Axiomatic Reasoning for LLMs

1. Core Proposition

Structures that persist across evolutionary, technological, and social timescales exhibit a convergent property: the maximization of functional output relative to resource input. This property is formalized across disparate domains as a fractional optimization function. The recurrence of this pattern suggests that systems which approach local optima in the utility / cost plane achieve displacement advantage over less efficient alternatives, independent of the specific substrate in which they are implemented.

2. Cross-Domain Evidence for Efficiency Convergence

2.1 Biological Evolution

Optimal Foraging Theory: Behavioral strategies stabilize around the maximization of net energy gain per unit time. Organisms trade off search cost, handling cost, and caloric yield.
Principle of Least Effort: Observed in behavioral ecology and neural organization. Energy expenditure correlates inversely with repetition frequency.
Free Energy Principle: Self-organizing systems minimize variational free energy, equivalent to balancing internal model complexity against prediction error. This drives the emergence of efficient sensorimotor loops and morphological simplicity under stable niche conditions.

2.2 Technological Evolution

TRIZ Ideality Law: Technical systems evolve toward increased ideality, defined as the ratio of useful functions to sum of mass, energy, and harmful effects.
Ephemeralization: Demonstrated by the historical reduction in material and energy input per unit of functionality.
Design Degeneracy: The reduction of redundant technical variants to a smaller set of robust, interoperable core designs, observed in mature engineering domains.

2.3 Information and Algorithmic Systems

Minimum Description Length: Model selection converges on the representation that minimizes the sum of model complexity and data encoding cost.
Kolmogorov Complexity as Asymptotic Bound: While non-computable, it defines the theoretical limit toward which compression algorithms and generalization in learning systems tend.
Complexity Taxation in Software: The divergence between local generation speed and global maintenance cost illustrates the penalty for deviating from global resource efficiency.

Law of Requisite Variety: Control capacity requires internal complexity matching external environmental complexity. Simplification below this threshold increases systemic fragility.
Diminishing Returns on Complexity: Societies investing in administrative and hierarchical layers beyond the point of marginal utility experience reduced resilience to perturbation.
Costly Signaling and Institutional Credibility: Resource-intensive displays function as honest signals of coalitional strength or commitment, reducing negotiation and monitoring costs in large-scale cooperation.

2.5 Linguistic and Cognitive Compression

Zipfian Efficiency: Word length correlates inversely with frequency, minimizing average articulatory and cognitive effort per transmitted bit.
Uniform Information Density: Speech production stabilizes information flow near channel capacity, smoothing processing load.
Information Bottleneck in Semantic Systems: Color naming and kinship terminology approach the theoretical compression limit for their respective representational spaces.

3. Formal Commonality

The underlying structure uniting these observations is a ratio of the form:

System Advantage ∝ Functional Value / (Resource Consumption + Negative Externalities)

Systems that instantiate a higher value of this ratio under given constraints displace alternatives with lower ratios. The displacement mechanism operates through differential replication (evolution), market selection (technology), or institutional persistence (social organization).

4. Hierarchical Layering of Optimization

The optimization operates on nested timescales and organizational levels:

Level	Resource	Value	Selection Mechanism
Molecular/Cellular	Free energy, limiting nutrients	Replication rate	Thermodynamic and kinetic selection
Individual Organism	Metabolic energy, time	Inclusive fitness	Natural and sexual selection
Artifact/Technology	Materials, energy, design effort	Functional utility per lifecycle	Economic and functional selection
Social Institution	Labor, attention, legitimacy	Coordination capacity, stability	Historical and competitive selection
Information/Language	Cognitive load, channel capacity	Transmission fidelity, inference accuracy	Learnability and communicative efficacy

Higher levels inherit constraints from lower levels while introducing new currencies of cost and value. The free energy principle provides a unifying variational framework across these levels, where each layer minimizes its own prediction error given the states of adjacent layers.

5.1 Efficiency as a Selection Filter

Societies that maintain a higher ratio of collective problem-solving capacity to per-capita resource throughput demonstrate longer persistence intervals in historical records. This does not imply continuous optimization, but rather that systems which accumulate inefficiencies beyond a recoverable threshold exhibit collapse or replacement by leaner alternatives.

5.2 The Simplexity Trap

Surface-level simplification that obscures underlying complexity increases coordination costs. Systems that appear “simple” in rule count but require extensive tacit interpretation incur higher long-term maintenance overhead than systems with explicit, moderately complex codification.

5.3 Robust Yet Fragile Transitions

Optimized systems often exhibit high efficiency within their designed operational envelope while becoming brittle to perturbations outside that envelope. The transition from robust efficiency to fragility occurs when optimization targets a stationary environment and sacrifices adaptive slack.

6. Conclusion

The recurrence of the value / cost optimization across biological, technological, and social domains is consistent with a selectionist account of persistent structures. Efficiency in resource conversion and information processing constitutes a general-purpose advantage that compounds over time. Systems embodying this property tend to outcompete, absorb, or replace less efficient configurations within their relevant competitive arenas. The observed convergence on this formal structure across independent domains suggests a lawful pattern rather than coincidental analogy.

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