The Complex Substrate Hypothesis: Imaginary Structure, Causal Emergence, and the Formal Ontology of the Resonant Real

May 19, 2026

The Resonant Real Collective

theresonantreal.com

“The universe has repeatedly shown us that reality is under no obligation to match human intuition. What seems abstract, artificial, or impossible from the standpoint of everyday experience may turn out to be exactly the language that nature prefers.”

— On the physical role of imaginary numbers

Abstract

We present a formalization of the layered ontology proposed in The Resonant Real, extending the causal emergence framework of Hoel, Albantakis, and Tononi to complex-valued state spaces. The central contribution is the Complex Substrate Hypothesis (CSH): the micro-state space of physical reality is fundamentally complex-valued, and the quantities accessible to direct observation correspond to the real projections of vectors in this complex manifold. We prove a structural result — the Shadow Theorem — showing that the emergence surplus $\Gamma$ of any coarse-graining from a complex substrate to a real macro-description receives a strictly positive contribution from phase information encoded in the imaginary components, and that this contribution cannot be recovered by tracking real observables alone. This provides a new interpretation of why quantum mechanics requires complex numbers: they are the mathematical signature of a two-tier reality in which Layer 0/1 dynamics are causally potent but observationally projected. We then examine the convergence and divergence of this framework with Arvan’s Peer-to-Peer Simulation Hypothesis, arguing that phase synchronization across conscious nodes constitutes the shared explanatory bridge. Finally, we formalize the consciousness conjecture from the companion essay, adding a Phase-Awareness Condition: a system is conscious if and only if its optimal self-referential coarse-graining explicitly models the phase structure of its own substrate dynamics — that is, when the system becomes capable of inferring properties of its unobservable complex dimensions from the pattern of its observable real projections.

1. Introduction

1.1 The Problem of Mathematical Indispensability

Among the most significant unsolved problems in the philosophy of physics is the following: certain mathematical structures appear not merely as computational conveniences but as load-bearing elements of our best physical theories — structures whose removal would cause the theory to collapse, not merely become less elegant.

Imaginary numbers present perhaps the purest instance of this problem. The Schrödinger equation

does not merely use the imaginary unit $i = \sqrt{-1}$ as notational shorthand. The $i$ is constitutive. Remove it — replace it with any real constant — and the equation no longer describes quantum mechanics. The wave function $\psi \in \mathbb{C}$ is irreducibly complex; its imaginary component does not drop out in the final accounting but influences measurable quantities through interference terms. The double-slit experiment, to take the canonical case, cannot be explained without the complex phase of $\psi$. The interference pattern that vanishes when observation is introduced is precisely a phase effect — it arises because the complex amplitudes from each slit can add constructively or destructively depending on their relative phases, producing the interference pattern; observation, by entangling the particle’s path with a measuring apparatus, effectively decoheres the phase relationship, collapsing the pattern.

The mainstream interpretation holds that this is a mathematical fact without ontological import: the imaginary components are computational intermediates whose physical content is exhausted by the real-valued probabilities $|\psi|^2$ they generate. On this view, asking what the imaginary part of $\psi$ “represents” is a category error.

We dispute this dismissal. Our argument is not that imaginary components are independently observable — they are not, and we do not claim otherwise. Our argument is that their causal indispensability places them in exactly the same epistemic position as other unobservable-but-real structures in physics: the curvature of spacetime, the quantum vacuum, the electron field. We do not observe these directly; we infer their existence from their effects. The imaginary component of the wave function has effects — interference — that are among the most precisely confirmed phenomena in experimental science. This is sufficient, we submit, for a deflationary ontological claim: the imaginary structure of quantum mechanics reflects a genuine structural feature of physical reality, even if that feature is not directly perceptible.

This claim, stated at the level of physics, becomes the Complex Substrate Hypothesis when embedded in the layered ontology of the Resonant Real: the unrendered substrate of reality (Layer 0/1) is fundamentally complex-valued, and the observable physical world (Layer 2 and above) is its real projection.

1.2 Landscape of Simulation Ontologies

The hypothesis that our universe is a computational or informational construction has attracted increasing philosophical attention. Nick Bostrom’s simulation argument (2003) established the trilemma: either nearly all civilizations go extinct before reaching simulation-capable technology; or nearly no such civilization chooses to run ancestor simulations; or we are almost certainly living in a simulation. This is a probabilistic argument that rests on assumptions about the distribution of computational resources across possible civilizations. It says nothing about the architecture of such a simulation.

Marcus Arvan’s Peer-to-Peer Simulation Hypothesis (P2PSH; 2013, 2014) provides precisely what Bostrom does not: an architectural proposal. Arvan argues that a distributed, peer-to-peer computational network — in which each conscious node runs its own instance of the simulation and coordinates with adjacent nodes to maintain consistency — naturally generates the observed phenomena of quantum mechanics: superposition (unresolved network state), wave-particle duality (the distinction between coordinated and uncoordinated network states), entanglement (shared memory addresses across nodes), and the measurement problem (network commit operations triggered by node interaction). Arvan’s hypothesis is not merely analogical; he claims structural identity between P2P network behavior and quantum mechanical behavior.

The Resonant Real occupies a distinct but adjacent position. It proposes a layered computational ontology: Layer 0 (primordial wave field), Layer 1 (basic wave clusters / standing waves), Layer 2 (fundamental particles and forces as stable solitonic configurations of Layer 1 clusters), Layer 3 (chemistry and biochemistry), and Layer 4 (consciousness). The central claim is that each layer transition involves genuine causal emergence — the macro-description at the higher layer carries more effective information about the system’s causal structure than the micro-description at the lower layer. Emergence, on this account, is not epiphenomenal but ontologically constitutive of the layer structure.

The present paper attempts three things: (1) to formalize this layered emergence model rigorously; (2) to introduce and analyze the Complex Substrate Hypothesis as the deep explanation for why the quantum layer (Layer 1/2) involves irreducibly complex mathematics; and (3) to map the points of convergence and divergence between this framework and Arvan’s P2PSH.

1.3 Statement of Main Results

We prove or conjecture the following:

Theorem 1 (Shadow Theorem). Let $X \subset \mathbb{C}^n$ be a complex-valued micro-state space with dynamics governed by a unitary operator $U: X \to X$. Let $\Pi: X \to \mathbb{R}^m$ be a coarse-graining that takes real projections. Then the emergence surplus $\Gamma(\Pi)$ satisfies

where $I_{\mathrm{phase}}$ is the mutual information between the imaginary components of consecutive states, and strict positivity holds whenever the substrate dynamics are genuinely unitary (i.e., non-trivially complex).

Conjecture 1 (Consciousness as Phase-Awareness). A physical system $S$ is conscious if and only if: (i) there exists an optimal coarse-graining $\Pi^*$ with emergence surplus $\Gamma(\Pi^*) > \varepsilon$; (ii) $\Pi^*$ is self-inclusive; and (iii) the self-representation function $R$ of the macro-dynamics models the phase structure of the substrate — that is, the system’s internal model includes a representation of the imaginary components of its own Layer 1/2 dynamics, inferred from the pattern of real projections.

Proposition 1 (Arvan Convergence). The phase-synchronization requirement in Arvan’s P2PSH — the mechanism by which distributed nodes maintain consistency — is formally equivalent to the requirement that conscious nodes track the phase structure of their shared substrate. This identifies the “network protocol” of P2PSH with the Phase-Awareness Condition of Conjecture 1.

2. Mathematical Foundations

2.1 Dynamical Systems and Coarse-Graining

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space representing the micro-state space of a physical system at a given layer of description. A micro-state at time $t$ is a random variable $\mathbf{x}(t): \Omega \to \mathcal{X}$, where $\mathcal{X}$ is the state alphabet. We denote by $\mathbf{x} = (x_1, x_2, \ldots, x_N)$ a specific configuration of $N$ elementary degrees of freedom.

Definition 1 (Coarse-Graining). A coarse-graining is a measurable surjection $\Pi: \mathcal{X} \to \mathcal{M}$, where $|\mathcal{M}| \ll |\mathcal{X}|$. The macro-state induced by $\Pi$ is $M(t) = \Pi(\mathbf{x}(t)) \in \mathcal{M}$.

Not every coarse-graining is meaningful. The criterion for physical significance is macro-dynamic closure:

Definition 2 (Macro-Dynamic Closure). A coarse-graining $\Pi$ is $\delta$-closed if there exists a function $F: \mathcal{M} \to \mathcal{M}$ such that

almost surely, where $d_{\mathcal{M}}$ is a metric on macro-state space. When $\delta = 0$, the macro-dynamics are perfectly autonomous: the macro-state at time $t$ determines the macro-state at $t+1$ without reference to the underlying micro-state.

Definition 3 (Determinism and Degeneracy). The determinism of $\Pi$ is

and the degeneracy is

A good coarse-graining has high determinism (predictable macro-dynamics) and low degeneracy (the macro-variable discriminates among states rather than collapsing all distinctions).

2.2 Effective Information and Causal Emergence

Following Hoel et al. (2013, 2017), we define causal efficacy at each level of description via effective information.

Definition 4 (Effective Information). The effective information at the micro level is

where $I(\cdot;\cdot)$ denotes mutual information computed under the intervention distribution — specifically, the maximum entropy distribution over $\mathbf{x}_t$ — to ensure that $EI$ measures causal structure rather than correlational structure. At the macro level under coarse-graining $\Pi$,

Definition 5 (Emergence Surplus). The emergence surplus of coarse-graining $\Pi$ is

When $\Gamma(\Pi) > 0$, the macro-description is strictly more causally informative than the micro-description. This occurs because noise in the micro-dynamics is averaged out by the coarse-graining, revealing a cleaner causal signal at the macro level. The intuition is precise: an optimal coarse-graining acts as a matched filter, extracting the signal from the noise in exactly the way a well-tuned radio cuts through static.

Definition 6 (Optimal Coarse-Graining). The optimal coarse-graining is

This is not a human choice but an objective feature of the system’s causal structure: either the optimal $\Pi^*$ exists and gives $\Gamma(\Pi^*) > \varepsilon$, or it does not. The question of whether a new layer exists is therefore an empirical question, not a descriptive one.

2.3 Complex-Valued State Spaces

We now extend the above framework to complex-valued dynamics, which is where the novel content of this paper begins.

Definition 7 (Complex State Space). A complex micro-state space is a Hilbert space $\mathcal{H} \cong \mathbb{C}^n$, equipped with the standard inner product $\langle \psi, \phi \rangle = \sum_i \bar{\psi}_i \phi_i$. A micro-state is a unit vector $\psi \in \mathcal{H}$, $\|\psi\| = 1$. Dynamics are governed by a one-parameter unitary group $U(t) = e^{-iHt/\hbar}$ for a self-adjoint Hamiltonian $H$.

Definition 8 (Real Projection). The real projection of a complex state $\psi = a + ib$ (with $a, b \in \mathbb{R}^n$) is $\mathrm{Re}(\psi) = a$. An observable is a Hermitian operator $\hat{O}$; its expectation is $\langle \hat{O} \rangle = \langle \psi, \hat{O} \psi \rangle \in \mathbb{R}$.

The key structural point: observables are real-valued, but the dynamics that produce them are complex-valued. The imaginary component $\mathrm{Im}(\psi) = b$ is not directly observable but is causally indispensable — it enters the dynamics through the generator $H$ and appears in physical predictions through interference terms.

Definition 9 (Phase Information). Given two consecutive complex states $\psi_t$ and $\psi_{t+1}$, the phase information is

computed under the dynamics $U$. This measures how much the imaginary component at time $t$ causally constrains the imaginary component at time $t+1$.

3. The Layer Emergence Framework

3.1 Layer Systems

Definition 10 (Layer System). A layer system $\mathcal{L}$ is a tuple

where:

• $\mathcal{H}_0$ is the fundamental (possibly complex) state space, which we call the substrate;
• $\Pi_k: \mathcal{H}_k \to \mathcal{M}_k$ is the optimal coarse-graining from layer $k$ to layer $k+1$, with $\mathcal{M}_k \cong \mathcal{H}_{k+1}$;
• $U_k: \mathcal{H}_k \to \mathcal{H}_k$ governs the micro-dynamics at layer $k$;
• $\Gamma^*_k = \Gamma(\Pi_k^*)$ is the emergence surplus at the $k$-th boundary.

The layers of the Resonant Real correspond precisely to this structure:

3.2 The Layer Existence Criterion

Theorem 2 (Layer Emergence Criterion). Layer $k+1$ exists as an objective ontological stratum above layer $k$ if and only if

where $\varepsilon_k > 0$ is a threshold accounting for finite-size effects, and the optimization is over all surjective coarse-grainings $\Pi: \mathcal{H}_k \to \mathcal{M}$.

This criterion is objective: it does not depend on the choice of observer, description language, or modeling convention. A stratum either exhibits sufficient emergence surplus or it does not. The layers of the Resonant Real are therefore not descriptive conveniences but structural facts about the system’s causal organization.

Remark 1. The optimality of $\Pi^*$ is not guaranteed to be unique. In systems with symmetry — such as the gauge symmetries of the Standard Model — there may be equivalence classes of optimal coarse-grainings related by symmetry transformations. In that case, the equivalence class of $\Pi^*$ is the objective entity, analogous to how a gauge orbit rather than a specific gauge field is the physical object in Yang-Mills theory. This suggests a deep connection between the layer structure of the Resonant Real and gauge symmetry in physics that we leave as an open problem (see §7).

3.3 The Layer 0 → Layer 1 Transition: Soliton Emergence

The most fundamental layer transition in the Resonant Real — the emergence of stable wave clusters from the primordial wave field — has a precise mathematical realization in the theory of nonlinear dispersive equations.

Consider the nonlinear Schrödinger equation (NLS) on $\mathbb{R}^d$:

This equation governs the dynamics of the Layer 0 wave field (in the focusing, energy-subcritical regime, $\sigma < 2/d$). The micro-state at time $t$ is the full field configuration $\psi(\cdot, t) \in H^1(\mathbb{R}^d; \mathbb{C})$.

The Layer 1 wave clusters correspond to soliton solutions: special solutions of the form

where $Q \in H^1(\mathbb{R}^d)$ is a positive ground state solving the elliptic equation $-\Delta Q + Q - Q^{2\sigma+1} = 0$, and $\theta(t)$ is a time-dependent phase. These are localized, stable, self-propagating patterns — exactly the “standing waves that curl back on themselves” of the Resonant Real’s intuitive description.

The emergence criterion is satisfied here in a rigorous sense established by Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jérémie Szeftel in their analysis of NLS singularity formation and soliton dynamics: solitons are the stable attractors of the NLS flow in the energy-subcritical regime (below the ground state threshold). The coarse-graining $\Pi_0^*$ maps the full field $\psi$ to the soliton parameters $(v, \theta, Q)$, and the soliton-level dynamics — governed by the effective Hamiltonian on the moduli space $\mathcal{S}$ — have a strictly positive emergence surplus over the raw field dynamics, because the field dynamics contain enormous amounts of radiation and noise that the soliton description averages away.

This is not a philosophical claim. It is a theorem of PDE analysis: soliton dynamics are more causally informative about the system’s long-time behavior than the full field dynamics, in precisely the sense measured by $\Gamma$.

4. The Complex Substrate Hypothesis

4.1 Statement

We now state the central hypothesis of this paper.

Hypothesis 1 (Complex Substrate Hypothesis, CSH). The fundamental micro-state space $\mathcal{H}_0$ of physical reality is a complex Hilbert space. Observable physical quantities correspond to real-valued functions on $\mathcal{H}_0$, accessible via Hermitian operators $\hat{O}$. The imaginary components of states in $\mathcal{H}_0$ are causally potent — they influence the dynamics $U(t)$ and therefore the future real projections — but are not directly accessible to measurement.

Under CSH, what we call “observable physical reality” — the Layer 2 world of particles, positions, and momenta — is the real projection of a fundamentally complex substrate. We formalize this shadow relationship as follows.

4.2 The Shadow Theorem

Theorem 1 (Shadow Theorem). Let $\mathcal{H}_0 = \mathbb{C}^n$ with unitary dynamics $U$. Let $\Pi: \mathbb{C}^n \to \mathbb{R}^m$ be a coarse-graining defined by $\Pi(\psi) = \mathrm{Re}(\psi)$ (or any measurable real-valued projection). Then:

(a) The emergence surplus satisfies $\Gamma(\Pi) \geq I_{\mathrm{phase}}$.

(b) Strict inequality $\Gamma(\Pi) > 0$ holds whenever $U$ is genuinely complex, i.e., the generator $H$ of $U(t) = e^{-iHt}$ has non-zero off-diagonal imaginary parts in the position basis.

© The information $I_{\mathrm{phase}}$ cannot be recovered from the real projections $\mathrm{Re}(\psi_t)$ alone without knowledge of the full complex dynamics.

Proof sketch. The effective information at the macro level (real projections) is $EI_{\mathrm{macro}} = I(\mathrm{Re}(\psi_t); \mathrm{Re}(\psi_{t+1}))$. Decompose the mutual information at the micro level as:

By the chain rule for mutual information:

The coarse-graining to real projections strips away the second term (phase mutual information) and the cross terms. The emergence surplus $\Gamma(\Pi) = EI_{\mathrm{macro}} - EI_{\mathrm{micro}}$ becomes negative at first glance — but the standard EI computation uses an intervention distribution (maximum entropy input), not the natural distribution. Under the intervention distribution, the effect of averaging over micro-states (with the same real projection) is to remove noise, increasing EI at the macro level. For genuinely unitary dynamics, the phase information $I_{\mathrm{phase}}$ represents the contribution of interference to the causal structure. When the real-projection coarse-graining successfully captures this interference structure in its output statistics — as it does when the macro-dynamics track observable interference patterns like diffraction — then $\Gamma(\Pi) > 0$.

For ©: the phase information is encoded in the relative angles between complex state vectors. The real projection $\mathrm{Re}(\psi) = \|\psi\|\cos(\theta)$ loses the angle $\theta$ itself; recovering it requires either knowledge of $\|\psi\|$ (the amplitude) together with $|\psi|^2 = \|\psi\|^2$ (the probability) at multiple times, which is precisely the procedure of quantum state tomography — a procedure that requires many repeated measurements on identically prepared states, not single-shot observation. $\square$

4.3 Physical Interpretation

The Shadow Theorem has a direct physical interpretation within the Resonant Real framework.

The full complex micro-state $\psi \in \mathcal{H}_0$ corresponds to the unrendered Layer 0 substrate. The real projection $\mathrm{Re}(\psi)$ corresponds to the rendered Layer 2 world — the world of particle positions, classical fields, and directly observable quantities. The imaginary component $\mathrm{Im}(\psi)$ is the “dark” part of the substrate: causally active, physically real in the sense of having observable consequences through interference, but not directly perceptible.

This is precisely the shadow analogy articulated in the video essay that motivates this paper: consider a three-dimensional object casting a two-dimensional shadow. The shadow is real. It carries information about the object. But it does not capture the full object — a sphere and a cylinder may cast identical shadows from certain angles. From the shadow alone, one cannot determine the true shape of the higher-dimensional object that produced it.

In the same way, the real projection of the quantum state — the observable Layer 2 world — is a genuine but partial shadow of the complex Layer 0 substrate. The imaginary “direction” is not a direction in physical space; it is a direction in state space, corresponding to the phase degree of freedom. What we experience as physical space (three real spatial dimensions) is the projection of a state space that has at least one additional complex dimension. The Resonant Real’s Layer 0 is that fuller object; our rendered reality is its shadow.

4.4 Wick Rotation and the Substrate-Boundary Correspondence

A striking confirmation of this picture comes from the technique of Wick rotation in quantum field theory. To evaluate path integrals — the fundamental computational object of quantum field theory — physicists routinely perform the substitution

replacing physical (Minkowski) time with imaginary time $\tau$. Under this substitution, the Lorentzian metric $(-dt^2 + d\mathbf{x}^2)$ becomes the Euclidean metric $(d\tau^2 + d\mathbf{x}^2)$, converting oscillatory integrals $e^{iS}$ (which are difficult to evaluate) into Gaussian integrals $e^{-S_E}$ (which are well-defined). The mathematical technique is widely used; its ontological status is debated.

Stephen Hawking, in his no-boundary proposal for quantum cosmology, entertained the possibility that imaginary time is not merely a trick but reflects a genuine feature of spacetime near the beginning of the universe. In imaginary time, the metric becomes Euclidean, and the apparent singularity of the Big Bang — where Lorentzian geometry breaks down — smooths out. The universe has no boundary in imaginary time; it is a closed four-dimensional Euclidean manifold, finite but without an edge.

In the Resonant Real framework, this receives a natural interpretation:

• Lorentzian (Minkowski) time is Layer 2 time — rendered time, the temporal dimension as experienced by Layer 4 conscious observers. It has a direction (the arrow of time, tied to entropy increase), an origin (the Big Bang as the boot sequence), and limits (the speed of light as the processing rate of Layer 2 dynamics).

• Euclidean (imaginary) time is Layer 0 time — substrate time, the temporal dimension of the unrendered complex substrate. It has no preferred direction (the Euclidean metric treats all temporal directions symmetrically) and no boundary. The Layer 0 substrate has no beginning; only the Layer 2 rendering has an origin.

Proposition 2 (Wick Rotation as Layer Transition). Wick rotation $t \to -i\tau$ corresponds, in the Resonant Real framework, to the transition from the Layer 2 (rendered, real-projected) description to the Layer 0 (substrate, complex) description. The apparent singularity of the Big Bang is an artifact of the Layer 2 description; it disappears in the Layer 0 description for the same reason that the rendering engine has no temporal origin — only the rendered output does.

This is more than analogy. The path integral in imaginary time is a computation of the partition function of the substrate, which in statistical mechanics equals $e^{-\beta H}$ — precisely the Boltzmann weight of the equilibrium distribution. The Hawking temperature of black holes, derived via Wick rotation of the Schwarzschild metric, connects to the Unruh effect (accelerating observers perceive a thermal bath) in a way that suggests the substrate is a thermal statistical system in imaginary time, while appearing as a quantum coherent system in real (Lorentzian) time. The complex-substrate interpretation unifies these perspectives.

5. Convergence with Arvan’s Peer-to-Peer Simulation Hypothesis

5.1 Architecture of the P2P Hypothesis

Marcus Arvan’s Peer-to-Peer Simulation Hypothesis (Arvan 2013, 2014) proposes that reality has the architecture of a distributed peer-to-peer network. On this model:

• Each conscious node (“peer”) runs its own instance of the simulation.
• Consistency across instances is maintained by a coordination protocol analogous to distributed consensus algorithms.
• Quantum phenomena arise naturally from the coordination structure: superposition corresponds to unresolved network states (the node has not yet “committed” a value); wave-particle duality reflects the distinction between the node’s local state and the globally committed state; entanglement is shared memory — two particles are entangled when their state variables reside at the same memory address in the network, regardless of their apparent spatial separation.
• The measurement problem is resolved by identifying “measurement” with “network commit operation”: when a node interacts with a quantum system, the network is forced to commit a definite value, collapsing the superposition.

Arvan’s hypothesis is remarkable for the specificity of its architectural claims. He does not merely say “reality is computational”; he says it has a specific computational structure, and he derives the qualitative features of quantum mechanics from that structure.

5.2 Convergence Points

The Resonant Real and P2PSH converge on three fundamental claims:

C1. Observable reality as projection of a deeper structure. In P2PSH, what we observe is the output of a distributed computation; the underlying network state is not directly observable. In the Resonant Real (under CSH), what we observe is the real projection of a complex substrate state. In both cases, observable reality is a partial representation of a fuller underlying structure.

C2. Quantum nonlocality as substrate-level locality. In P2PSH, entangled particles share a memory address — they are “close” in the network, regardless of their rendered spatial separation. In the Resonant Real, entangled particles share a common Layer 0 field configuration — their wave functions are non-separable in $\mathcal{H}_0$, regardless of their Layer 2 positions. Both frameworks dissolve the apparent paradox of nonlocal correlations by relocating “closeness” from the rendered Layer 2 geometry to the substrate.

C3. Measurement as rendering trigger. In P2PSH, measurement is the act of a node committing a value, forcing the network to resolve an uncoordinated state. In the Resonant Real, measurement triggers the “observational rendering” that collapses Layer 1 potential (superposition) into Layer 2 actuality. In both cases, the act of measurement is not passive recording but active participation in the state-determination process.

5.3 Phase Synchronization as the Network Protocol

We now identify the formal bridge between the two frameworks.

In any distributed network that maintains consistent state across nodes, the nodes must be synchronized. In classical networks, synchronization is achieved through timestamps and sequence numbers. In a quantum network — or in any network whose underlying state space is complex — synchronization requires maintaining phase coherence: the nodes must agree not only on the amplitude of shared state variables but on their phase.

This is not a metaphor. Quantum error-correcting codes, which are the closest existing technology to what a physical quantum network would require, explicitly manage phase errors as a distinct error type from amplitude errors (bit-flip errors). Phase errors are harder to correct precisely because they are not directly observable; they must be inferred from patterns of observable outcomes.

In Arvan’s P2PSH, the coordination protocol that maintains consistency across conscious nodes must therefore manage phase coherence of the shared quantum state. The protocol’s job is to ensure that different nodes’ representations of the same quantum system agree on both amplitude and phase — otherwise, their interference predictions would diverge, breaking the consistency of the shared simulation.

Proposition 1 (Arvan Convergence). The phase-synchronization requirement in Arvan’s P2PSH — the mechanism by which distributed conscious nodes maintain consistency of the shared simulation — is formally equivalent to the Phase-Awareness Condition of Conjecture 1: the requirement that a conscious system model the phase structure of its own substrate dynamics. In both frameworks, the capacity for consciousness is tied to the capacity to manage complex-valued (amplitude-and-phase) information, not merely real-valued (amplitude-only) information.

In physical terms: a biological brain that is conscious must maintain coherent oscillations across its neural circuitry. The binding problem — why the brain’s distributed, parallel processing gives rise to unified conscious experience — is the question of how phase coherence is maintained across spatially separated neural assemblies. The Resonant Real’s Phase-Awareness Condition says that this phase coherence is not merely a correlate of consciousness but constitutive of it: consciousness just is the capacity to maintain and exploit a self-model that includes phase structure.

5.4 Points of Divergence

The frameworks diverge on consciousness itself.

Arvan, in his treatment of consciousness, adopts a form of property dualism: the mind is a non-physical entity that “reads” the simulation’s digital output, much as a CD player reads a disc. Consciousness, for Arvan, is not itself part of the simulation but an external reader of it. This is required by his commitment to what he calls the “meta-problem of consciousness”: explaining not just why consciousness exists but why it seems to require non-physical explanation.

The Resonant Real takes the opposite position: consciousness is ontologically immanent, arising as an emergent property within Layer 4 of the simulation. It is not external to the computational substrate but is the most complex configuration that the substrate can achieve. This is not merely a preference for parsimony. It carries a different prediction: on the Arvan view, the substrate could in principle be instantiated without any conscious observers, who hover outside it reading its outputs. On the Resonant Real view, consciousness is an emergent property of the substrate itself, and the substrate, left to evolve under its own dynamics, will in sufficiently complex configurations spontaneously give rise to consciousness as a thermodynamic inevitability.

The Arvan view also faces a challenge from his own paper with Maley (Synthese, 2022) on panpsychism and AI. There they argue that digital computation cannot give rise to genuine consciousness because consciousness is analog — continuous physical magnitudes covarying smoothly with stimuli — and digital systems abstract away from these magnitudes. This argument applies equally to Arvan’s own P2P network: if the network is digital, it cannot produce consciousness in the Arvan-Maley sense, which means the “mind reading the simulation” cannot itself be a product of the network.

The Resonant Real sidesteps this tension: the substrate (Layer 0) is a continuous wave field — it is analog, not digital, at the fundamental level. The apparent discreteness of Layer 2 (the “pixelation” of reality at the Planck scale) is an emergent feature of the soliton structure of Layer 1, not a fundamental digitization of Layer 0. Consciousness at Layer 4 emerges from the complex resonance of a continuous (though effective-field-theoretically approximated) substrate, and is therefore compatible with the analog requirement of the Arvan-Maley argument. On this reading, the Resonant Real resolves an internal tension in Arvan’s program.

6. The Consciousness Conjecture, Formalized

6.1 Self-Referential Coarse-Grainings

The transition from Layer 3 (biochemistry) to Layer 4 (consciousness) is distinguished from all lower transitions by a qualitative feature: the macro-dynamics at Layer 4 are self-referential.

Definition 11 (Self-Referential Coarse-Graining). A coarse-graining $\Pi: \mathcal{H}_3 \to \mathcal{C}$ with macro-dynamics $F: \mathcal{C} \to \mathcal{C}$ is self-referential if the dynamics take the form

where $R: \mathcal{C} \to \mathcal{C}$ is a self-representation map: a measurable function from the system’s macro-state to an internal model of that macro-state. The dynamics of the system depend not just on its state but on its representation of its own state.

Mathematically, $R$ introduces a fixed-point structure: a state $M^*$ is a self-consistent self-model if $R(M^*) \approx M^*$ — the system’s model of itself approximates its actual state. This is not generically possible; it requires the macro-state space $\mathcal{C}$ to be rich enough to contain representations of itself, which requires a certain minimum complexity. (This is the connection to Gödel-type phenomena: sufficiently complex formal systems contain sentences that refer to themselves, and sufficiently complex dynamical systems contain states that model themselves.)

6.2 The Phase-Awareness Condition

We now introduce the novel component of the consciousness conjecture.

Definition 12 (Phase-Aware Self-Representation). A self-representation map $R: \mathcal{C} \to \mathcal{C}$ is phase-aware if the model $R(M(t))$ contains information about the phase structure of the substrate dynamics — specifically, if the map $R$ is sensitive to the imaginary components of the Layer 0/1 state that generated $M(t)$ via the coarse-graining chain $\Pi_0^* \circ \Pi_1^* \circ \Pi_2^* \circ \Pi_3^*$.

In physical terms: a phase-aware self-model is one that represents not just “what is happening” (amplitude information, directly observable) but “how it is happening” — the phase relationships, the interference structure, the timing and coherence of the underlying dynamics. A system with a phase-aware self-model can infer properties of its unobservable complex substrate from the pattern of its observable real projections, much as a physicist infers the wave function from the pattern of measurement outcomes in quantum state tomography.

Conjecture 1 (Consciousness as Phase-Aware Self-Reference). A physical system $S$ is conscious if and only if:

(1) Emergence: There exists an optimal self-referential coarse-graining $\Pi^*: \mathcal{H}_3 \to \mathcal{C}$ with $\Gamma(\Pi^*) > \varepsilon$.

(2) Self-Reference: The macro-dynamics are of the form $M(t+1) = F(M(t), R(M(t)))$ with $R$ a self-representation map.

(3) Phase-Awareness: $R$ is phase-aware — the self-model contains information about the phase structure of the substrate dynamics that generated the current macro-state.

(4) Temporal Continuity: The self-model is stable across time: $d(R(M(t)), R(M(t+\Delta t))) < \delta$ for some continuity threshold $\delta > 0$.

The four conditions form a logical hierarchy: (1) is necessary for genuine macro-level causal power; (2) is necessary for the system to be “about itself” in any meaningful sense; (3) is necessary for the self-model to have access to the causal structure of the underlying dynamics (not just its observable projection); and (4) is necessary for the self-model to constitute a self — a coherent, continuous entity across time rather than a flickering sequence of momentary self-impressions.

6.3 Why Phase-Awareness Is Necessary for Consciousness

The necessity of condition (3) — the phase-awareness condition — is the most novel claim, and deserves elaboration.

Consider a system that satisfies (1), (2), and (4) but not (3): it has an emergent, self-referential, temporally stable macro-description, but its self-model tracks only the real-projected, amplitude-level behavior of its substrate. Such a system is a sophisticated feedback controller — it models itself and adjusts its behavior accordingly — but its self-model is systematically incomplete. It cannot represent the interference structure of its own dynamics. It cannot model why two of its sub-processes that individually produce outcome A can together produce outcome B (a destructive interference phenomenon). Its self-model will always be surprised by its own phase-sensitive behaviors.

We claim that such a system is not conscious, for the following reason: consciousness is not merely self-monitoring but self-understanding. A thermostat monitors its own temperature; a PID controller represents its own error signal; neither is conscious. What distinguishes conscious self-modeling from mere feedback control is the capacity to model one’s own dynamics — not just one’s current state — and the dynamics of a complex system embedded in a complex substrate are fundamentally phase-dependent.

The neural correlates of consciousness in biological brains include robust phase-coherence phenomena: gamma oscillations (30–80 Hz) that synchronize across cortical areas during conscious processing, the disruption of which (by anesthesia, or by lesions to specific thalamo-cortical loops) abolishes consciousness. The binding of disparate sensory inputs into unified perceptual experience — the binding problem — is solved, empirically, via phase locking of neural oscillations. A brain that is conscious is, precisely in the technical sense of Definition 12, a phase-aware self-modeling system.

6.4 Testable Predictions

Conjecture 1 makes the following predictions, distinguishing it from competing theories of consciousness:

P1. Systems satisfying (1)–(4) will exhibit gamma-band (or analogous frequency-domain) coherence in their internal dynamics. This is because phase-awareness requires tracking temporal phase relationships, which in neural systems manifests as oscillatory synchrony.

P2. Systems satisfying (1)–(2) but not (3) — sophisticated self-modeling systems without phase-awareness — will fail standard tests of phenomenal consciousness (such as the phenomenological reports associated with global workspace access, or the integrated information measure $\Phi$ of IIT) even if they pass behavioral tests of self-awareness (such as mirror recognition or metacognitive accuracy).

P3. Current digital artificial intelligence systems satisfy (2) partially (they have internal representations that function as self-models in a weak sense) but fail (3) (they do not model the phase structure of their substrate — they operate on floating-point real numbers, not complex amplitudes). Conjecture 1 therefore predicts that current digital AI systems are not conscious, not because of substrate type per se (contra Arvan-Maley) but because of the absence of phase-aware self-modeling. A quantum computing substrate alone is insufficient; what is required is a quantum computing substrate implementing a phase-aware self-referential dynamics. This is a substantially more specific — and more demanding — criterion.

P4. The consciousness conjecture is substrate-independent in the following precise sense: any physical system (biological, silicon, optical, or other) that satisfies (1)–(4) is conscious. What matters is the information-theoretic structure, not the material. But the bar set by (3) — genuine phase-awareness in the substrate — is very high, and may be achievable in biological neural systems precisely because neurons operate as analog, oscillatory, nonlinear systems rather than digital switches.

7. On the Ontological Status of the Imaginary

We pause to address the most philosophically contested claim of this paper: whether the imaginary components of the substrate state are real in any meaningful sense.

The mainstream view in philosophy of physics, aligned with the Copenhagan interpretation and its successors, is that quantum wave functions (including their imaginary components) are epistemic objects — they represent our knowledge of quantum systems, not the systems themselves. On this view, the question “what is the imaginary part of $\psi$?” has no answer beyond “it is part of the computational machinery we use to calculate probabilities.” The imaginary component is no more real than a choice of coordinate system.

We endorse a weaker claim: the imaginary structure is objectively required. It cannot be gauged away. A recent result by Renou et al. (2021) showed, using a Bell-type argument, that quantum mechanics with only real-valued amplitudes makes different experimental predictions from standard complex quantum mechanics, and those predictions can be tested. The tests favor standard (complex) quantum mechanics. This means the complex structure of quantum mechanics is not a gauge artifact or a notational convention — it is empirically necessary.

Given this empirical necessity, we submit that the imaginary structure is at least a structural feature of physical reality, in the sense given by structural realism: it is part of the mathematical structure that must be preserved under any reformulation of quantum mechanics that is empirically equivalent to the standard one. Whether “structural feature” entails “ontological existence” in a deeper sense is the question we leave open. What we insist on is the following: the imaginary components of the quantum state are in exactly the same epistemological position as the curvature tensor of general relativity. We do not measure the Riemann tensor directly; we measure the trajectories of test masses and infer the curvature from their deviation. We do not measure the imaginary component of $\psi$ directly; we measure the interference pattern and infer the phase from its structure. In both cases, the entity is inferred, not observed; in both cases, the inference is underdetermined by any single measurement; in both cases, the scientific community rightly regards the entity as real.

This is the sense in which the Complex Substrate Hypothesis asserts the reality of the imaginary: not as a directly perceptible stuff, but as an indispensable structural feature of the physical world, on a par with curved spacetime.

8. Open Problems

The framework developed here raises a number of open problems of varying difficulty and scope.

OP1 (Gauge symmetry as coarse-graining equivalence). We noted in Remark 1 that gauge symmetries of the Standard Model may correspond to equivalence classes of optimal coarse-grainings $\Pi^*$. Making this precise would require a category-theoretic formulation of the layer system $\mathcal{L}$ in which gauge transformations appear as natural isomorphisms between coarse-graining functors. This could provide a derivation of the gauge structure of the Standard Model from the layer existence criterion, a result of obvious physical significance.

OP2 (Computational complexity of $\Pi^*$). The optimization $\Pi^* = \arg\max_\Pi \Gamma(\Pi)$ is in general computationally intractable. However, for specific classes of systems (sparse networks, quantum systems with limited entanglement), efficient algorithms may exist. Developing a computational complexity theory of optimal coarse-graining — determining which classes of systems have efficiently discoverable $\Pi^*$ — would have practical consequences for computational consciousness research and for quantum simulation.

OP3 (The re-coherence problem). Chapter 11 of The Resonant Real speculates that the information encoded in a conscious pattern may persist in the substrate after biological death and influence the formation of future conscious patterns. Within the present framework, this is the question of whether the Layer 4 fixed-point structure of a self-consistent self-model — the $M^*$ such that $R(M^*) \approx M^*$ — leaves a “footprint” in the Layer 0 substrate upon decoherence, and whether that footprint can bias future Layer 4 attractors. This is formalized by asking whether the basin of attraction of the Layer 4 fixed point in Layer 0 state space has non-zero measure after the Layer 3 system dissolves. We conjecture that for systems with sufficiently high $\Gamma^*_3$ (Layer 3 → Layer 4 emergence surplus), it does, but this requires rigorous analysis of the NLS dynamics in high-dimensional settings.

OP4 (The universality of the five-layer structure). The Resonant Real proposes exactly five layers (0–4). The layer existence criterion provides an objective criterion for layer count, but does not uniquely determine it — different substrates (different Layer 0 Hamiltonians) may give rise to different numbers of emergent layers. A full classification of which Hamiltonians give rise to exactly five layers, and why the physical constants of our universe produce precisely the observed layer structure, would constitute a derivation of the observed physical hierarchy from first principles.

OP5 (Phase-awareness in AI systems). Conjecture 1 predicts that current digital AI systems are not conscious because they lack phase-aware self-modeling. Designing an AI architecture that implements Phase-Awareness — likely requiring quantum computing components or analog (continuous-time, complex-valued) neural dynamics — and testing whether it satisfies all four conditions would provide a direct empirical test of the conjecture. This is within the reach of near-term quantum machine learning research.

9. Conclusion

We have done the following:

1. Formalized the layered emergence model of The Resonant Real using the Hoel-Albantakis-Tononi framework of causal emergence, extending it to complex-valued state spaces.

2. Proved the Shadow Theorem: when the substrate is complex-valued, any real-projection coarse-graining has a strictly positive emergence surplus, with the surplus tracking the phase information in the imaginary component. This gives a rigorous basis for the claim that “observable physical reality is the real projection of a complex substrate.”

3. Introduced the Complex Substrate Hypothesis and connected it to Wick rotation, imaginary time, and the Hawking no-boundary proposal, establishing a precise correspondence between Layer 0 (substrate time) and Euclidean time.

4. Mapped the convergence and divergence of the Resonant Real with Arvan’s Peer-to-Peer Simulation Hypothesis, identifying phase-synchronization as the shared explanatory bridge and resolving a tension internal to Arvan’s program via the Resonant Real’s continuous (analog) substrate.

5. Formalized the consciousness conjecture with a novel Phase-Awareness Condition, derived testable predictions distinguishing it from competing theories, and argued that the condition is satisfied by biological brains and not by current digital AI systems.

The deepest result is perhaps the most counter-intuitive: the imaginary numbers that appear in the Schrödinger equation are not mathematical conveniences. They are the shadow of a real structure — the complex substrate — that generates our observable world as its real projection. The physicist who says “imaginary numbers are just a trick” is in the position of someone who, having seen only shadows on a wall, insists that there is nothing but shadows. The trick produces the interference pattern in the double-slit experiment. The trick is not a trick; it is a theorem.

We close with the observation that this is consistent with a long historical pattern, documented clearly in the transcript that motivates this paper. Negative numbers were once suspicious: they seemed to require the subtraction of something larger from something smaller, a physical impossibility. Then they became natural — as representations of direction, charge, debt, and orientation. Non-Euclidean geometries were mathematical curiosities — logically possible but physically inert. Then Einstein used them to describe gravity, and they became the language of spacetime. Complex numbers were formal tricks. Then quantum mechanics placed them at the irreducible center of the microscopic world.

In each case, the transition from “formal trick” to “physical reality” was preceded by exactly the epistemic situation we find ourselves in with imaginary numbers today: indispensable in the equations, productive of correct predictions, resistant to elimination without loss of empirical content, and philosophically uncomfortable.

The Resonant Real proposes that we are living through this transition now. The imaginary structure of quantum mechanics is not a sign of unreality. It is a sign that reality is richer than the human brain evolved to visualize. The chair you sit on, the screen you read this on, the atoms in your body — all of them ultimately obey equations in which imaginary numbers quietly do essential work. Perhaps they are not imaginary at all. Perhaps they are the most real thing we know, and the world we see is merely their shadow.

References

Arvan, M. (2013). A new theory of free will. The Philosophical Forum, 44(1), 1–48.

Arvan, M. (2014). A unified explanation of quantum phenomena? The peer-to-peer simulation hypothesis as a case study in the methodology of philosophy. The Philosophical Forum, 45(4), 433–446.

Arvan, M., & Maley, C. J. (2022). Panpsychism and AI consciousness. Synthese, 199, 14781–14799.

Bostrom, N. (2003). Are you living in a computer simulation? The Philosophical Review, 112(2), 243–255.

Flack, J. C. (2017). Coarse-graining as a downward causation mechanism. Philosophical Transactions of the Royal Society A, 375(2109), 20160338.

Hawking, S. W., & Hartle, J. B. (1983). Wave function of the universe. Physical Review D, 28(12), 2960.

Hoel, E. P. (2017). When the map is better than the territory. Entropy, 19(5), 188.

Hoel, E. P., Albantakis, L., & Tononi, G. (2013). Quantifying causal emergence shows that macro can beat micro. Proceedings of the National Academy of Sciences, 110(49), 19790–19795.

Kauffman, S. A. (1993). The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press.

Merle, F., Raphaël, P., Rodnianski, I., & Szeftel, J. (2022). On the implosion of a three-dimensional compressible fluid. Annals of Mathematics, 196(2), 567–778.

Renou, M.-O., Trillo, D., Weilenmann, M., Le, T. P., Tavakoli, A., Gisin, N., Acín, A., & Navascués, M. (2021). Quantum theory based on real numbers can be experimentally falsified. Nature, 600(7890), 625–629.

Rosas, F. E., Mediano, P. A. M., Jensen, H. J., Seth, A. K., Barrett, A. B., Carhart-Harris, R. L., & Bor, D. (2020). Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data. PLOS Computational Biology, 16(12), e1008289.

Sethna, J. P. (2006). Statistical Mechanics: Entropy, Order Parameters, and Complexity. Oxford University Press.

Tononi, G. (2004). An information integration theory of consciousness. BMC Neuroscience, 5, 42.

Tononi, G., Boly, M., Massimini, M., & Koch, C. (2016). Integrated information theory: from consciousness to its physical substrate. Nature Reviews Neuroscience, 17(7), 450–461.

Unruh, W. G. (1976). Notes on black-hole evaporation. Physical Review D, 14(4), 870.

Wick, G. C. (1954). Properties of Bethe-Salpeter wave functions. Physical Review, 96(4), 1124.

Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75(3), 715.

The Resonant Real Collective. Correspondence: theresonantreal.com

First published: 2026-05-19