The Complexity Bound: A Unified Mathematical Framework for Universe Formation and Consciousness Emergence
Ivars Vilums
Wimberley, Texas - December 2025
Abstract
We propose that coherent self-organizing complexity, whether manifest as physical universes or conscious entities, exists only within a bounded range of information density on appropriately structured substrates. Below a critical threshold, insufficient information prevents coherent organization; above a critical ceiling, the system undergoes phase transition through gravitational collapse or coherence fragmentation. This habitable band for complexity is governed by a single mathematical framework that unifies the Bekenstein-Hawking entropy bound, the emergence of physical forces from geometric structure, and the conditions for conscious experience. We demonstrate that black holes do not collapse to singularities but to blitzon shells, transaction endpoint surfaces that represent the complexity floor. The nested hierarchy of possible universes follows necessarily from information conservation across phase transitions, with each interior level constrained to lower complexity than its exterior. The same mathematics describes consciousness as complexity emerging on neural transaction substrates, suggesting that universe-formation and mind-formation are scale-variant manifestations of a single underlying principle. The framework makes specific predictions about the relationship between black hole mass and interior complexity, the minimum substrate requirements for consciousness, and the accessibility of alternate paths through block-universe structure.
The previous papers in this series established three interconnected results: (1) photons undergo gravitational collapse at wavelengths near the square root of 2 times the Planck length, implying that photon propagation is not fundamental; (2) the Transaction-Geometric Interpretation reconceptualizes quantum phenomena as observational perspectives on direct spacetime connections rather than propagating particles; and (3) Blitzon Cosmology proposes that our universe exists within a black hole, with fundamental forces emerging from geometric rotation modes of a compactified manifold.
Here we develop the mathematical framework unifying these results with the theory of consciousness. The central insight is that coherent self-organizing complexity, the kind that produces universes with physics or minds with experience, can only exist within a bounded range. Too little information yields no structure; too much triggers phase transitions that redistribute complexity into nested interior structures.
This framework suggests that black holes are not endpoints but transitions, reality's mechanism for managing complexity that exceeds local carrying capacity. Consciousness, similarly, emerges when neural substrates achieve sufficient complexity for self-referential distinction processing, but faces its own coherence limits. The mathematics governing both domains is identical.
We define the complexity function C for a system with information content I organized on substrate S:
C = f(I, S) = I times Gamma(S) times Phi(I, S)
where I is the information content measured in bits, Gamma(S) is the substrate capacity factor encoding the geometric and topological structure available for organization, and Phi(I, S) is the coherence function measuring how effectively information integrates across the substrate.
The complexity function is not simply information content. A random bit string has high information but low complexity. Complexity requires that information be organized in self-referential, causally integrated patterns on an appropriate substrate.
The substrate capacity Gamma(S) depends on the geometric and topological properties of the organizing medium. For transaction-geometric substrates (universes), this involves the ratio of horizon radius to Planck length, manifold topology, and spacetime dimensionality. For distinction-processing substrates (consciousness), it involves the number of processing nodes, their connectivity, and the depth of temporal integration.
The coherence function Phi(I, S) measures the degree to which information integrates into a unified structure. Crucially, this function approaches zero both as I approaches zero (insufficient information for coherence) and as I approaches infinity (information exceeds substrate capacity, coherence breaks down). Phi has a maximum at intermediate I where complexity is optimized.
This mathematical structure, with both a floor and a ceiling, is the key to the habitable band for complexity.
Below a critical information threshold, coherent self-organization cannot occur. For physical universes, this corresponds to black holes with Bekenstein-Hawking entropy below approximately 100 bits, too few transaction endpoints to support any coherent interior geometry. For consciousness, this corresponds to neural substrates with insufficient nodes and connectivity for recursive self-modeling.
Physical interpretation:
The blitzon shell is the complexity floor for gravitational systems. A collapsing black hole does not reach a singularity because at Planck density, there is no further
down to go. The blitzon endpoints are the substrate itself. Below approximately 1 bit of Bekenstein-Hawking entropy, no interior structure is possible.
Above a critical information density, coherence breaks down and the system undergoes phase transition. For physical systems, this manifests as gravitational collapse. Attempting to pack more information into a region than the Bekenstein bound permits triggers black hole formation. The ceiling is:
I_ceiling = A / (4 times Planck length squared)
For conscious systems, exceeding coherence capacity produces fragmentation rather than collapse. The unified field of experience breaks into disconnected processes, or the system fails entirely through seizure, unconsciousness, or death.
Physical interpretation:
Black hole formation is complexity management. When a region's information content approaches the Bekenstein bound, gravitational collapse creates a new interior spacetime, redistributing the excess complexity into a nested structure. The exterior sees only the horizon; the interior gains a new universe.
Coherent self-organizing complexity exists only in the band between floor and ceiling. Within this band, the complexity function varies from minimal near floor to maximal at some optimal value to declining as it approaches the ceiling. The optimal complexity occurs at:
I_optimal = square root of (I_min times I_max)
|
System |
I_floor |
I_ceiling |
Emergent Complexity |
|
Planck-scale BH |
~1 bit |
~100 bits |
None (at floor) |
|
Stellar BH |
~100 bits |
~10^77 bits |
Simple geometry |
|
Our Universe |
~100 bits |
~10^122 bits |
4 forces, matter, life |
4 forces, matter, life
Human brain
~10^4 bits
~10^16 bits
Rich consciousness
When a system exceeds its complexity ceiling and undergoes gravitational collapse, the resulting black hole contains at most the information that fell in. This establishes a strict hierarchy:
I_interior is less than or equal to I_infalling which is less than I_exterior
The interior universe cannot be more complex than the exterior universe that created it. Each level of nesting necessarily has reduced information capacity and therefore reduced maximum complexity.
Theorem (Hierarchy Bound):
For any nested sequence of universes U_0, U_1, U_2, and so on, where each U_(n+1) forms from gravitational collapse within U_n, the maximum complexity is strictly decreasing at each level.
Since complexity decreases at each nesting level and there exists an absolute floor at the Planck scale with approximately 1 bit, the nesting depth is finite. For our universe with information content of approximately 10^122 bits and the floor at approximately 1 bit, even with one order of magnitude reduction per level, approximately 122 levels of nesting are possible in principle.
However, for complex physics and consciousness to exist at level n, we need information content much greater than the floor. This dramatically reduces the number of interesting levels. Perhaps only a few nestings can support physics rich enough for complex structures.
The hierarchy constraint allows inference about the external universe. Since our universe has complexity of approximately 10^122 bits, the external reality must have at least this much information available for collapse into our region. The external universe is necessarily richer than ours, though we cannot determine by how much.
This establishes an epistemological horizon: we can infer lower bounds on external complexity but never measure it directly, as all our observations are transaction geometries within our interior structure.
Classical general relativity predicts singularities, points of infinite density where physics breaks down. The complexity framework resolves this: as matter collapses and approaches Planck density, it reaches the complexity floor, not a singularity.
At the Planck scale, all energy becomes blitzon endpoints, transaction termini that cannot be further compressed. The singularity is replaced by a blitzon shell with a radius that depends on the black hole mass and the geometric structure of the shell's transaction network.
The blitzon shell is not empty structure. It encodes all information that fell into the black hole. The Bekenstein-Hawking entropy S = A/(4 times Planck length squared) counts the shell's degrees of freedom: each Planck area supports approximately one bit of transaction endpoint configuration.
This information is not lost or destroyed. It constitutes the interior universe's initial conditions. Hawking radiation represents transactions routing through the shell, gradually returning information to the exterior as the black hole evaporates.
Above the complexity floor, the blitzon shell's transaction structure self-organizes into interior geometry. The emergent spacetime dimension, topology, and force content depend on the shell's information. Our universe's compactification emerges from approximately 10^122 bits organized on a shell with appropriate transaction geometry. A stellar-mass black hole's approximately 10^77 bits would produce radically simpler interior structure, perhaps a single force analog, limited spatial extent, no stable matter.
Consciousness, in the Distinction Presencing Machine (DPM) framework, arises from recursive self-modeling through nested distinctions. A conscious system maintains a model that includes itself-as-modeler, creating the loop structure characteristic of subjective experience.
This requires: (1) Sufficient information capacity for nested self-representation. (2) Appropriate substrate structure supporting distinction formation and integration. (3) Temporal depth for the presencing operation to complete. (4) Coherence across the system, as fragmented processing does not produce unified experience.
These are precisely the requirements captured by the complexity function with neural substrate parameters.
Below a critical complexity, consciousness cannot emerge. The floor is set by the minimum information required for recursive self-modeling. A system must have enough states to represent its environment, its internal state, its model of its internal state, and its model of itself modeling. This recursion demands exponential state space, setting a firm floor.
Estimate: A minimal conscious system requires 10^4 to 10^6 bits of integrated information, roughly the complexity of small nervous systems. Below this, information processing occurs but presencing does not.
Can consciousness grow without bound? The coherence function suggests not. As information content increases on a fixed substrate, integration becomes harder. The system fragments into semi-independent processes, each potentially conscious but none containing the whole.
The human brain may operate near optimal complexity for its substrate, rich enough for deep self-modeling, not so rich that coherence fails. Pathological states such as psychosis, dissociation, and seizure may represent approaches to the ceiling where integration breaks down.
A crucial observation links consciousness directly to TGI: all sensory distinctions ultimately arise from photon transactions. Light hitting retinas, electrons triggering neural signals (mediated by virtual photons), chemical bonds forming and breaking, all are electromagnetic, all are transactional.
The DPM does not merely parallel TGI. It is implemented by it. Consciousness is transaction geometry becoming aware of itself through a sufficiently complex neural substrate. The presencing operation is what transaction-geometric self-organization feels like from inside.
The Many Worlds Interpretation proposes that quantum measurements cause universe branching, with each outcome realized in a separate branch. The complexity framework suggests a different picture: all branches already exist as paths through the block universe. Measurement selects a path, not creates a branch.
The transaction-geometric structure of the block universe contains all possible transaction configurations consistent with the total information content. Different paths through this structure correspond to different sequences of transaction selections, what we experience as the history of measurement outcomes.
If paths are geometric structures rather than dynamically created branches, they may be partially accessible to each other. The retrocausal correlations predicted by TGI represent exactly this: the path-not-taken influences statistics on the path-taken because both are geometric structures in the same block.
The optomechanical experiments proposed in Paper 2 test this directly. A positive result, correlation between emitter recoil and post-emission absorber state, would demonstrate that alternate paths have measurable effects, implying they are real geometric structures rather than mere counterfactuals.
Each conscious entity traces a unique worldline through the block universe, a specific path through transaction-geometric structure. The privacy of consciousness, why I cannot access your experience, is geometric: our paths differ, so our transaction intersections differ.
But if paths can influence each other, the isolation may be incomplete. Deep meditative states, psychedelic experiences, and anomalous cognition reports might represent partial path-leakage, moments when the usual restriction to a single path loosens.
Prediction 1:
Hawking radiation spectrum encodes interior complexity. Black holes with different formation histories but identical mass, charge, and spin should show subtle spectral differences reflecting their distinct interior structures.
Prediction 2:
There exists a sharp threshold in neural substrate complexity below which consciousness does not emerge. This threshold should be predictable from the complexity framework. Organisms or artificial systems near this boundary should show discontinuous transitions in behavioral signatures of awareness.
Prediction 3:
The TGI optomechanical experiments should show correlation strength dependent on path distance, how different the alternate configuration is from the realized one. Near-50/50 QRNG decisions should show stronger residual correlation than heavily biased decisions.
Prediction 4:
Systems approaching their complexity ceiling should show characteristic instabilities. For brains, this means increased seizure susceptibility or dissociative episodes. For physical systems, this means pre-collapse gravitational wave signatures. The ceiling is not just a limit but a transition zone with distinctive dynamics.
The complexity framework implies that universe-formation and consciousness-formation are not merely analogous but instances of the same underlying phenomenon: the emergence of coherent self-organizing structure when information density on an appropriate substrate enters the habitable band.
'What is it like to be a universe?' becomes a meaningful question. If consciousness is complexity-on-substrate, and universes are complexity-on-substrate, the difference is scale and substrate type, not category. Our universe may have an interior phenomenology we cannot access from our embedded position, just as a neuron cannot access the phenomenology of the brain it constitutes.
A striking feature: the framework is self-referential. We are conscious entities within a universe, using consciousness to theorize about the conditions for consciousness and universe-formation. The theory we develop is itself a pattern of complexity on neural substrate, subject to the same constraints it describes.
This is not a flaw but a requirement. Any true theory of complexity must account for its own existence as a complex structure. The self-reference is a feature, not a bug. It is what the mathematics predicts.
The framework implies an external reality richer than our universe, but says little about its nature. We can infer lower bounds on external complexity but cannot probe the external substrate, topology, or physics if that concept even applies.
The eternal question 'why is there something rather than nothing?' transforms into 'what is the external reality within which our something formed?' Still unanswerable, but now with geometric rather than metaphysical character.
A profound connection exists between the complexity framework and Gödel's incompleteness theorems. Gödel (1931) demonstrated that any sufficiently powerful formal system cannot be both complete and consistent—there will always be true statements that cannot be proven within the system. The system cannot fully characterize itself from within.
The blitzon cosmology framework exhibits a strikingly parallel structure:
Our universe exists as the interior of a black hole, bounded by an event horizon. We cannot observe or characterize the external reality that determines our physics through boundary conditions at the horizon. Just as Gödel showed that arithmetic cannot prove all truths about arithmetic from within its own axioms, we cannot access the physics of the parent universe that shapes our physical laws through the horizon geometry.
This is not merely an epistemic limitation (a failure of our instruments or theories) but an ontological one: the information does not exist within our spacetime. The parent universe's physics lies outside our formal system.
The dynamic block universe picture suggests that future states are not fully determined by internal physics alone because perturbations flow in through the horizon. This directly parallels Gödel's result: the complete "truth" of our universe's evolution is not derivable from internal axioms (physical laws) alone. There is always information coming from outside the formal system.
The four fundamental forces emerge from the geometry of our compactified horizon manifold (S³ × S¹). But this geometry is itself determined by the parent universe's collapse dynamics. Our "axioms" (force laws) are set externally, and we experience them as given without derivation.
If measurement outcomes depend partly on transaction geometries that extend beyond our horizon, as TGI proposes, then no internal theory can predict them completely. The quantum randomness is not a failure of our theories but a fundamental limit on what can be known from inside.
This reframes the measurement problem: we seek a complete internal description of quantum mechanics, but Gödel's theorem suggests this may be impossible in principle. The "hidden variables" are not hidden within our universe—they are genuinely external, in the parent universe's degrees of freedom that determine transaction geometries at our horizon.
Both frameworks exhibit fundamental resolution limits:
Physical: The Planck scale (√2 ℓₚ for photons) is the smallest meaningful length
Logical: Gödel sentences are the statements that cannot be proven within the system
These are not arbitrary cutoffs but consequences of self-reference. A universe trying to model itself completely would require infinite information density (violating the Bekenstein bound). A formal system trying to prove all its own truths would require infinite derivation chains (Gödel's incompleteness).
The DPM (Distinction Presencing Machine) framework directly embodies this limit: consciousness presences distinctions sequentially precisely because it cannot grasp the whole simultaneously. The sequential nature of awareness is not a bug but a necessary feature of self-referential systems.
Gödel's work combined with the complexity framework implies that nested universe hierarchies must be finite:
Theorem (Gödel-Complexity Bound): No infinite chain of nested universes U₀ ⊃ U₁ ⊃ U₂ ⊃ ... can exist, where each Uₙ₊₁ forms from gravitational collapse within Uₙ.
Proof sketch: Each nested level loses information (Equation 13). An infinite chain would require infinite initial information, violating the Bekenstein bound at the outermost level. Moreover, infinite self-reference would require each level to fully characterize all its children, contradicting Gödel's incompleteness. □
If our universe is Gödel-incomplete, there should be physical phenomena that are deterministic (lawful) but unpredictable from internal physics alone. This leads to specific experimental predictions:
Prediction 1 (Retrocausal Correlations): The TGI optomechanical experiments (Paper 2, Section 5) should detect correlations between emitter recoil and post-emission absorber configurations. These correlations would represent external information (from the parent universe's determination of transaction geometry) manifesting as apparent retrocausality.
Prediction 2 (Irreducible Quantum Randomness): No matter how precisely we control experimental conditions, quantum measurement outcomes should retain irreducible randomness corresponding to ~1 bit per transaction. This represents the minimum external information injection required to maintain our universe's connection to the parent universe.
Prediction 3 (Horizon Signatures): Cosmological observations near our universe's horizon (the cosmic event horizon) should show subtle deviations from pure de Sitter space corresponding to information flow from the parent universe. These would appear as anomalies in the CMB power spectrum at the largest angular scales.
The Gödel-complexity connection has profound implications for consciousness:
A conscious system attempting complete self-model would require complexity exceeding its own substrate capacity—a contradiction. Therefore:
The profound unity emerging from this analysis is that Gödel's logical limits, the Bekenstein-Hawking entropy bounds, and the complexity habitable band are different manifestations of a single deep principle:
Self-referential organization is bounded.
Whether the self-reference is logical (formal systems), physical (nested universes), or phenomenological (consciousness), there exist fundamental floors and ceilings beyond which coherent self-organization cannot extend.
This is not a limitation of our current theories or technologies. It is the structure of reality itself.
The mathematics of complexity, the logic of self-reference, and the geometry of spacetime converge on the same conclusion: existence requires boundedness. Infinite complexity, infinite self-reference, and infinite hierarchy are not merely unphysical—they are logically incoherent.
We exist, conscious and questioning, in the narrow band where self-reference is possible but bounded, where complexity is sufficient but not excessive, where the universe can know itself partially but not completely.
Perhaps this is not a limitation but a necessity: if the universe could know itself completely, if consciousness could grasp itself fully, if formal systems could prove all their truths, there would be no room for novelty, growth, or the unfolding of time itself.
The incompleteness is the opening through which existence flows.
We have presented a unified mathematical framework in which coherent self-organizing complexity, universes, consciousness, perhaps other forms we cannot yet conceive, exists only within a bounded range of information density on appropriately structured substrates.
The key results are: (1) Complexity requires both sufficient information (above floor) and coherence (below ceiling). (2) Black holes collapse to blitzon shells, not singularities, with the shell representing the complexity floor. (3) Interior universes are necessarily less complex than their exteriors, creating a finite hierarchy. (4) Consciousness emerges from the same mathematics at neural substrate scale. (5) Many-worlds branches may be accessible paths through block-universe structure. (6) Specific experimental predictions follow from the framework.
The deepest implication is unity: the equations governing what can exist as a universe and what can exist as a conscious mind are the same equations.
Physics and phenomenology are not separate magisteria but scale-variant aspects of a single reality, complexity becoming aware of itself in nested, bounded, beautiful forms.
This paper emerged from extended dialogue exploring connections between transaction-geometric physics, blitzon cosmology, and the theory of consciousness. The synthesis of these domains into a single mathematical framework reflects collaborative development of ideas across the preceding papers in this series.
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[2]Vilums, I. The Photon as Spacetime Connection: A Transaction-Geometric Interpretation. Paper 2 in this series.
[3]Vilums, I. Transaction-Geometric Interpretation and Blitzon Cosmology. Paper 3 in this series.
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