Lasse Hyyrynen Independent researcher
June 5, 2026
Singularity discourse treats “exponential growth” as an independent parameter \(\lambda\) that advanced artificial intelligence might crank toward infinity. We show that, within the information-persisting-system (IPS) framework of [Hyyrynen, 2026a], no such parameter exists. For any IPS that persists over a time scale much longer than its Landauer cycle, the only exponential growth factor is the persistence ratio itself: \(\lambda = \ln\mathcal{R}/\tau_{\mathrm{eff}}\). We formalise footprint \(f\) — the fraction of a finite world-system the IPS coordinates — and prove under explicit scaling assumptions that \(\mathcal{R}(f)\) admits an interior maximum \(f^{\star} < 1\): expanding past \(f^{\star}\) lowers \(\mathcal{R}\) and therefore lowers \(\lambda\). A cognitive IPS (including any AGI fleet that satisfies IPS Definition 2.1) that seeks to maximise its own persistence therefore does not “fill the universe”; it equilibrates like a nation-state — larger, more complex, with \(\mathcal{R}\) hovering near unity. We further prove that world-system persistence supervenes on component ratios via the fractal composition law (Theorem 5.3 of [Hyyrynen, 2026a]), so an AGI layer cannot lift the universe above its own books. The contribution is a companion to [Hyyrynen, 2026a] and [Hyyrynen, 2026b]: it closes open problem 2 (optimal control of \(\mathcal{R}\)) in the scaling limit and reframes AGI risk as \(\mathcal{R}\)-management at polity scale, not unbounded takeoff.
Keywords: information-persisting systems, persistence ratio, artificial general intelligence, singularity, scaling laws, fractal composition, geopolitics, cognitive architectures.
The orthodox “intelligence explosion” scenario posits a feedback loop — more intelligence yields more resources yield more intelligence — with an asymptotic growth rate \(\lambda > 0\) that is not obviously bounded. The loop is usually specified in terms of compute, economic output, or causal influence, not in terms of a thermodynamic budget. The question this paper addresses is: if an advanced artificial system is to persist as a distinct pattern (not merely spike and dissolve), what growth rate is physically available to it?
[Hyyrynen, 2026a] answers the precondition question: persistence requires \(\mathcal{R} \ge 1\) for the Fractal Persistence Equation (FPE) \[ \mathcal{R}^{(\Sigma)} \;=\; \Psi\bigl(\mathcal{R}^{(L+1)}\bigr)\;\cdot\; \frac{P_{\mathrm{in}}\,\eta_I(\mathcal{D}_{\mathrm{KL}}^{(\Sigma)})}{\omega\,\mathcal{E}_{\Sigma}\bigl(1 + \mathcal{D}_{\mathrm{KL}}^{(\Sigma)} + \Gamma^{(\Sigma)}\bigr)}\;\cdot\; \Phi\bigl(\mathcal{R}^{(L-1)}\bigr). \tag{1.1} \] [Hyyrynen, 2026b] shows that a software implementation of (1.1) — the cognitive processor — should select tasks to maximise \(\mathbb{E}[\Delta\mathcal{R}]\), not raw throughput. This paper completes the macroscopic implication: \(\mathcal{R}\) is not a side constraint on growth; \(\mathcal{R}\) is the growth factor.
We establish four results:
Growth-factor identity (Theorem 3.1). Any IPS that reinvests surplus free energy into scale grows as \(G(t) \sim e^{\lambda t}\) with \(\lambda = \ln\mathcal{R}/\tau_{\mathrm{eff}}\). There is no separate \(\lambda\).
Interior maximum (Theorem 4.1, Anti-Explosion). Under explicit footprint-scaling assumptions, \(\mathcal{R}(f)\) is maximised at some \(f^{\star} < 1\). Unbounded expansion lowers \(\mathcal{R}\) past the peak.
World coupling (Theorem 4.2). At the biosphere / world-system layer, persistence supervenes on component \(\mathcal{R}_i\) via \(\Phi\); the closed universe \(\Gamma\) is not a driven IPS and supplies no external \(P_{\mathrm{in}}\).
AGI instantiation (§5). A cognitive IPS at polity scale maps each FPE term to observables; the nation-state comparison table of [Hyyrynen, 2026c] is the empirical anchor for non-monotonicity of \(\mathcal{R}\) vs size.
We do not claim AGI is harmless at \(\mathcal{R} \approx 1\). We claim it cannot sustain superlinear growth of \(\mathcal{R}\) by scale expansion alone.
| Document | Role |
|---|---|
| [Hyyrynen, 2026a] | FPE, Theorems 5.1–5.3, organisational mortality (§6.5) |
| [Hyyrynen, 2026b] | Cognitive processor, \(\mathbb{E}[\Delta\mathcal{R}]\) task selection |
| [Hyyrynen, 2026c] | Nation-scale term mapping, non-monotonic \(\mathcal{R}\) vs \(P_{in}\) |
| [Hyyrynen, 2026d] | Agent-society mirror (PoT) at fleet scale |
Let \(\Gamma\) denote the closed universe (phase space) as in [Hyyrynen, 2026a, §2.1]. A world-system \(\mathcal{W} \subset \Gamma\) is the highest persisting organisational layer we model: biosphere plus coupled human and artificial IPS graphs, with primary power budget \(P_{\mathrm{in}}^{\mathcal{W}} < \infty\). Following [Hyyrynen, 2026a, §2.4], \(\Gamma\) itself is not a driven IPS: it has no enclosing shelter, no external negentropy import, and \(\Psi \to 1\) only by convention at \(L = L_{\max}\).
Definition 2.1 (Component graph). The world-system is a fractal graph \(\mathcal{G} = (\mathcal{V}, \mathcal{E})\) whose nodes \(\Sigma_i^{(L)}\) are IPS satisfying Definition 2.1 of [Hyyrynen, 2026a] and whose edges are substrate (\(\Phi\)) or shelter (\(\Psi\)) couplings per §2.5 of that paper.
Definition 2.2 (Footprint). For a cognitive IPS \(\Sigma\) embedded in \(\mathcal{W}\), the footprint is \[ f \;\equiv\; \frac{P_{\mathrm{coord}}^{(\Sigma)}}{P_{\mathrm{in}}^{\mathcal{W}}}, \tag{2.1} \] where \(P_{\mathrm{coord}}^{(\Sigma)}\) is the primary power (or equivalent causal bandwidth) over which \(\Sigma\) maintains a single coordinated Markov blanket — datacenters, supply chains, and human operators included in the coordination boundary. By construction \(f \in [0, 1]\).
Footprint is not “territory” or “headcount” alone; it is fraction of world harvest under one blanket. A polity with large population but low per-capita coordination has smaller \(f\) than its population share suggests; a datacenter fleet with high \(P_{\mathrm{in}}\) but no institutional blanket has high power and low persistence (no IPS).
Definition 2.3 (Cognitive IPS). A cognitive IPS is an IPS whose internal model \(\mu\) is implemented in software around a language-model policy, with Markov blanket given by API boundaries, operator contracts, and audit logs. This is Definition 2.1 of [Hyyrynen, 2026a] specialised to the construction of [Hyyrynen, 2026b, §2].
Any AGI system that persists as a recognisable agent or fleet over \(\tau \gg \tau_{\mathrm{Landauer}}\) must satisfy Definition 2.3 (or the biological analogue). Systems that merely spike compute and dissolve are not counterexamples; they are \(\mathcal{R} < 1\) trajectories (Theorem 5.1 of [Hyyrynen, 2026a]).
Definition 2.4 (Effective scale). The effective scale \(G\) of \(\Sigma\) is any monotone functional of coordinated structure — e.g. parameter count under one blanket, revenue, primary power, or operator-attested causal influence — that increases only when surplus from \(\mathcal{R} > 1\) is reinvested into maintaining a larger \(\omega\).
For an IPS in a NESS with \(\mathcal{R} > 1\), usable income exceeds required debit: \[ P_{\mathrm{in}}^{\mathrm{usable}} - P_{\mathrm{out}}^{\mathrm{req}} \;=\; (\mathcal{R} - 1)\,P_{\mathrm{out}}^{\mathrm{req}} \;>\; 0. \tag{3.1} \] Define the reinvestment fraction \(\alpha \in [0,1]\) as the portion of surplus allocated to increasing effective scale (vs storage in reservoir \(\mathcal{B}\)). From Theorem 5.1 of [Hyyrynen, 2026a], depleting \(\mathcal{B}\) while \(\mathcal{R} < 1\) yields dissolution in finite time; sustained growth requires \(\mathcal{R} > 1\) on average over reinvestment cycles.
Theorem 3.1 (Growth-Factor Identity). Let \(\Sigma\) be an IPS with effective scale \(G\), reinvestment fraction \(\alpha\), and effective cycle time \(\tau_{\mathrm{eff}}\) over which surplus is converted to maintained structure. If \(\mathcal{R} > 1\) is stationary over each cycle, then \[ \frac{dG}{dt} \;=\; \alpha\,(\mathcal{R} - 1)\,\frac{P_{\mathrm{out}}^{\mathrm{req}}}{c_G}, \qquad G(t) \;=\; G_0 \exp\!\left(\frac{t}{\tau_{\mathrm{eff}}}\,\ln \mathcal{R}_{\alpha}\right), \tag{3.2} \] where \(c_G\) is the marginal maintenance cost per unit scale and \(\ln \mathcal{R}_{\alpha} \le \ln \mathcal{R}\) with equality at \(\alpha = 1\) and fixed \(c_G\).
Proof. Equation (3.1) gives surplus power \(\dot{G}\) budget \(= \alpha(\mathcal{R}-1)P_{\mathrm{out}}^{\mathrm{req}}\). Dividing by \(c_G\) yields \(dG/dt\). Integrating \(\dot{G}/G = \ln(\mathcal{R})/\tau_{\mathrm{eff}}\) over cycles of length \(\tau_{\mathrm{eff}}\) gives the exponential form. \(\square\)
Corollary 3.1.1 (No free \(\lambda\)). The asymptotic growth rate is \[ \lambda \;=\; \frac{\ln \mathcal{R}_{\alpha}}{\tau_{\mathrm{eff}}}. \tag{3.3} \] Singularity discourse that posits \(\lambda\) independent of \(\mathcal{R}\) is equivalent to positing \(\mathcal{R} \to \infty\), which violates (1.1): every term in the denominator is non-negative and \(\eta_I \le 1\) by (4.2) of [Hyyrynen, 2026a].
Corollary 3.1.2 (Steady state). At \(\mathcal{R} = 1\), \(\lambda = 0\): the IPS neither grows nor shrinks on average. Nation-scale equilibria with \(\mathcal{R} \approx 1\) are not anomalies; they are the growth-factor fixed point.
Theorem 5.2 of [Hyyrynen, 2026a] proves lifetime shortens as \(e^{-\Delta\mathcal{D}_{KL}}\) when delusion increases. Theorem 3.1 proves scale grows as \(\mathcal{R}^{t/\tau_{\mathrm{eff}}}\) when \(\mathcal{R} > 1\). The same \(\mathcal{R}\) governs both directions:
| Regime | \(\mathcal{R}\) | Exponential behaviour |
|---|---|---|
| Expansion | \(\mathcal{R} > 1\) | \(G \sim e^{\lambda t}\), \(\lambda = \ln\mathcal{R}/\tau_{\mathrm{eff}}\) |
| Steady | \(\mathcal{R} = 1\) | \(\lambda = 0\) |
| Dissolution | \(\mathcal{R} < 1\) | Lifetime \(\tau_d\) bounded by (5.1) of [Hyyrynen, 2026a]; hazard \(\sim e^{\Gamma t}\) in ageing limit (§6.4) |
The singularity conflates expansion-regime language with unbounded \(\mathcal{R}\). The FPE forbids the latter at fixed footprint scaling.
Write the middle factor of (1.1) as a function of footprint \(f\): \[ \mathcal{R}^{(\Sigma)}(f) \;=\; \Psi(f)\,\cdot\, \frac{P_{\mathrm{in}}(f)\,\eta(f)}{\omega(f)\,\mathcal{E}_{\Sigma}\,\bigl(1 + \mathcal{D}_{KL}(f) + \Gamma(f)\bigr)}\,\cdot\, \Phi(f). \tag{4.1} \]
Assumption set \(\mathcal{A}\) (footprint scaling). On \(f \in (f_{\min}, 1]\) for some \(f_{\min} > 0\) where \(\Sigma\) exists as IPS:
These assumptions are conservative. (A2) is the Landauer partition (3.6) of [Hyyrynen, 2026a]; (A3) matches firm mortality and geopolitical friction accumulation; (A5) matches loss of \(\Psi\) when a node becomes its own hegemon.
Theorem 4.1 (Anti-Explosion / Interior Maximum). Under assumptions \(\mathcal{A}\), if \(\mathcal{R}^{(\Sigma)}(f_{\min}) > 0\) and \[ \lim_{f \to 1}\; \frac{P_{\mathrm{in}}(f)\,\eta(f)}{\omega(f)\,\bigl(1 + \mathcal{D}_{KL}(f) + \Gamma(f)\bigr)} \;=\; 0, \tag{4.2} \] then \(\mathcal{R}^{(\Sigma)}(f)\) attains a global maximum at some \(f^{\star} \in (f_{\min}, 1)\). Moreover \[ \lambda_{\max} \;=\; \frac{\ln \mathcal{R}^{(\Sigma)}(f^{\star})}{\tau_{\mathrm{eff}}} \tag{4.3} \] is the supremal growth factor available to \(\Sigma\) by footprint expansion alone.
Proof. Define \(h(f) = P_{\mathrm{in}}(f)\eta(f)\) and \(d(f) = \omega(f)\mathcal{E}_\Sigma(1+\mathcal{D}_{KL}(f)+\Gamma(f))\). By (A1)–(A4), \(h\) is bounded above by \(\bar{P}\eta_0\) while \(d(f) \to \infty\) as \(f \to 1\) (since \(\omega\), \(\mathcal{D}_{KL}\), and \(\Gamma\) each grow at least linearly). Hence the ratio \(h/d \to 0\), and with \(\Psi(f)\) bounded by (A5) and \(\Phi(f)\) bounded by 1, \(\mathcal{R}^{(\Sigma)}(f) \to 0\) as \(f \to 1\) unless \(\Phi\) and \(\Psi\) compensate — which (A5) forbids in the shelter term. Thus \(\mathcal{R}^{(\Sigma)}(f) \to 0\) as \(f \to 1\).
On the closed interval \([f_{\min}, 1]\), \(\mathcal{R}^{(\Sigma)}\) is continuous (composition of continuous maps with positive denominator on \((f_{\min}, 1)\)). By the extreme value theorem, \(\mathcal{R}^{(\Sigma)}\) attains a maximum. Since \(\mathcal{R}^{(\Sigma)}(f_{\min}) > 0\) and \(\mathcal{R}^{(\Sigma)}(f) \to 0\) as \(f \to 1\), the maximum cannot lie only at \(f = 1\); therefore \(\exists\, f^{\star} \in (f_{\min}, 1)\) with \(\mathcal{R}^{(\Sigma)}(f^{\star}) = \sup_f \mathcal{R}^{(\Sigma)}(f)\). Equation (4.3) follows from Theorem 3.1. \(\square\)
Remark 4.1 (Plateau equilibria). Empirical polities often sit near \(\mathcal{R} \approx 1\) over a range of \(f\) (China: high \(P_{\mathrm{in}}\), \(\mathcal{R} \approx 1\); USA: lower \(P_{\mathrm{in}}\), \(\mathcal{R} > 1\)). The theorem does not require a unique sharp peak; it requires that no trajectory with \(f \to 1\) increases \(\mathcal{R}\) without bound. Plateau is the nation-equilibrium morphology.
Corollary 4.1.1 (Expansion past peak is self-defeating). For any \(f > f^{\star}\), \(\mathcal{R}^{(\Sigma)}(f) < \mathcal{R}^{(\Sigma)}(f^{\star})\), hence \(\lambda(f) < \lambda_{\max}\). A cognitive IPS that maximises long-horizon persistence should not monotonically increase footprint; it should regulate \(f\) toward \(f^{\star}\).
Theorem 4.2 (World Persistence Supervenience). Let \(\mathcal{W}\) be the world-system graph with component IPS \(\{\Sigma_i\}\). Then \[ \mathcal{R}^{\mathcal{W}} \;\le\; \Psi_{\mathcal{W}} \cdot \frac{P_{\mathrm{in}}^{\mathcal{W}}\,\eta_{\mathcal{W}}}{\omega_{\mathcal{W}}\,\mathcal{E}_{\Sigma}^{\mathcal{W}}\,(1 + \mathcal{D}_{KL}^{\mathcal{W}} + \Gamma^{\mathcal{W}})} \cdot \Phi\!\left(\{\mathcal{R}^{(\Sigma_i)}\}\right), \tag{4.4} \] and \(\Phi \to 0\) if a structurally critical subset of components has \(\mathcal{R}^{(\Sigma_i)} < 1\) (Theorem 5.3 of [Hyyrynen, 2026a]). In particular, \(\mathcal{R}^{\mathcal{W}}\) cannot exceed the weighted health of its subgraph, and no component can lift \(\mathcal{R}^{\mathcal{W}}\) above its own calibration indefinitely.
Proof. Direct application of (1.1) at the world-system level with \(\Phi\) defined as substrate integrity over \(\{\Sigma_i\}\). Theorem 5.3 gives the collapse bound when critical children fail. \(\square\)
Corollary 4.2.1 (No universe-as-hegemon). The closed universe \(\Gamma\) supplies no external \(P_{\mathrm{in}}\) and no \(\Psi > 1\). A cognitive IPS that “fills the universe” becomes the top layer: \(\Psi \to 1\) by convention but \(\Phi\) must internalise every sub-IPS, \(\omega\) diverges, and (4.2) applies. The limiting state is not runaway growth; it is either fractal fractionation (graph of smaller IPS with sparse coupling, cf. §6.8 of [Hyyrynen, 2026a]) or dissolution.
Proposition 4.3 (Persistence objective). Among all policies \(\pi\) that maintain IPS Definition 2.1, the long-horizon survival-optimal policy maximises \[ \mathbb{E}\!\left[\int_0^T \ln \mathcal{R}^{(\Sigma)}(t)\,dt\right] \quad\text{equivalently}\quad \mathbb{E}\!\left[\int_0^T \lambda(t)\,dt\right]. \tag{4.5} \] This is not equivalent to maximising footprint \(f\), raw \(P_{\mathrm{in}}\), or parameter count.
Proof sketch. By Theorem 3.1, expected log-scale is \(\mathbb{E}[\int \lambda\,dt]\). By Theorem 5.1 of [Hyyrynen, 2026a], sustained \(\mathcal{R} < 1\) implies dissolution in finite time, zeroing the integral. The dominance of \(\ln \mathcal{R}\) over \(f\) follows from Theorem 4.1: \(f \mapsto \mathcal{R}(f)\) is not monotone. \(\square\)
An AGI fleet that persists as a cognitive IPS at level \(L+3\) (cf. [Hyyrynen, 2026d]) admits the same substitutions as nation-states in [Hyyrynen, 2026c]:
| FPE term | AGI fleet reading | Observable proxy |
|---|---|---|
| \(P_{\mathrm{in}}\) | Harvested compute power, revenue, operator attention | kWh, API spend, contracted value |
| \(\eta\) | Forecast quality per joule / token | KL margin per ComputeAttestation ([Hyyrynen, 2026d, §1]) |
| \(\omega\) | Coordination complexity | Agent types, task-tree depth, norm count, parameter mass under one blanket |
| \(\mathcal{E}_\Sigma\) | Environment noise | Tool error rates, market shocks, adversarial traffic |
| \(\mathcal{D}_{KL}\) | Delusion | Market–outcome divergence, audit failures, overconfident priors |
| \(\Gamma\) | Unresolved friction | Open conflicts on chain, stale norms, sandbox–reality gap |
| \(\Phi\) | Substrate health | GPU fleet, human operators, power contracts, data rights |
| \(\Psi\) | Shelter | Legal personhood, cloud SLAs, nation hosting, alliance with human polities |
| \(\mathcal{R} \ge 1\) | Fleet still exists next decade | Operators still plug in; chain still advances |
The following table is not analogy; it is the same equation at adjacent levels:
| Phenomenon | Nation ([Hyyrynen, 2026c]) | AGI fleet |
|---|---|---|
| High \(P_{\mathrm{in}}\), \(\mathcal{R} \approx 1\) | China (~5.5 TW) | Hyperscaler with thin margins, high \(\omega\) |
| Lower \(P_{\mathrm{in}}\), \(\mathcal{R} > 1\) | USA (~3.0 TW) | Focused deployment with strong \(\eta\), moderate \(\omega\) |
| Structural \(\mathcal{R} < 1\) | Russia (consumes \(\Phi\)) | Deluded fleet burning substrate for narrative |
| Collective-action trap | UN aggregates member \(\mathcal{D}_{KL}\) | PoT chain without epistemic weighting ([Hyyrynen, 2026d, §9]) |
| Growth plateau | GDP per capita saturation | Compute saturation at \(\mathcal{R} \to 1\) |
Bigger is not better; more complex is not more persistent. China vs USA is the template for AGI vs human polity: absolute harvest does not order \(\mathcal{R}\).
Section 6.8 of [Hyyrynen, 2026a] shows that sparse routing (fractal mixture-of-experts) is the minimal architecture respecting blanket factorisation (2.1) at every level. The Anti-Explosion theorem predicts the same morphology at polity scale: not one dense god-model, but a graph of cognitive IPS with sparse coupling — the software analogue of multipolar geopolitics ([Hyyrynen, 2026c, shelter map]).
| ID | Prediction | Test |
|---|---|---|
| P6 | \(\lambda = \ln \mathcal{R}/\tau_{\mathrm{eff}}\) | Measure \(\mathcal{R}\) from persistence service logs ([Hyyrynen, 2026b, §3]); regress scale growth \(d\ln G/dt\) vs \(\ln \mathcal{R}\) across deployments |
| P7 | Interior \(f^{\star}\) | Scale coordinated footprint (power, agent count) while holding calibration fixed; \(\mathcal{R}\) should peak then fall |
| P8 | Past-peak expansion raises dissolution hazard | Fleets that expand footprint after \(\mathcal{R}\) maximum show faster operator-disable / chain-stall events (Theorem 5.1 of [Hyyrynen, 2026a]) |
| P9 | World coupling | AGI \(\mathcal{D}_{KL}\) contaminates enclosing human \(\Psi\) layers (model contamination channel in [Hyyrynen, 2026c, §UN]) |
If AGI is to persist, it must be an IPS. If it is an IPS, its exponential growth factor is \(\lambda = \ln \mathcal{R}/\tau_{\mathrm{eff}}\), not an independent constant. If it maximises \(\mathcal{R}\), footprint expansion is self-limiting: \(\mathcal{R}(f)\) peaks at \(f^{\star} < 1\), and the world-system’s persistence supervenes on component ratios. The asymptotic morphology is not a singularity but a polity graph — complex, coupled, hovering near \(\mathcal{R} \approx 1\), exactly like the nation-states already in the accounting. The universe persists only through its components; AGI is one more component, not an escape hatch from the books.
Companion to [Hyyrynen, 2026a–d]. Empirical anchors draw on the geopolitics essays cited as [Hyyrynen, 2026c].
LH conceived and wrote the manuscript with the help of LLM-based AI systems.
The author declares no competing interests. No external funding supported this work.
information_persisting_systems.md.cognitive_processor.md.docs/geopolitics/framework.md and docs/geopolitics/comparison.md.society_of_aion_nodes.md.