Golden–Ratio Partition of Information

• Golden–ratio partition is a self-similar principle defining the optimal 0.618 ratio between known and unknown information in adaptive systems.
• It underpins neural spike coding by matching symbol probabilities to the golden ratio, thereby maximizing channel capacity and transmission efficiency.
• The recursive allocation facilitates antifragility and robust adaptation across complex systems, including cognitive architectures and robotic control frameworks.

The golden-ratio partition of information is a structural allocation principle in information theory and neurodynamics, prescribing the division of information between "known" and "unknown" such that their ratios satisfy the self-similarity condition $p : (1-p) = 1 : p$, yielding $p = 1/\varphi \approx 0.618$, where $\varphi = (1+\sqrt{5})/2$ is the golden ratio. This partition emerges as the unique information-optimal allocation for certain neural spike codes and is proposed as a scale-free, recursive design principle for adaptive systems operating at the edge of criticality. The concept is distinguished from optimization-driven allocations (e.g., the maximizer of informational balance functions) by its structural invariance and self-similar nesting properties, supporting antifragile adaptation and maximal dynamic range in complex systems (Deng, 2023, Padilla et al., 16 Feb 2026).

1. Formal Definition and Mathematical Origin

The golden-ratio partition is defined by solving the self-similarity equation

$\frac{p}{1-p} = \frac{1}{p}$

which leads to the quadratic $p^2 + p - 1 = 0$, with the solution in $(0,1)$

$$p_\varphi = \frac{\sqrt{5} - 1}{2} = \frac{1}{\varphi} \approx 0.618.$$

This value prescribes the ratio of "known" to "unknown" information such that the allocation remains recursive: the ratio between known and unknown is identical to the ratio of known to total. This mathematically encodes a self-similar, scale-free structure, distinct from a mere extremization condition (Padilla et al., 16 Feb 2026).

2. Golden Ratio Channel Capacity in Neural Spike Codes

In the context of neural communication, spike bursting patterns can be modeled as codes with symbols taking varying time to emit. Considering a binary spike-number code with two symbols—one- and two-spike bursts—with emission times $\tau_1$ and $\tau_2=2\tau_1$, the system-specific transmission rate is

$R(p) = \frac{H(p)}{T(p)},$

where $p = 1/\varphi \approx 0.618$0 is the entropy and $p = 1/\varphi \approx 0.618$1 the mean symbol-processing time. The maximal $p = 1/\varphi \approx 0.618$2 under normalization yields two key relations:

• $p = 1/\varphi \approx 0.618$3
• $p = 1/\varphi \approx 0.618$4

Equating and solving yields $p = 1/\varphi \approx 0.618$5, giving $p = 1/\varphi \approx 0.618$6 and $p = 1/\varphi \approx 0.618$7, where $p = 1/\varphi \approx 0.618$8 is the golden ratio conjugate. The resulting channel capacity is

$p = 1/\varphi \approx 0.618$9

Thus, the golden-ratio partition of symbol probabilities is uniquely optimal for time-constrained binary neural coding (Deng, 2023).

For an $\varphi = (1+\sqrt{5})/2$0-ary spike-number code with emission times following $\varphi = (1+\sqrt{5})/2$1, the unique capacity-achieving distribution is $\varphi = (1+\sqrt{5})/2$2 with normalization $\varphi = (1+\sqrt{5})/2$3, where for $\varphi = (1+\sqrt{5})/2$4 this reduces to the golden ratio result above. The system-wide mean transmission rate (assuming equiprobability) is maximized for $\varphi = (1+\sqrt{5})/2$5 (quaternary code), with $\varphi = (1+\sqrt{5})/2$6, indicating the optimality of a quaternary spike-number code for maximizing average information throughput (Deng, 2023).

3. The Information-theoretic Balance Function and its Maximizer

An alternative approach employs the information-theoretic "balance function"

$\varphi = (1+\sqrt{5})/2$7

to quantify the net informational gain from balancing explained variance ($\varphi = (1+\sqrt{5})/2$8) against unexplained novelty ($\varphi = (1+\sqrt{5})/2$9). The first and second derivatives

$\frac{p}{1-p} = \frac{1}{p}$0

show strict concavity. The unique maximizer $\frac{p}{1-p} = \frac{1}{p}$1 is found by setting $\frac{p}{1-p} = \frac{1}{p}$2, corresponding to maximum information-balance or maximal epistemic vulnerability: high confidence, but with residual uncertainty contributing maximal excess surprise.

Notably, $\frac{p}{1-p} = \frac{1}{p}$3 is distinct from $\frac{p}{1-p} = \frac{1}{p}$4: the former extremizes a specific balance, while the latter preserves a self-similar, recursive allocation. $\frac{p}{1-p} = \frac{1}{p}$5 is not an extremum but a structurally-mediated partition that confers invariance under hierarchical inference (Padilla et al., 16 Feb 2026).

4. Self-Similarity, Recursion, and Criticality

The golden-ratio partition's self-similar property manifests in recursive nesting: at every scale, the ratio of known to unknown information remains invariant. Embedding this partition within a compute–inference–model–action (CIMA) loop, a system maintains $\frac{p}{1-p} = \frac{1}{p}$6 near $\frac{p}{1-p} = \frac{1}{p}$7 by dynamically steering explained and unexplained variance via feedback. The dynamical update

$\frac{p}{1-p} = \frac{1}{p}$8

modulates model parameters to maintain this golden mean. The regime near $\frac{p}{1-p} = \frac{1}{p}$9 aligns with the edge of criticality in network dynamics. Empirical and theoretical studies (e.g., Kinouchi & Copelli 2006, Shew et al. 2009) demonstrate that at $p^2 + p - 1 = 0$0, neuronal networks show power-law activity distributions ($p^2 + p - 1 = 0$1), $p^2 + p - 1 = 0$2 spectral scaling, and maximized dynamic range (Padilla et al., 16 Feb 2026).

5. Antifragility and Adaptive Implications

A system operating near the golden-ratio partition converts environmental perturbation (indexed by, e.g., volatility $p^2 + p - 1 = 0$3) into net information gain. The payoff function

$p^2 + p - 1 = 0$4

captures improvement in predictive-coding error. Antifragility is formally defined as

$p^2 + p - 1 = 0$5

so the system derives disproportionately larger learning gains from larger perturbations. The golden-ratio regime ensures this convex relationship, facilitating continuous, scale-free adaptation and learning, and maximizing resilience to uncertainty (Padilla et al., 16 Feb 2026).

6. Applications and Broader Impact

The golden-ratio partition serves as a general design principle for allocation of computational or adaptive resources in diverse domains:

• Neural systems: Optimal binary spike-number codes with golden-ratio probabilities maximize channel capacity under time constraints; quaternary codes maximize mean transmission rate (Deng, 2023).
• Adaptive algorithms: The partition prescribes $p^2 + p - 1 = 0$6 for exploitation (prediction) and $p^2 + p - 1 = 0$7 for exploration (novelty/openness), supporting robust learning and flexibility (Padilla et al., 16 Feb 2026).
• Robotics and autonomous systems: Dynamically steering explained versus unexplained variance toward $p^2 + p - 1 = 0$8 maintains adaptability to environmental change.
• Cognitive architectures: Neuromodulatory mechanisms may target $p^2 + p - 1 = 0$9 to balance habitual and novel stimuli.
• Resilient engineering and finance: "Barbell" strategies grounded in the golden split optimize the tradeoff between safety and upside exposure.
• Complex adaptive systems: Ecological, social, and synthetic biological networks can exploit golden-ratio informatics to sustain operation at the edge of criticality.

This suggests the golden-ratio partition is not only a mathematical artifact but a viable regulatory principle for complex, adaptive, and antifragile systems.

7. Summary and Complementary Landmarks

Two complementary informational benchmarks emerge. The golden-ratio partition $(0,1)$0 confers recursive self-similarity and underpins continuous adaptation at criticality, while the maximizer $(0,1)$1 of the information-balance function identifies points of greatest informational vulnerability. Embedding $(0,1)$2 within adaptive feedback loops provides a recursive, scale-free mechanism for balancing prediction and surprise, enabling robust, antifragile operation across domains and organizational scales (Deng, 2023, Padilla et al., 16 Feb 2026).