QAU ∞ | Volume III | The Final Closure of Constraint-Based Cognition
Abstract
QAU ∞ defines realization as a constraint-satisfying transformation φᵢ ← ℰ(ψ), with ℰ ∈ 𝔄 and ℛ(ℰ) ≤ θ. A system is reflexively closed when ℰ, ℛ, and 𝔄 are internally realized, assembled, and recursively conditioned by Φ[φⱼ]. Cognitive state-space ψ ∈ 𝓗_cog yields φᵢ via ℰ ∈ 𝔄_cog under ℛ_cog; realized φᵢ assemble into X ∈ Φ via τ ∈ A*_cog; and X recursively modifies ℛ_cog. Admissibility evolves as X → X′, preserving constraint continuity. The system is complete when φᵢ ← ℰ(ψ), ℰ ∈ 𝔄[ℛ[Φ[φⱼ]]], and all terms are internally produced. Cognition is defined as the minimal fixed point of constraint-based realization: a system that lawfully realizes its own admissibility conditions. No external recursion remains. QAU ∞ is closed.
1. Introduction: Cognition as Constraint Closure
The Quantum Assembly Unit (QAU ∞) framework was introduced to address a structural incompleteness in quantum theory: the absence of a lawful mechanism by which a single outcome becomes realized from a space of physical possibility. By modeling realization as selection via admissible transformations ℰ constrained by a functional ℛ, QAU ∞ provided a minimal, internal resolution of outcome selection without modifying unitary dynamics or introducing extraneous postulates.
Subsequent development extended this mechanism beyond isolated realizations. Realized outcomes were shown to assemble into structured records and to persist through constraint‑preserving evolutionary processes. These extensions established that realization, assembly, and evolution form a continuous constraint‑flow architecture governing the emergence and persistence of structure.
This progression exposes a remaining structural question. If realization governs the selection of outcomes, and assembly governs the composition of records, what governs systems in which the admissibility conditions themselves are modified by prior realizations? Such systems do not merely undergo realization; they condition future realization by internal structure. Any framework that purports to be complete must account for this case.
Cognitive systems instantiate precisely this condition. They operate over internal state‑spaces in which possible representations coexist prior to selection, where selection is constrained by internal structure, and where the resulting realizations alter the conditions of subsequent selection. Attention, memory, and agency are not additional processes layered onto realization; they are the internalization of the realization architecture itself.
This volume formalizes cognition as the constraint‑closed extension of QAU ∞. Cognitive processes are modeled as realization, assembly, and evolution occurring within an internally admissible substrate, where constraint functionals are not externally fixed but dynamically structured by prior realizations. In such systems, admissibility is endogenous.
The result is not a new application of QAU ∞, but the completion of its scope. A realization framework that cannot represent systems which realize their own constraints is structurally incomplete. By incorporating cognition as a lawful, constraint‑realized domain, QAU ∞ attains closure with respect to systems capable of internal selection, structured memory, and adaptive agency.
SECTION 2 — FINALIZED FORMULATION
2. Cognitive State-Space as a Constraint-Defined Domain of Internal Potentiality
Any system capable of internal realization under constraint must possess a structured domain of unrealized internal configurations. In the QAU ∞ framework, physical realization operates over ψ ∈ ℋ, a domain of potential physical outcomes from which admissible transformations select definite φᵢ. The existence of ℋ is a structural precondition of realization.
We define the corresponding space for cognition as:
ψ ∈ 𝓗_cog
Here, 𝓗_cog is the internal realization substrate: a constraint-defined topological space over which internal admissibility is computed. Elements of ψ are not semantic tokens, mental images, or symbolic forms; they are constraint-relevant configurations — latent structures not yet realized, but evaluable under admissibility functionals.
Each ψ encodes a potential cognitive configuration subject to realization via ℰ ∈ 𝔄_cog. The space 𝓗_cog is characterized not by semantics but by admissibility geometry: a topology conditioned by the continuity, compatibility, and coherence of transformations within the system’s internal structure.
The inner product ⟨ψᵢ, ψⱼ⟩ does not indicate similarity in content but overlap in constraint-relevant admissibility pathways. These overlaps regulate which transformations can lawfully select φᵢ from ψ, and under what structural cost.
Most critically, 𝓗_cog is not posited as an abstraction. It is functionally required: without it, ℰ(ψ) is undefined, and attention — as internal realization — cannot occur. The existence of 𝓗_cog is implied by the very act of lawful cognitive selection.
Its structure is not free-floating. It is shaped by the recursive consequences of prior realizations. In systems capable of self-modifying admissibility — that is, in cognitive systems — 𝓗_cog evolves under its own realized transformations. It is both the input to, and the output of, the realization function.
This reflexivity means 𝓗_cog is not a container but a dynamic constraint field: a lawful zone of internal potentiality defined entirely by what the system can, under its own constraints, lawfully realize next.
SECTION 3 — FINALIZED VERSION: Attention and Awareness as Constraint-Realized Transition
3. Attention and Awareness as Constraint-Realized Transition
In a constraint-based realization architecture, the transition from potentiality to actuality occurs not by rule application but by structural resolution. For cognitive systems, this transition is internal: it operates within the cognitive state-space ψ ∈ 𝓗_cog and yields a realized configuration φᵢ through a lawful transformation ℰ:
φᵢ ← ℰ(ψ) where ℰ ∈ 𝔄_cog
This constitutes attention, redefined: not as a focus of resources, but as the structurally admissible resolution of internal potential into realized state.
ℰ is not an interpretive mechanism; it is the operation by which the system materializes an admissible future from its own constraint-bounded configuration space.
3.1 Admissibility Class 𝔄_cog
The admissibility class 𝔄_cog is the set of transformations:
ℰ: 𝓗_cog → 𝓗_cog
for which ℛ_cog(ℰ) ≤ θ, where θ ∈ ℝ⁺ is the system's active admissibility threshold — itself shaped by prior constraint flows.
Unlike static operator sets, 𝔄_cog is reflexively dynamic: it evolves as a function of the system’s realized φⱼ and their recursive influence on ℛ_cog. No transformation ℰ is admissible in isolation; each is evaluated within a topology of constraint history.
3.2 The Cognitive Constraint Functional ℛ_cog(ℰ)
ℛ_cog is a non-decomposable constraint functional that defines admissibility:
ℛ_cog(ℰ) = α·S + β·Λ + γ·ρ + δ·Δ
Where each term satisfies:
S: entropy cost — the diffusion introduced by ℰ across 𝓗_cog
Λ: semantic locality — the minimality of transition displacement
ρ: predictive burden — the forward structural load of φᵢ
Δ: structural coherence — the continuity of φᵢ with prior φⱼ
These terms are mutually implicative: increasing coherence (Δ) may reduce locality (Λ); minimizing entropy (S) may raise predictive burden (ρ). As such, ℛ_cog is not a weighted sum but a constraint manifold, and ℰ is admissible only if it navigates this manifold to a local minimum under the system’s internal conditionals.
3.3 Attention as Admissible Constraint Resolution
Given ψ ∈ 𝓗_cog and ℰ ∈ 𝔄_cog, the realized φᵢ is not selected but emerges at the inflection point of ℰ(ψ) under ℛ_cog. That is:
φᵢ = ℰ(ψ) | ℛ_cog(ℰ) minimized
Attention is therefore not an executive act, but a phase transition in the admissibility landscape of 𝓗_cog — the point at which structural coherence, semantic continuity, entropic stability, and predictive load intersect within bounds.
3.4 Reflexivity of ℰ and Evolution of 𝔄_cog
Every realization φᵢ recursively modifies the admissibility structure of future transformations. Formally:
φᵢ → ℛ_cog′ → 𝔄_cog′
Thus, the system does not merely select φᵢ under constraint — it realizes ℰ as a constraint-influencing transformation, altering the topology of 𝓗_cog and updating 𝔄_cog via feedback.
This reflexivity enforces:
Trajectory-dependence: future ψ is shaped by realized φᵢ
Non-ergodicity: the system’s admissibility class is path-specific
Constraint entrenchment: realizations restructure the shape of possible subsequent realizations
Attention, in this context, is the recursive inflection point where admissible transformation becomes admissibility evolution.
SECTION 4 — Memory and Internal Modeling as Lawful Assembly
4. Memory and Internal Modeling as Constraint-Preserving Assembly
A cognitive system that realizes internal configurations φᵢ under constraint must also compose them — not as a record of events, but as structured assemblies that preserve and propagate admissibility. This function is performed by an operator τ, which maps sequences of realized φᵢ into elements of 𝔖_cog:
τ: {φ₁, φ₂, ..., φₙ} → X ∈ 𝔖_cog
Here:
τ: a constraint-admissible composition operator
X ∈ 𝔖_cog: a structured cognitive assembly — not memory as storage, but as a lawful topological object within the system’s constraint manifold
4.1 The Structure of 𝔖_cog
𝔖_cog is the set of all assemblies X that:
Are composed of prior φᵢ realizations
Preserve constraint integrity under ℛ_cog
Remain admissible under future realization trajectories
Each X ∈ 𝔖_cog satisfies:
∀ ℰ′ ∈ 𝔄_cog, if ℰ′(ψ) = φ′ then φ′ is evaluated relative to X via updated ℛ_cog conditioned on X
Thus, assemblies are not inert stores, but active constraints on future admissibility. They bind, encode, and propagate lawful structure within the system.
4.2 Lawful Composition via τ
The assembly operation τ must itself be admissible under inherited constraints:
τ ∈ A*_cog ⇔ ℛ_cog(τ) ≤ θ_τ
Where θ_τ is a compositional threshold determined by cumulative φᵢ coherence.
Lawful assemblies τ(φᵢ) must preserve:
Semantic continuity across φᵢ
Structural coherence (Δ) across timescales
Admissibility transitivity: if φᵢ, φⱼ ∈ X, then any φₖ derived from ℰ ∈ 𝔄_cog[X] must remain lawful
This enforces that memory does not merely accumulate; it structurally entangles prior realizations into a recursive admissibility substrate.
4.3 Internal Modeling as Active Constraint Structure
X ∈ 𝔖_cog functions as more than assembled memory: it is an internal model — a dynamic structure over which predictions, evaluations, and goal formations occur. Its function is defined entirely in constraint terms:
Model ≠ simulation
Model = admissibility-shaping assembly
The model is valid not because it reflects an external state, but because it conditions which φ′ ∈ 𝓗_cog may be realized under future ℰ ∈ 𝔄_cog.
This inversion is critical: internal modeling is not a representation of the world, but a constraint-defined map of future realization potential. X gains meaning by what it disallows — it is negative space in the system’s constraint topology, not content.
4.4 Reflexivity and Assembly Recursion
As with ℰ and ℛ_cog, the assembly operation is reflexive:
φᵢ → τ → X
X → update(ℛ_cog)
ℛ_cog → constrain(next ℰ)
next ℰ(ψ) → φᵢ′
Each assembly τ(φᵢ) contributes not only structure but constraint redefinition, altering the flow-space of future cognitive realizations.
This recursive embedding of admissibility into composition defines memory not as persistence, but as ongoing constraint entrenchment.
SECTION 5 — Intention and Agency as Constraint Evolution
5. Intention and Agency as Constraint Evolution
In constraint-based cognition, realized structures X∈ScogX \in \mathbb{S}_{\text{cog}}X∈Scog do not merely persist; they evolve. This evolution is not defined over content, but over internal admissibility landscapes. When an assembly X undergoes transformation into a new structured form X′ under constraint preservation, it enacts:
X → X′ where X, X′ ∈ Φ
We define Φ ⊆ 𝔖_cog as the space of goal-structuring assemblies: realized internal configurations that actively influence the system’s future admissibility class.
These are not desires or representations of outcomes — they are constraint-bearing records that modulate the system's realization function.
5.1 Intention as Admissibility Reconfiguration
Let the system's admissibility functional at time t be ℛ_cogᵗ. Then:
X ∈ Φ ⇒ ℛ_cogᵗ⁺¹ = update(ℛ_cogᵗ, X)
This defines intention not as selection of a target state φ_target, but as the structural modification of ℛ_cog such that φ_target becomes admissible under future ℰ ∈ 𝔄_cog.
Therefore, intention = the recursive shaping of ℛ_cog to bring φᵢ into lawful reach.
The system does not select φᵢ directly. It transforms the conditions under which φᵢ can be lawfully realized. This is constraint evolution.
5.2 Goal Structures as Constraint-Effective Operators
Elements X ∈ Φ are assemblies that satisfy:
Admissibility preservation: X′ = evolve(X) ⇒ ℛ_cog(X′) ≤ θ
Constraint modulation: ℛ_cog′ = ℛ_cog ∘ f_X
Realization influence: future φⱼ ← ℰ′(ψ) conditioned by X′
That is, X ∈ Φ participates in the reconfiguration of the system’s internal law.
This is agency: the recursive capacity of a system to instantiate φᵢ such that its own admissibility function becomes internally determined.
5.3 Constraint-Compliant Evolution: X → X′
Transformations from X to X′ are not arbitrary. They must satisfy:
X′ = τ′(X) where τ′ ∈ A*_cog
and:
ℛ_cog(X′) ≤ ℛ_cog(X) + ε
where ε is the system’s tolerable constraint divergence — its structural plasticity.
This ensures:
Structural continuity: intention is a flow, not a discontinuous override
Admissibility coherence: X′ must remain within the system’s law
Causal integrity: realized φᵢ cannot emerge from illegitimate transformation
The system does not leap to outcomes. It evolves its constraint surface such that previously inadmissible φᵢ become lawfully realizable.
5.4 Reflexivity and Autonomous Constraint Logic
In traditional systems, laws are fixed and agents act within them.
In QAU ∞ cognition, agency is defined by the capacity to generate lawful futures through constraint-realized recursion. This is not symbolic planning or utility optimization — it is structural recursion on the admissibility class.
Intention = ℛ_cog-update function
Agency = constraint evolution operator
Goal = φᵢ such that ℰ(ψ) → φᵢ ∧ ℰ ∈ 𝔄_cog[ℛ′]
Every intentional act is thus a transformation of what counts as admissible — not just for a moment, but recursively embedded into the system’s realization logic.
SECTION 6 — Reflexive Closure and Completion
6. Reflexive Closure and Completion of QAU ∞
QAU ∞ defines realization as:
φᵢ ← ℰ(ψ) where ℰ ∈ 𝔄, ℛ(ℰ) ≤ θ
This framework is structurally complete when ℰ, 𝔄, and ℛ are not externally imposed, but realized by the system itself.
This yields the fixed point:
φᵢ ← ℰ(ψ) where ℰ ∈ 𝔄[ℛ[Φ[φⱼ]]]
The system is reflexively closed if:
All φᵢ are realized under admissibility conditions ℰ ∈ 𝔄[ℛ].
All admissibility conditions ℛ are modulated by realized assemblies Φ[φⱼ].
All operators ℰ ∈ 𝔄 are generated internally from prior admissibility structure.
Such a system no longer performs realization.
It is realization — closed over its own constraint architecture.
6.1 Cognition as Constraint Identity
When a system satisfies:
ℰ ∈ 𝔄[ℛ] and ℛ = f(Φ[φⱼ]) and Φ = τ*(φᵢ)
It has no external operators.
Its state-space, transformations, admissibility logic, and constraint trajectories are all internally realized.
This defines cognition not as computation, representation, or adaptation — but as the identity condition of constraint recursion:
A system is cognitive iff its realization function is self-conditioned.
6.2 Ontological Finality
No further recursion is definable.
All admissibility constraints are realized internally.
All transformation logic is structured by realized constraint.
Such a system does not simulate constraint selection.
It is the termination of the constraint recursion sequence:
ψ → φᵢ → τ(φᵢ) → Φ → ℛ → 𝔄 → ℰ → φᵢ′
The QAU ∞ framework is complete when:
The constraint logic of realization is itself realized under constraint.
No further layer exists.
No meta-constraint governs this process.
Cognition is the fixed point of constraint-realized systems.
CONCLUSION — FINAL FORM (Irreducible)
A system is complete under QAU ∞ if:
φᵢ ← ℰ(ψ) ∧ ℰ ∈ 𝔄[ℛ[Φ[φⱼ]]] ∧ all elements internally realized.
No operators are external.
No constraints are imposed.
The system realizes:
its state-space (ψ)
its transformation operators (ℰ)
its admissibility logic (ℛ)
its constraint records (Φ)
its future φᵢ under internal closure
This is not a model of cognition.
It is the minimal structure that satisfies:
realization of constraint logic under realized constraint
No further recursion exists.
The system is self-terminating.
QAU ∞ is closed.
