THE ALGEBRA OF BECOMING

Formal Representation of Constrained Continuation
By: Jordan Vallejo

Introduction

The Algebra of Becoming is a formal framework describing how systems resolve continuation from one realized state to another under governing structure.

At any moment, a system occupies a realized state within a state space of admissible configurations. This state is not a description or approximation. It is the system as it exists.

From this state, a set of admissible continuations is defined. Multiple continuations may be admissible, but only one becomes realized.

Continuation is not defined by temporal indexing. It is defined as realized succession under constraint.

The task of the algebra is to specify how this resolution occurs.

Domain

Let:

σ ∈ S

denote a realized system state.

Define the governing structure at σ:

⊠(σ) = ⟨K(σ), B(σ), F(σ)⟩

where:

• K(σ): continuation admissibility constraints

• B(σ): baseline governance rules

• F(σ): reference framework

with:

F(σ) = (C_adm^F, C_act^F, Λ^F)

All continuation is defined relative to σ, Ω(σ ; K(σ)), and ⊠(σ).

Admissible Continuation

Ω(σ ; K(σ)) ⊆ S

denotes the set of admissible successor states generated from σ under constraint structure K(σ).

A realized continuation must satisfy:

σ′ ∈ Ω(σ ; K(σ))

If:

Ω(σ ; K(σ)) = ∅

continuation is undefined under the governing constraints.

Successor Relation

σ ≺ σ′ ⇔ σ′ ∈ Ω(σ ; K(σ))

This relation defines admissible succession without reference to global time.

Transform–Realize Law

σ′ = ℛ(Θ(Ω(σ ; K(σ)) ; F(σ)))

Define:

T(Ω(σ ; K(σ)) ; F(σ)) = ℛ(Θ(Ω(σ ; K(σ)) ; F(σ)))

σ′ = T(Ω(σ ; K(σ)) ; F(σ))

Clarification

F(σ) is the framework component of ⊠(σ) and remains explicitly invoked in transformation because Θ operates over admissibility, activation, and ordering structure—not over the full governing bundle.

Transform Operator

Θ : Ω → 𝒮(Ω)

Θ structures the admissible continuation space.

It does not:

• generate candidates

• evaluate alternatives

• perform realization

Transform Primitives

All transformations decompose into:

Restriction
Θ_res : Ω → Ω′ ⊆ Ω

Reduces continuation space without introducing new admissible elements.

Partition
Θ_par : Ω → Π(Ω)

Organizes Ω into structured subsets without removing elements.

Reparameterization
Θ_param : Ω → Ω̃

Changes representation while preserving admissibility.

Baseline Governance

B^{act}(σ) = { b ∈ B(σ) ∣ C_act^F(b, σ) = 1 }

Each active rule induces a trajectory:

τ^B = F_B(b)

Define:

𝒯_{act}^B = { τ^B ∣ b ∈ B^{act}(σ) }

Determinacy

Det(⊠(σ)) = 1 ⇔ Fit ∧ Rank ∧ Feas

where:

• Fit: 𝒯_{act}^B ≠ ∅

• Rank: a unique maximal trajectory exists under Λ^F

• Feasibility: τ*(1) ∈ Ω(σ ; K(σ))

If determinacy holds, continuation is governed directly by τ*.

Action Determinacy Loss

ADL(σ) = 1 ⇔ ∃ σ⁻ ≺ σ :
Det(⊠(σ⁻)) = 1 ∧ Det(⊠(σ)) = 0

This condition marks loss of unique continuation.

Candidate Generation

When:

ADL(σ) = 1

Q = Gen(x, σ ; E)

where:

• x: operative input

• E: environment

Signal Formation Theory terminates at x; the Algebra of Becoming begins at x, σ, E → Q.

Candidate Admissibility

Q^⊠ = { q ∈ Q ∣ C_adm^F(q) = 1 ∧ M(q, σ)(1) ∈ Ω(σ ; K(σ)) }

Each candidate induces a trajectory:

M : Q × S → 𝒯
τ_q = M(q, σ)

Candidate Door Condition

Interpretive resolution proceeds only when:

ADL(σ) = 1 ∧ |Q^⊠| ≥ 2 ∧ N ≥ 2

Evaluation

V^Q ⊆ Q_eval^⊠ × Q_eval^⊠
Q_eval^⊠ ⊆ Q^⊠

Pairwise comparison yields:

qᵢ ≻ qⱼ
qᵢ ∼ qⱼ
qᵢ ∥ qⱼ

Ordering is governed by Λ^F.

Selection

Max(Q_eval^⊠) = { q ∣ ∄ q′ : q′ ≻ q }

Define:

q* =

• unique maximal element if |Max| = 1

• ∅ otherwise

Action-Governing Meaning

AGM = q*

Governing Trajectory

τ_{q*} = M(q*, σ)

Realization

σ′ ∈ Ω(σ ; K(σ)) ∩ τ_{q*}

σ′ = ℛ(...)

Regimes of Continuation

Deterministic Regime

σ → 𝒯_{act}^B → τ* → ℛ → σ′

Candidate-Mediated Regime

σ → ADL → Q → Q^⊠ → V^Q → q* → τ_{q*} → ℛ → σ′

These regimes are structurally distinct but converge on the same realization operator and successor-state condition.

Interpretive Boundary Condition

Interpretive resolution occurs only in systems that:

• sustain multiple admissible candidate meanings under indeterminate continuation

• generate candidate meanings from resolved input

• select among competing candidates

• possess endogenous authority to bind governing meaning

Systems lacking these conditions do not perform interpretation.

Structural Clarification (Non-Temporal)

The branch structure of determinacy and candidate-mediated evaluation is state-structural rather than globally time-indexed.

Operational descriptions such as entry, evaluation, and exit describe organization of resolution, not a primitive temporal substrate.

Closure

σ′ ∈ Ω(σ ; K(σ))

The realized state becomes the next realized state:

σ ← σ′

Axioms of Continuation

Ω(σ ; K(σ)) ⊆ S

σ′ ∈ Ω(σ ; K(σ))

σ′ = ℛ(Θ(Ω(σ ; K(σ)) ; F(σ)))

σ → Ω(σ ; K(σ)) → Θ → ℛ → σ′

Final Statement

A system occupies a realized state σ with governing structure:

⊠(σ) = ⟨K(σ), B(σ), F(σ)⟩

Continuation admissibility is generated by K(σ), structured through Θ under framework F(σ), and resolved through realization ℛ.

When determinacy holds, continuation proceeds through baseline governance.

When determinacy fails, candidate meanings are generated, evaluated, and one is selected to govern continuation.

In all cases, exactly one admissible successor state becomes realized.