Theory of Energy Fractals (TEF): A Unified Framework for Systemogenesis Across Physical, Biological, and Cognitive Domains

# Theory of Energy Fractals (TEF)## A Formal Meta-Theory of Admissible Descriptions Under Finite Resources Author: Yaroslav Ikhnenko ikhnenko.yaroslav@gmail.comStatus: Canonical / Append-Only Category: Meta-theory of causally admissible descriptions ================================================================ABSTRACT================================================================ The Theory of Energy Fractals (TEF) is a meta-theoretical framework thatformalizes the structural conditions under which descriptions of realityremain causally admissible under finite resources. TEF does not introduce new physical entities, dynamics, or laws.Instead, it constrains the space of possible descriptions, models,and computations by enforcing resource-bounded admissibility. Two foundational constraints define the framework:Delta (irreducible asymmetry) and Omega (prohibition of global closure).From these follow the necessity of interfaces, regimes, levels,operator-governed transitions, and stable incomputability. TEF applies uniformly across physical, mathematical, biological,cognitive, and computational domains without reduction between them. ================================================================PART I — CORE THEORY================================================================ ## 1. Scope and Intent TEF is a meta-theory.It specifies conditions on descriptions, not on physical reality directly. TEF does not:- propose new ontological primitives,- replace domain-specific theories,- provide predictive dynamics. TEF constrains when a description is causally admissible, meaning itsupports reproducible causal action under finite resources. --- ## 2. Finite Resources as a Structural Constraint Axiom 2.1 (Finite Resources) All physically realizable systems operate under finite resources,including energy, time, memory, precision, control bandwidth,and cognitive capacity. Corollary 2.1 Any description whose causal use requires unbounded resourcesis physically inadmissible. This constraint is structural, not practical. --- ## 3. Omega (Omega): Prohibition of Global Closure Axiom 3.1 (Omega Constraint) No description of reality can be simultaneously:- globally complete,- causally exhaustive,- computable,- and realizableunder finite resources. Definition 3.1 (Global Closure) Global closure is the existence of a single description that fullydetermines all admissible states and transitions of reality without loss. Proposition 3.1 Any attempt at global closure constitutes an Omega-violation. Omega forbids:- final theories of everything,- total microstate control,- global reversibility,- zero-loss global optimization. Local computability is admissible.Global computability is forbidden. --- ## 4. Delta (Delta): Irreducible Asymmetry Axiom 4.1 (Delta Constraint) Delta is the irreducible asymmetry that constitutes the condition ofpossibility for difference itself. Delta is not an event, fluctuation, or stochastic disturbance. Formal statement: Delta : X -> P(X) From any admissible state space X, more than one future remains admissible. Delta guarantees impossibility of total closure.Delta does not guarantee realization of all possibilities. Delta generates admissible futures.Omega prevents their global exhaustion. --- ## 5. Minimal Interaction Grammar Axiom 5.1 All interactions reduce to two primitive modes:- (+) amplification, coupling, coherence- (-) dissipation, decoupling, dispersion Proposition 5.1 There are no neutral interactions.Apparent neutrality arises only through averaging. Stability arises from structured compositions of (+)constrained by (-). Selection follows asymmetry of reproducibility, not optimization. --- ## 6. Impossibility of Microstate Causality Proposition 6.1 Microstate dynamics are:- high-dimensional,- sensitive to perturbations,- incomputable under finite resources. Corollary 6.1 Causality cannot be defined at the microstate level. Any causal definition at that level would be unreproducibleand operationally meaningless. --- ## 7. Interfaces and the Origin of Causality Definition 7.1 (Interface) An interface is a mapping: I : X -> Y where:- X is a microstate space with C_micro(X, eps, T) > B,- Y is an observable space with bounded reproducibility cost. Proposition 7.1 Interfaces are necessarily lossy. Corollary 7.2 Lossless interfaces violate Omega. Principle 7.1 Causality exists only at interfaces.All causal action is interface reconfiguration. --- ## 8. Regimes and Levels Definition 8.1 (Regime) A regime is a stabilized behavioral pattern observablethrough an interface. Definition 8.2 (Level) A level is a regime that functions as an effective degreeof freedom for higher-order descriptions. Levels are forced compressions required to preserve predictabilityunder finite resources. Level Transition Criterion A transition X -> Y is admissible iff: C_micro(X, eps, T) > B C_micro(Y, eps, T) = theta Cost(I) B- Cl (Collapse): reduce degrees of freedom when Rep < theta- Rw (Rewrite): reconfigure interface when correlations change- Mrg (Merge): unify interfaces under stable correlations Constraint 10.1 Rewrite is admissible only if predictive power increases.Repeated rewrite without improvement invalidates the model. Omega functions as a global stop-rule. --- ## 11. Stable Incomputability Definition 11.1 A system exhibits stable incomputability iff: C_micro(X, eps, T) > B Rep(I) >= theta Cost(I)