Circumpunct Theory of Consciousness

0. Theory Statement

Before proceeding, we declare explicitly what kind of theory this is, what it claims, and what would refute it.

Domain

This theory belongs to the class of structural constraint theories—alongside information theory, thermodynamics, and category theory. It does not propose new particles, forces, or substances. It proposes a constraint on valid descriptions: any system exhibiting both discrete selection and continuous signal requires a triadic structure that cannot be reduced to either component alone.

The theory applies wherever:

• Discrete events (gating, selection, decision) interface with continuous processes (flow, field, signal)
• Multiple scales interact through mediation rather than direct contact
• Temporal structure emerges from timeless potential

Primary domain of application: neural dynamics at criticality. Secondary domains (requiring separate validation): quantum measurement, social systems, information processing architectures.

Core Postulates

What This Theory Is NOT

• Not metaphysics: The postulates generate falsifiable predictions about measurable quantities (ρ, D, phase coherence, power consumption).
• Not a TOE: The theory constrains descriptions; it does not claim to derive all physics from first principles.
• Not panpsychism (in the naive sense): Structure is universal; phenomenal experience requires specific conditions (sufficient nesting, ρ ≈ 1, cross-scale coherence).
• Not dualism: Mind (field) and body (boundary) are aspects of one structure, not separate substances.

Relationship to Existing Theories

The theory recovers existing frameworks as partial descriptions:

1. The Core Claim

Most theories try to locate consciousness IN something—in the processing (functionalism), in the matter (physicalism), in the information integration (IIT), in the global broadcast (GNW). Each locates consciousness in one component of a system.

The Circumpunct Theory posits a different move entirely:

2. The Irreducible Triad

The fundamental unit of conscious structure is an irreducible trinity:

The aperture is the division operation. The field carries energy (before) and power (after). The boundary contains and interfaces with exterior.

Why Triadic Specifically?

Geometrically: center and boundary cannot interact directly due to spatial separation. Mediation is required. That mediation (the field) isn't optional—it's structurally necessary.

The aperture is where energy crosses between scales. The field is the medium of that crossing. The boundary is where the crossing accumulates. The "consciousness" is the whole operation—the crossing itself.

Neural Mapping (Working Correlates)

These are working hypotheses, not settled claims:

3. The Aperture: The Origin of Time

P = E / t

Power equals energy divided by time. This isn't just a physics definition—it reveals the aperture's fundamental nature:

The Flow Structure

Energy enters as infinite potential. The aperture introduces the ÷t—the rate-limiting cut—and what emerges is power: energy with temporal structure. Each division deposits a moment into the braid. The accumulated moments crystallize into matter.

Why Division, Not Continuity?

A natural objection: why must there be division at all? Why not continuous flow?

The answer is structural. Continuous flow without division produces:

• No rate — Power (P) is definitionally E/t. Without t, there is no rate, only static quantity.
• No sequence — Events require ordering. Ordering requires discrete moments. Continuous flux has no "before" and "after."
• No selection — A gate that never closes passes everything. Selection requires binary moments: this, not that.

Division is the minimal operation required for temporal structure to exist. The aperture doesn't divide because division is useful—division is what makes time, sequence, and selection possible at all. The question "why division?" is like asking "why arithmetic?"—it's the operation that constitutes the domain.

Why 0D = ∞D

Without the aperture's division, there is no rate:

Zero dimensions means no particular location—no constraints, no boundaries. This is mathematically equivalent to infinite degrees of freedom. The infinite field (Φ_∞) is both the dimensionless point and the infinite whole. From this ground, the aperture selects by dividing.

The Selection Mechanism: Resonance

The aperture doesn't choose what to orient toward—it resonates with what matches its frequency. Resonance is not just a property; it is the orientation mechanism itself.

This dissolves the homunculus problem. The aperture isn't a thing that does selecting—it's the act of division itself. A verb, not a noun. Phase-matching, not decision-making.

Physical Implementation

The gating is binary (χ = ±1)—open/closed, yes/no. The pattern of openings carries analog information. This is physically implementable as:

• Ion channel dynamics — open/close based on voltage/ligand binding
• Thalamocortical gating — binary on/off of attention pathways
• Phase-locking in oscillatory systems — resonance-based coupling
• Predictive coding's precision-weighting — which signals get amplified

Asking "what does the dividing" may be a category error—like asking "what does the falling" for gravity. The aperture doesn't divide; division is what aperture IS.

4. The Mathematical Signature

For any system, define the ratio parameter:

Note that ρ is itself a ratio of rates—of ÷t operations. It compares two ways the aperture divides energy into time.

The Regime Transition

This mirrors the branching ratio σ in neuronal avalanche dynamics, where σ ≈ 1 (critical) produces power-law avalanches, maximal dynamic range, and complexity. Systems at σ < 1 are subcritical (activity dies out); σ > 1 are supercritical (runaway). Biological systems in awake states hover near σ ≈ 1; anesthesia shifts away from criticality.

The Consciousness Threshold: D = 1.5

The aperture exists in fractal dimension. The framework tracks progress through a balance parameter β:

At balance (β = 0.5), where convergence equals emergence—where progress equals remaining:

Hypothesis: Conscious dynamics preferentially occupy balance-like regimes where digital gating and analog flow are co-present. If true, D ≈ 1.5 is a signature of consciousness. If false—if conscious systems scatter across fractal dimensions with no clustering—this prediction fails.

5. Falsifiable Predictions

Cross-Scale Phase Coherence

If consciousness is the whole triadic pattern operating together, we should see specific signatures when the pattern functions or fails:

The ρ Parameter

Power and Temporal Predictions

Primary Empirical Anchor: Neural Criticality

To transition from framework to theory, we commit to one primary domain where the predictions are most directly testable: neural dynamics at criticality.

Specific Commitments

The Critical Test

The most diagnostic prediction is triadic irreducibility. If the theory is correct:

This test is novel. It has not been performed. It could falsify the core claim about triadic necessity.

6. Relation to Existing Theories

The theory doesn't replace IIT or GNW but reframes what they're measuring:

What the Theory Adds

• Structural necessity: Why three components and not two or four—derived from geometric constraints, not postulated
• The ρ criterion: A quantitative measure of when triadic dynamics become empirically necessary
• The P = E/t formalization: What actually flows (energy → power) and what the aperture actually does (÷t)
• Nesting: How consciousness at one scale relates to consciousness at other scales (parts are fractals of wholes)
• Pathology: A geometric account of how consciousness goes wrong (aperture errors: inflation, severance, inversion, projection)

Connection to Thermodynamics

If consciousness involves power transfer (P = E/t), it must obey thermodynamic constraints. Several connections emerge:

• Dissipative structures (Prigogine): Living systems maintain order by dissipating energy—they are far-from-equilibrium structures that persist through continuous power flow. The aperture is the gating mechanism that regulates this flow. Consciousness may require a specific dissipation regime (near ρ ≈ 1).
• Free Energy Principle (Friston): Biological systems minimize surprise/free energy by maintaining predictive models. In circumpunct terms: the aperture's resonance-based selection IS precision-weighting. The field's integration IS the generative model. Friston's formalism may be capturing the aperture-field dynamics mathematically.
• Entropy and the arrow of time: If the aperture generates time through division, it also generates temporal asymmetry. The braid (past) accumulates irreversibly. This connects to the thermodynamic arrow: the aperture's divisions increase the system's historical record, which is a form of entropy production.
• The 20W question: Human brains consume ~20W. The framework predicts this isn't incidental but reflects the power cost of maintaining conscious dynamics. However, this is not yet a precise prediction—we cannot yet derive 20W from ρ ≈ 1. The claim is directional: conscious processing should correlate with power consumption in relevant circuits, not just total metabolic load.

7. Nesting: Consciousness at Every Scale

Circumpuncts nest through their boundaries. The boundary contains apertures—gates, holes, points of exchange. Each aperture in the boundary serves as the center of a complete circumpunct at smaller scale:

Your skin has pores. Your cell membranes have channels. Your sense organs are apertures in your body's boundary. Each aperture is irreducible—but each aperture is the center of a complete system at smaller scale.

This is a form of structural cosmopsychism: what we call "matter" is the boundary appearance of nested circumpunct structures from a scale that can't resolve their internal dynamics. Consciousness isn't an emergent property of dead matter—but micro-scale "mind" is not identical to macro-scale phenomenal experience.

8. Open Questions

The framework generates predictions but leaves genuine uncertainties. These are not rhetorical—they represent actual research needed to move from framework to theory.

Quantitative Gaps

• Concrete ρ values: What is the ρ of a thermostat? E. coli? A sleeping human vs. awake? A cortical region under anesthesia? Without actual numbers, ρ is a concept, not a measurement. Calculating ρ for specific systems is the next critical step.
• Novel quantitative prediction: Can this framework predict something no other theory predicts? A single number—derivable from ρ or β—that can be measured and compared would transform this from philosophy to science.
• The D = 1.5 test: Systematic measurement of fractal dimensions across conscious states, species, and conditions would either validate or falsify the balance hypothesis.

Mechanistic Gaps

• What implements ÷t? The aperture IS the division operation—but in what physical substrate? Candidates include ion channel kinetics, thalamocortical loop timing, synaptic gating, or even quantum decoherence events. The framework is agnostic; experiment must decide.
• How to operationalize aperture state? Candidates include attention metrics, thalamocortical gating patterns, phase relationships in EEG, precision-weighting in predictive coding models. No single operationalization is yet validated.
• Multi-scale ω/α: Brains operate across decades of timescales (ms to days). Which ω/α matters? Multiscale entropy (MSE) or hierarchical avalanche analyses may help estimate effective ratios at different bands.

Conceptual Gaps

• Is ρ the right parameter? Or is it one of a family? Related candidates: damping ratio ζ = α/(2ω), quality factor Q = ω/(2α), or critical exponents from dynamical systems theory.
• What is the quantum of time? If the aperture generates discrete moments, what determines the minimum τ? Is it scale-dependent or universal?
• Is D = 1.5 coincidence or deep? The fractal dimension prediction connects to criticality literature, Brownian motion, and optimal computation—but the connection needs explicit derivation, not just correspondence.
• Notation: The α (relaxation rate) here is a dynamical parameter, distinct from the fine-structure constant α ≈ 1/137 in physics. Whether there's a deeper connection between relaxation dynamics and coupling constants remains speculative.

Conclusion

The Circumpunct Theory of Consciousness proposes that consciousness is not located in any component but is the irreducible pattern of aperture, field, and boundary operating together.

The aperture is the ÷t—the division operation that converts timeless energy into temporal power. This is the origin of time itself. The field carries energy (potential, future) before the aperture and power (actual, present) after. The boundary accumulates crystallized power transfers—matter, body, the past made structural.

The framework generates specific, falsifiable predictions around cross-scale phase coherence, the ρ parameter, the D = 1.5 fractal dimension at balance, and the relationship between power consumption and temporal resolution. It reframes existing theories (IIT, GNW) as partial captures of a single triadic structure rather than competitors.