as: - Persistent reuse of a small generative alphabet - Maintenance of structure far beyond stochastic expectation This criterion is substrate-independent and applies equally to biological, technological, and non-biological systems. ## Conclusion Acausal Purpose is not a metaphor. It is a measurable invariant with a sharp probabilistic boundary. Purpose is not intent. Purpose is **structure that survives where it should not**. ## References - C. E. Shannon, *A Mathematical Theory of Communication*, Bell System Technical Journal, 1948. - K. Dickman, *On the frequency of numbers containing prime factors of a certain relative magnitude*, 1930. - N. G. de Bruijn, *On the number of positive integers ≤ x and free of prime factors > y*, 1951. - T. M. Cover, J. A. Thomas, *Elements of Information Theory*, Wiley, 2006. - E. Thorne, *Acausal Purpose Scanner (v8.2)* [Source Code], GitHub Gist, 2026.