The Algebra of Becoming

By: Jordan Vallejo

Introduction

The Algebra of Becoming is a formal framework describing how systems move from one realized state to the next under constraint.

At any moment a system occupies a realized state within a state space of admissible configurations. From that state multiple continuations may be possible, but only one successor state becomes realized. The Algebra of Becoming specifies the structural relations governing this transition.

Most systems continue through established governance structures such as rules, habits, or learned procedures. Under these conditions continuation is determined directly. When governance cannot determine the next action, the system must resolve indeterminacy before continuation can occur.

The Algebra of Becoming formalizes this resolution. It specifies the conditions under which interpretation becomes necessary and the process by which a governing trajectory is selected from competing possibilities.

The framework therefore provides a unified mathematical description of constraint-governed continuation and interpretive resolution.

Scope

The Algebra of Becoming is a structural theory of system continuation. It specifies formal relations governing:

• system states and state spaces

• constraint-governed continuation spaces

• determinacy of baseline governance

• conditions under which interpretation becomes necessary

• evaluation and selection of candidate meanings

• realization of successor states

The algebra describes the evaluation and selection of candidate interpretations once they exist. It does not specify the mechanisms by which candidates are generated.

Candidate generation depends on domain-specific processes that vary across systems and belongs to fields such as neuroscience, cognitive science, artificial intelligence, and organizational theory.

The Algebra of Becoming therefore restricts its scope to the structural conditions governing evaluation, selection, and continuation.

The Algebra of Becoming begins with the present system state σₜ. This symbol denotes the realized state of the system at time t. Formally, σₜ ∈ S, meaning the current state belongs to the system’s state space S, which is the set of all admissible configurations the system can occupy.

Formal Specification

From any state, the system can continue in multiple possible ways. The set of possible next states is the continuation space Ωₜ, defined as Ωₜ = Ω(σₜ ; K). Here Ω is the continuation operator that generates the possible next states, and K represents the system’s constraint structure. Constraints include physical laws, environmental limits, institutional rules, or any conditions that restrict how the system may evolve. The fundamental continuation law of the Algebra of Becoming states that the next realized state must lie within this space: σₜ₊₁ ∈ Ω(σₜ ; K).

Most of the time, the system does not need interpretation to continue because it already possesses baseline governance, denoted B. Baseline governance is the set of established response rules, habits, or learned procedures that normally determine how the system acts. Each rule in this set is evaluated against the current state through the determinacy test, which checks three conditions: Fit(B, σₜ, R), Rank(B, σₜ, R), and Feas(B, σₜ, R). The Fit condition asks whether a baseline rule applies to the current state. Rank determines whether one rule clearly dominates competing rules. Feas determines whether the trajectory implied by that rule is actually possible within the continuation space. These evaluations occur relative to the system’s reference structure R, which includes the system’s beliefs, goals, internal models, norms, or evaluative frameworks. If these conditions succeed, the next action is determined and the system proceeds directly to σₜ₊₁, the next realized state, without interpretation.

If these conditions fail, the system experiences Action Determinacy Loss, written ADLₜ = 1. ADL means that baseline governance cannot determine what action should occur next. At this point interpretation becomes necessary. The interpretive process begins when the system encounters a signal xₜ, where xₜ ∈ X and X denotes the signal space, the set of possible signals the system can receive.

The system then generates possible meanings for the signal through the candidate generation operator, producing a candidate set Qₜ defined by Qₜ = Gen(xₜ , σₜ ; E). Here Gen is the candidate generation function, and E is the generation environment, which includes the system’s memories, experiences, emotional states, cultural knowledge, and contextual associations that influence how meanings are produced.

Not all candidates are admissible. The candidate set is filtered through the system’s reference structure R, producing a subset Q_{R,t}, which contains only those interpretations consistent with the system’s reference framework and continuation constraints.

Interpretation becomes structurally necessary when the Candidate Door opens. This occurs when three conditions hold simultaneously: ADLₜ = 1, meaning determinacy is lost; |Q_{R,t}| ≥ 2, meaning at least two viable interpretations remain; and χₜ ≥ 2, where χₜ denotes trajectory divergence, the number of distinct continuation trajectories implied by the remaining candidates. In other words, multiple interpretations would lead the system to different future paths.

The remaining candidates are then evaluated using the candidate evaluation operator Θ_val^Q, which assesses each candidate relative to the system’s reference structure. This evaluation produces Y_Q, the set of scored or ranked candidate meanings.

A selection operator Γ then chooses the governing interpretation from this evaluated set. The selected candidate is written q*, representing the interpretation that becomes action-governing meaning.

This selected meaning implies a trajectory through the meaning-to-trajectory mapping M, written τ* = M(q*). The trajectory τ* is the sequence of states that the system will follow if that meaning governs behavior.

Finally, the next realized system state is produced from the first step of this trajectory, written σₜ₊₁ = τ*(1). At that point the system has moved into its next state and the continuation process begins again.

In summary, the Algebra of Becoming shows that interpretation is not simply guessing meaning. It is the structural process a system uses to determine continuation when baseline governance cannot determine the next action.