Interiority Hierarchy: Formal Specification

Terminological Revision for Unitary Holonomic Monism

On notation

In this document:

• $\mathcal{H}_E$ — Hilbert space of the Interiority dimension. Not to be confused with $H$ — the Hamiltonian.
• $D_{\text{diff}}$ — differentiation measure. Not to be confused with $D$ — the Dynamics dimension.
• $\Phi$ — integration measure. Not to be confused with CPTP channels $\Phi$.
• $R$ — reflection measure.

Motivation

The problem

The term "Qualia" has historically been associated with conscious subjective experience (Nagel, 1974; Chalmers, 1996). UHM uses it to describe a fundamental property of any system, including atoms, which creates:

1. Terminological conflict: Philosophers of mind understand qualia as "the redness of red," "the painfulness of pain" — phenomena requiring a conscious subject.

2. Anthropomorphism: Attributing "qualia" to an atom implicitly transfers to it the properties of conscious experience.

3. Conceptual dilution: If everything has qualia, the term loses its discriminative force.

The solution

Introduction of a five-level hierarchy (L0→L1→L2→L3→L4), where each level has:

• A strict mathematical definition
• Explicit conditions of applicability
• Examples of systems at that level

---

Part I: Formal Definitions

Level 0: Interiority

Definition 0.1 (Interiority)

Interiority is a fundamental topological property of the Coherence Matrix $\Gamma$ of having an "inner side."

Formally, a system $S$ possesses interiority if and only if:

$$\mathrm{Int}(S) := \exists \mathcal{H}_E, \exists \rho_E \in \mathcal{L}(\mathcal{H}_E) : \rho_E = \mathrm{Tr}_{-E}(\Gamma_S)$$

where:

• $\mathcal{H}_E$ — Hilbert space of the Interiority dimension
• $\rho_E$ — reduced density matrix of dimension $E$
• $\mathrm{Tr}_{-E}$ — partial trace over all dimensions except $E$
• $\Gamma_S$ — full coherence matrix of system $S$

Theorem 0.1 (Universality of Interiority)

Statement: Any system described by a coherence matrix $\Gamma$ in the extended formalism possesses interiority.

Precondition: tensor structure

The theorem requires the extended tensor formalism (see Two levels of formalization):

$$\mathcal{H} = \bigotimes_{i \in \{A,S,D,L,E,O,U\}} \mathcal{H}_i$$

In the minimal 7D formalism ($\mathcal{H} = \mathbb{C}^7$) the partial trace $\mathrm{Tr}_{-E}$ is not defined, since 7 is prime. Interiority in the minimal formalism should be understood as potential: any system can be described in the extended formalism where interiority is defined.

Proof (in the extended formalism):

1. By the Ω⁷ Axiom, any system $S$ is characterized by $\Gamma_S \in \text{Ob}(\mathcal{C})$
2. In the extended formalism the state space $\mathcal{H} = \bigotimes_i \mathcal{H}_i$ includes $\mathcal{H}_E$
3. The operation $\mathrm{Tr}_{-E}(\Gamma_S)$ is defined for any $\Gamma_S \geq 0$ given tensor structure
4. Therefore $\rho_E := \mathrm{Tr}_{-E}(\Gamma_S)$ exists
5. Ergo, $\mathrm{Int}(S) = \mathrm{true}$ ∎

Characteristics of Level 0

Aspect	Specification
Definition	Topological property of "having an inner side"
Mathematics	Existence of $\mathcal{H}_E$ and operator $\rho_E$
Ontological status	Fundamental primitive
System requirements	$\Gamma \geq 0$, $\mathrm{Tr}(\Gamma) = 1$
Reflection requirements	$R \geq 0$ (may be zero)
Integration requirements	$\Phi \geq 0$ (may be minimal)

Examples of systems with Interiority (Level 0)

1. Hydrogen atom

   • $\rho_E = \mathrm{diag}(p_{1s}, p_{2s}, p_{2p}, \ldots)$ — distribution over energy levels
   • $R \approx 0$ (no self-modeling)
   • $\Phi \approx 0$ (minimal integration)

2. NaCl crystal

   • $\rho_E$ — describes phonon modes
   • $R \approx 0$
   • $\Phi \approx 0.1$ (weak integration through lattice)

3. Thermostat

   • $\rho_E$ — classical temperature distribution
   • $R \approx 0$
   • $\Phi \approx 0$

What Level 0 does NOT claim

Interiority does not imply:

• Presence of "sensations"
• Presence of "experiences"
• Presence of a "subject"
• Capacity for reflection
• Consciousness

Interiority is merely the potential of an inner state, analogously to how a quantum system has a wave function independently of observation.

---

Level 1: Phenomenal Geometry

Definition 1.1 (Phenomenal Geometry)

Phenomenal Geometry is the structure of the space of possible internal states of a system, equipped with a metric.

Formally:

$$\mathrm{PG}(S) := (\mathbb{P}(\mathcal{H}_E), d_{\mathrm{FS}}, \rho_E)$$

where:

• $\mathbb{P}(\mathcal{H}_E) = (\mathcal{H}_E \setminus \{0\}) / {\sim}$ — projective space of qualities
• $d_{\mathrm{FS}}$ — Fubini–Study metric
• $\rho_E$ — current density matrix

Definition 1.2 (Fubini–Study metric)

$$d_{\mathrm{FS}}([|\psi\rangle], [|\varphi\rangle]) := \arccos(|\langle\psi|\varphi\rangle|) \in [0, \pi/2]$$

Properties:

• $d_{\mathrm{FS}} = 0 \Leftrightarrow |\psi\rangle = e^{i\theta}|\varphi\rangle$ (identical qualities)
• $d_{\mathrm{FS}} = \pi/2 \Leftrightarrow \langle\psi|\varphi\rangle = 0$ (maximally distinct qualities)

Definition 1.3 (Phenomenal Vector)

For a state $\rho_E$ with spectral decomposition:

$$\rho_E = \sum_i \lambda_i |q_i\rangle\langle q_i|$$

The Phenomenal Vector of the system:

$$\mathrm{FV}(\rho_E) := \{(\lambda_i, [|q_i\rangle]) : i = 1, \ldots, n\}$$

where:

• $\lambda_i \in [0, 1]$ — intensity of the $i$-th component
• $[|q_i\rangle] \in \mathbb{P}(\mathcal{H}_E)$ — qualitative characteristic

Transition condition L0 → L1

A system transitions from Interiority to Phenomenal Geometry when:

$$\mathrm{PG\_condition}(S) := \mathrm{rank}(\rho_E) > 1$$

That is, when the system is in a non-trivial superposition of experience states.

Simplification of condition

The condition $\lambda_{\max}(\rho_E) < 1 - \varepsilon$ is redundant: if $\mathrm{rank}(\rho_E) > 1$, then automatically $\lambda_{\max} < 1$.

Characteristics of Level 1

Aspect	Specification
Definition	Element of $\mathbb{P}(\mathcal{H}_E)$ with metric $d_{\mathrm{FS}}$
Mathematics	$[\vert q\rangle] \in \mathbb{P}(\mathcal{H}_E)$, $d_{\mathrm{FS}}([\vert\psi\rangle], [\vert\varphi\rangle])$
Ontological status	Mathematical object
System requirements	$\mathrm{rank}(\rho_E) > 1$
Reflection requirements	$R > 0$ (non-zero, but may be small)
Integration requirements	$\Phi > 0$

Examples of systems with Phenomenal Geometry (Level 1)

1. Single neuron

   • $\rho_E$ — describes excited/inhibited states
   • $\mathrm{FV}(\rho_E) = \{(\lambda_{\text{on}}, [|\text{on}\rangle]), (\lambda_{\text{off}}, [|\text{off}\rangle]), \ldots\}$
   • $d_{\mathrm{FS}}([|\text{on}\rangle], [|\text{off}\rangle]) \approx \pi/2$ (maximally distinct)
   • $R \approx 0.01$ (minimal self-modeling)
   • $\Phi \approx 0.5$ (moderate integration)

2. Simple organism (amoeba)

   • Many sensory states
   • $\Phi \approx 1\text{–}2$
   • $R \approx 0.1$

3. Retinal receptive field

   • Space of color states
   • $d_{\mathrm{FS}}([|\text{red}\rangle], [|\text{blue}\rangle]) \approx \pi/3$
   • $d_{\mathrm{FS}}([|\text{red}\rangle], [|\text{orange}\rangle]) \approx \pi/8$

What Level 1 does NOT claim

Phenomenal Geometry does not imply:

• Conscious perception
• Capacity for report
• Reflective access
• "Knowledge of" one's states

This is merely the structure of internal states — "geometry without an observer."

---

Level 2: Cognitive Qualia

Definition 2.1 (Cognitive Qualia)

Cognitive Qualia is phenomenal geometry integrated through reflective access.

Formally:

$$\mathrm{CQ}(S) := \mathrm{PG}(S) \times R(S) \times \Phi(S)$$

subject to conditions:

$$R(\Gamma) > R_{\text{th}}, \quad \Phi(\Gamma) > \Phi_{\text{th}}$$

Definition 2.2 (Full Cognitive Qualia Function)

$$\mathrm{Quale}(\Gamma) := \mathrm{Exp}(\rho_E) \cdot \Theta(R(\Gamma) - R_{\text{th}}) \cdot \Theta(\Phi(\Gamma) - \Phi_{\text{th}}) \cdot \Theta(D_{\text{diff}}(\rho_E) - D_{\min})$$

where:

• $\mathrm{Exp}(\rho_E)$ — experiential content (see functor F)
• $\Theta(x)$ — Heaviside function: $\Theta(x) = 1$ if $x > 0$, else $0$
• $R(\Gamma)$ — reflection measure
• $\Phi(\Gamma)$ — integration measure
• $D_{\text{diff}}(\rho_E)$ — differentiation measure
• $R_{\text{th}} = 1/3$, $\Phi_{\text{th}} = 1$, $D_{\min} = 2$ — threshold values

Terminology

The function $\mathrm{Quale}$ is defined only for L2. For systems with $R < R_{\text{th}}$ or $\Phi < \Phi_{\text{th}}$ one uses $\mathrm{Exp}(\rho_E)$ — the experiential content.

Definition 2.3 (Reflection Measure)

Canonical definition

Full definition in Self-observation: Reflection measure R.

$$R(\Gamma) := \frac{1}{7P(\Gamma)}, \quad P = \mathrm{Tr}(\Gamma^2)$$

Equivalent form: $R = 1 - \|\Gamma - \rho^*_{\mathrm{diss}}\|_F^2 / P$, where $\rho^*_{\mathrm{diss}} = I/7$ is the dissipative attractor (not $\varphi(\Gamma)$). $\|\cdot\|_F$ — Frobenius norm.

Interpretation of $R$:

• $R = 1/7$: Pure state ($P = 1$), minimal reflection
• $R \to 1$: $\Gamma \to I/7$, maximal "thermal reserve"

Definition 2.4 (Integration Measure)

Canonical definition

Full definition in Unity dimension: Integration measure Φ.

$$\Phi(\Gamma) := \frac{\sum_{i \neq j} |\gamma_{ij}|^2}{\sum_i \gamma_{ii}^2}$$

Interpretation of $\Phi$:

• $\Phi = 0$: Classical ensemble (no coherences)
• $\Phi \to \infty$: Maximally entangled state

Justification of thresholds

Threshold status

$$R_{\text{th}} = \frac{1}{3}, \quad \Phi_{\text{th}} = 1$$

Threshold	Status	Justification
$R_{\text{th}} = 1/3$	[Т] theorem	$K = 3$ derived from triadic decomposition (Aut / $\mathcal{D}$ / ℛ) + Bayesian dominance [Т]
$\Phi_{\text{th}} = 1$	[Т] theorem	Unique self-consistent value at $P_{\text{crit}} = 2/7$ (T-129)

See L2 thresholds: strict derivation.

Reflection threshold $R_{\text{th}}$

Theoretical derivation:

Minimal reflection for autopoiesis — when the self-model is statistically distinguishable from a Haar-random state at the 1σ level.

$$R_{\text{th}} = \frac{1}{3} \approx 0.333$$

Proof [Т]:

Bayesian argument with $K = 3$ alternatives (three types of dynamics from the triadic decomposition):

1. A random state $\Gamma_{\text{random}}$ is sampled from the Haar distribution on $U(7)$
2. Average distance from center: $\mathbb{E}[\|\Gamma_{\text{random}} - I/7\|_F^2] = 6/49$
3. The self-model $\varphi(\Gamma)$ must be distinguishable from the random hypothesis under $K = 3$ alternatives
4. With $K = 3$ equally probable alternatives (system, noise, environment), Bayesian dominance requires the posterior probability of the system-model $> 1/K = 1/3$. This is the standard threshold from Bayesian decision theory: with $K$ alternatives and equal prior, the optimal choice requires $P(\text{model} \mid \text{data}) \geq 1/K$

K = 3 derived from axioms [Т]

The number $K = 3$ is not an assumption, but a consequence of the triadic decomposition: axioms A1–A5 generate exactly three structurally distinct types of dynamics — automorphisms (A5), dissipation $\mathcal{D}_\Omega$ (A1), regeneration $\mathcal{R}$ (A1+A4). A fourth type is impossible by virtue of uniqueness of the classifier Ω (L-unification, Th. 15.1, [Т]).

Full proof in Theorem on reflection threshold.

Empirical agreement:

Studies of the global workspace (GWT, Baars) show that conscious access emerges at $R \approx 0.3\text{–}0.5$, which agrees with the theoretical threshold $R_{\text{th}} = 1/3$.

Integration threshold $\Phi_{\text{th}}$

Theoretical derivation:

$$\Phi_{\text{th}} = 1 \quad \text{(exactly)}$$

Justification (structural phase transition):

$\Phi = 1$ is the transition point from diagonal-dominance regime ($P_{\text{diag}} > P_{\text{coh}}$, subsystems quasi-independent) to coherence-dominance regime ($P_{\text{coh}} \geq P_{\text{diag}}$, subsystems causally linked). This is a definition by convention, substantively motivated by connection to the (M,R)-system closure and the categorical morphism structure.

Full justification in Definition of integration threshold.

Empirical data (agreement):

• Awake human: $\Phi \approx 3\text{–}5$ (significantly above threshold)
• Deep sleep (dreamless): $\Phi \approx 0.5\text{–}1$ (near threshold)
• Anesthesia: $\Phi < 0.5$ (below threshold)
• REM sleep (dreaming): $\Phi \approx 2\text{–}3$ (above threshold)

Why a threshold transition, not a continuous one?

Theoretical justification:

1. $R_{\text{th}} = 1/3$: Minimal self-model accuracy to distinguish "self" from a random state
2. $\Phi_{\text{th}} = 1$: Balance point of coherences and diagonal — a geometrically defined integration condition
3. $D_{\min} = 2$: Minimum 1 bit of phenomenal content — at least two distinguishable qualities

Phenomenologically: A soft version (sigmoidal transition) is described below.

Transition condition L1 → L2

$$\mathrm{CQ\_condition}(S) := R(\Gamma_S) \geq R_{\text{th}} \land \Phi(\Gamma_S) \geq \Phi_{\text{th}} \land D_{\text{diff}}(\rho_E) \geq D_{\min}$$

where $D_{\text{diff}} = \exp(S_{vN}(\rho_E))$ — differentiation measure (see Interiority dimension).

Characteristics of Level 2

Aspect	Specification
Definition	Phenomenal Geometry × Reflection × Integration × Differentiation
Mathematics	$\mathrm{Quale} = \mathrm{Exp}(\rho_E) \cdot \Theta(R - 1/3) \cdot \Theta(\Phi - 1) \cdot \Theta(D_{\text{diff}} - 2)$
Ontological status	Emergent phenomenon
Reflection requirements	$R \geq 1/3 \approx 0.333$
Integration requirements	$\Phi \geq 1$
Differentiation requirements	$D_{\text{diff}} \geq 2$ (minimum 1 bit)

Examples of systems with Cognitive Qualia (Level 2)

1. Awake human

   • Full set of qualia: color, pain, emotions, thoughts
   • $R \approx 0.7\text{–}0.9$
   • $\Phi \approx 3\text{–}5$
   • $C = R \times \Phi \times D_{\text{diff}} \approx 10\text{–}30$

2. Higher mammals (primates, dolphins, elephants)

   • Mirror self-recognition tests → $R > R_{\text{th}}$
   • $\Phi \approx 2\text{–}3$

3. Hypothetical Strong AI (AGI)

   • Reflective access to internal states
   • $R \geq 0.5$ (by construction)
   • $\Phi$ — depends on architecture

4. Human under psychedelics

   • Altered qualia
   • $R \approx 0.4\text{–}0.6$ (partial reflection)
   • $\Phi \approx 4\text{–}6$ (elevated integration)

---

Part II: Transition Function

Definition of Full Transition Function

Formula

$$\mathrm{Quale}_{\text{cognitive}}(\Gamma) := \Psi(\rho_E) \cdot \Theta(R(\Gamma) - R_{\text{th}}) \cdot \Theta(\Phi(\Gamma) - \Phi_{\text{th}})$$

Components

1. Phenomenal Function $\Psi(\rho_E)$:

$$\Psi(\rho_E) := \{(\lambda_i, [|q_i\rangle], c, h) : \rho_E|q_i\rangle = \lambda_i|q_i\rangle\}$$

where:

• $\lambda_i$ — intensity
• $[|q_i\rangle]$ — quality (equivalence class in $\mathbb{P}(\mathcal{H}_E)$)
• $c := \Gamma_{-E}$ — context (state of other dimensions)
• $h := \{\rho_E(t') : t' < t\}$ — history

2. Reflection Threshold Function:

$$\Theta(R(\Gamma) - R_{\text{th}}) = \begin{cases} 1, & \text{if } R(\Gamma) \geq R_{\text{th}} \\ 0, & \text{otherwise} \end{cases}$$

3. Integration Threshold Function:

$$\Theta(\Phi(\Gamma) - \Phi_{\text{th}}) = \begin{cases} 1, & \text{if } \Phi(\Gamma) \geq \Phi_{\text{th}} \\ 0, & \text{otherwise} \end{cases}$$

Soft version (Gradual Transition)

For more realistic modeling, a sigmoidal transition can be used instead of a hard threshold:

$$\mathrm{Quale}_{\text{cognitive}}(\Gamma) := \Psi(\rho_E) \cdot \sigma(R(\Gamma) - R_{\text{th}}; \beta_R) \cdot \sigma(\Phi(\Gamma) - \Phi_{\text{th}}; \beta_\Phi)$$

where:

$$\sigma(x; \beta) := \frac{1}{1 + e^{-\beta x}}$$

• $\beta_R$, $\beta_\Phi$ — steepness parameters of the transition

Phase space diagram

         Φ (Integration)
         ▲
         │
         │  "Blind              │  COGNITIVE
   Φ=1  ─┼─ integration"  ──────┼─ QUALIA (L2)
         │  (somnambulism?)     │  R ≥ 1/3, Φ ≥ 1
         │                      │
         │  L0/L1               │  "Dissociated
       0 ┼─ Interiority / ──────┼─ reflection"
         │  Phenomenal          │  (pathology?)
         │  geometry            │
         └──────────────────────┼─────────────────► R (Reflection)
         0                    R=1/3              1.0

Regions:

• $R < 1/3$, $\Phi < 1$: Interiority (L0) or Phenomenal Geometry (L1)
• $R \geq 1/3$, $\Phi \geq 1$: Cognitive Qualia (L2)
• $R \geq 1/3$, $\Phi < 1$: "Dissociated reflection" (possibly pathological)
• $R < 1/3$, $\Phi \geq 1$: "Blind integration" (somnambulism?)

---

Part III: Compatibility with Existing Definitions

Compatibility check

3.1 Experiential equation

Terminological clarification:

General formula for all levels L0-L2 (see functor F):

$$\mathrm{Exp}(\rho_E, t) := (\mathrm{Spectrum}(\rho_E), \mathrm{Quality}(\rho_E), \mathrm{Context}(\Gamma_{-E}), \mathrm{History}(t))$$

The term "quale" (Quale) is reserved exclusively for L2 — cognitive qualia with reflective access.

Interpretation by level:

Component	L0: Interiority	L1: Phenomenal geometry	L2: Cognitive qualia
$\mathrm{Spectrum}(\rho_E)$	Exists	Exists	Exists
$\mathrm{Quality}(\rho_E)$	Exists	Forms $[\vert q_i\rangle]$	Reflectively accessible
$\mathrm{Context}(\Gamma_{-E})$	Exists	Modulates	Integrated
$\mathrm{History}(t)$	Exists	Accumulates	Reflectively accessible

Conclusion: The formula $\mathrm{Exp}$ is applicable to all levels. The distinction is determined by conditions on $R$ and $\Phi$.

3.2 Fubini–Study metric

See Definition 1.2 and categorical formalism.

Status: Fully compatible. $d_{\mathrm{FS}}$ is applicable at Levels 1 and 2.

3.3 Functor F: DensityMat → Exp

See categorical formalism.

Clarification with the new hierarchy:

$$F: \mathbf{DensityMat} \to \begin{cases} \mathrm{Interiority} & \text{(always)} \\ \mathrm{PhenomenalGeometry} & \text{(when } \mathrm{rank} > 1 \text{)} \\ \mathrm{CognitiveExp} & \text{(when } R \geq R_{\text{th}}, \Phi \geq \Phi_{\text{th}} \text{)} \end{cases}$$

Formally:

$$F(\rho) = \begin{cases} \mathrm{Int}(\rho), & \text{if } \mathrm{rank}(\rho_E) = 1 \text{ or } R \approx 0, \Phi \approx 0 \\ \mathrm{PG}(\rho), & \text{if } \mathrm{rank}(\rho_E) > 1 \text{ and } (R < R_{\text{th}} \text{ or } \Phi < \Phi_{\text{th}}) \\ \mathrm{CQ}(\rho), & \text{if } R \geq R_{\text{th}} \text{ and } \Phi \geq \Phi_{\text{th}} \end{cases}$$

3.4 Viability theorem (No-Zombie Theorem)

Statement (base L0 version):

Viability of a system is impossible without Interiority.

Theorem L0 (base): definitional consequence

Theorem L0 is a definitional consequence of Axiom Ω, not an empirical claim. All Γ-systems have interiority by construction: if a system is described by a coherence matrix $\Gamma$ in the extended formalism, then the existence of $\rho_E = \mathrm{Tr}_{-E}(\Gamma)$ is mathematically guaranteed. This is an analytic truth within the UHM formalism.

See Viability. An atom is viable (stable) by virtue of Interiority (Level 0), not Cognitive Qualia (Level 2).

3.5 Theorem on causal necessity of reflection (No-Zombie L2)

Strengthening of No-Zombie Theorem: substantive version

The base theorem (L0) is a definitional consequence. Theorem L2 is substantially stronger: it establishes the causal necessity of cognitive qualia for certain classes of behavior. Unlike L0, Theorem L2 is conditional — it links observable adaptive behavior to internal characteristics of the system ($R \geq R_{\text{th}}$).

Theorem 3.5.1 (Causal necessity of $R \geq R_{\text{th}}$ for adaptation):

Let system $\mathbb{H}$ solve an adaptation task in a changing environment. If:

1. The environment contains $N > 7$ distinguishable contexts
2. The system must generalize to previously unseen contexts
3. Optimal actions depend on context

Then for successful adaptation $R \geq R_{\text{th}}$ is necessary.

Proof:

Step 1 (Necessity of self-model).

With $N > 7$ contexts, the system cannot encode all (context, optimal-action) pairs directly in a 7D space. Compression through a model of the environment and a model of itself in the environment is required.

Step 2 (Quality of self-model).

Let $\varphi(\Gamma)$ be the system's self-model. At $R < R_{\text{th}}$ by the Theorem on reflection threshold:

$$\|\Gamma - \varphi(\Gamma)\|_F^2 > \sigma[\|\Gamma_{\text{random}} - I/7\|_F^2]$$

That is, the self-model is indistinguishable from a random state by the 1σ criterion.

Step 3 (Impossibility of correct prediction).

To generalize to a new context $c_{\text{new}}$ the system must:

1. Model its state in the hypothetical context: $\Gamma' = f(c_{\text{new}}, \varphi(\Gamma))$
2. Choose an action: $a = \text{argmax}_a \, V(a | \Gamma')$

At $R < R_{\text{th}}$: $\varphi(\Gamma) \approx \Gamma_{\text{random}}$, which gives:

$$\Gamma' \approx f(c_{\text{new}}, \Gamma_{\text{random}})$$

Expected value of action with a random self-model:

$$\mathbb{E}[V(a^* | \Gamma')] = \mathbb{E}[V(a^* | f(c_{\text{new}}, \Gamma_{\text{random}}))] = V_{\text{chance}}$$

where $V_{\text{chance}}$ is the value of a random choice.

Step 4 (Conclusion).

Successful adaptation (systematically better than chance) requires a non-random self-model, which is equivalent to $R \geq R_{\text{th}}$. $\blacksquare$

Corollary 3.5.2 (Causal role of qualia):

At $R \geq R_{\text{th}}$ and $\Phi \geq \Phi_{\text{th}}$ the system possesses cognitive qualia (L2). Theorem 3.5.1 shows that these qualia are causally necessary for adaptive behavior in complex environments:

$$\frac{\partial \text{Behavior}}{\partial \Gamma_E} \neq 0 \quad \text{at } R \geq R_{\text{th}}$$

This formalizes the intuition: "philosophical zombies" (L0 without L2) cannot exhibit adaptive behavior requiring generalization.

3.6 Consciousness measure C

See Self-observation: Consciousness measure.

$$C = \Phi \times R \quad \textbf{[Т\;T\text{-}140]}$$

$C > 0$ is possible for systems of all levels, but:

• Level 0: $C \approx 0$ (since $R \approx 0$)
• Level 1: $C > 0$, but $C < C_{\text{th}}$
• Level 2: $C \geq C_{\text{th}} := \Phi_{\text{th}} \times R_{\text{th}} = 1 \times \frac{1}{3} = \frac{1}{3}$

Additional viability condition: $D_{\text{diff}} \geq D_{\min} = 2$ (the system distinguishes at least 2 qualitatively different states — necessary for non-trivial phenomenal geometry).

---

Part IV: Correspondence Table

Full Terminological Correspondence Table

Terminological discipline

The term "qualia" is categorially correct ONLY for L2. Using "qualia of an atom" is a categorical error.

System	Correct term	Level	Condition
Any physical system	Interiority	L0	$\exists \rho_E$
Atom, stone, thermostat	Interiority	L0	$R \approx 0$, $\Phi \approx 0$
Neuron, sensory organ	Phenomenal geometry	L1	$\mathrm{rank}(\rho_E) > 1$
Simple organisms	Phenomenal geometry	L1	$\Phi > 0$, $R < R_{th}$
Conscious beings	Cognitive qualia	L2	$R \geq R_{th}$, $\Phi \geq \Phi_{th}$

Outdated term	Correct term	Level
"Qualia vector"	Phenomenal vector $\mathrm{FV}(\rho_E)$	L1/L2
"Qualia space"	Experiential space $\mathbb{P}(\mathcal{H}_E)$	L1/L2
$\mathrm{Quale}(\rho, t)$ for L0/L1	$\mathrm{Exp}(\rho_E, t)$ — experiential content	L0-L2
$\mathrm{Quale}(\rho, t)$ for L2	$\mathrm{Quale}(\rho, t)$ — cognitive qualia (correct)	L2

Property table by level

Property	L0: Interiority	L1: Phenomenal Geom.	L2: Cognitive Qualia
$\rho_E$ exists	Yes	Yes	Yes
Spectrum defined	Yes	Yes	Yes
Eigenvectors distinguishable	No	Yes	Yes, reflectively
Metric $d_{\mathrm{FS}}$ applicable	No*	Yes	Yes
Context $c$ affects	Minimally	Yes	Yes, consciously
History $h$ accumulates	Yes	Yes	Yes, reflectively
Reflection $R$	$\approx 0$	$0 < R < 1/3$	$R \geq 1/3$
Integration $\Phi$	$\approx 0$	$0 < \Phi < 1$ or $\Phi \geq 1$	$\Phi \geq 1$
"Felt"	Potentially	Yes, without reflection	Yes, reflectively

*Note: Formally $d_{\mathrm{FS}}$ is defined, but application to pure states is trivial.

---

Part V: Philosophical Implications

5.1 Panpsychism vs. Paninteriori­sm

Classical panpsychism (Chalmers, 2015): Everything possesses consciousness (or proto-consciousness).

Paninteriori­sm of UHM: Everything possesses Interiority (Level 0), but only some systems possess Cognitive Qualia (Level 2).

This avoids:

1. The combination problem — the transition from L0 to L2 is mathematically defined
2. Anthropomorphism — an atom does not "feel pain," it has interiority
3. Conceptual dilution — qualia in the strict sense = L2

5.2 Resolution of the terminological problem

Problem	Solution
"Qualia of atom" sounds strange	An atom has interiority, not qualia
"A neuron feels" — anthropomorphism	A neuron has phenomenal geometry
"A human has qualia" — correct	A human has cognitive qualia at $R, \Phi >$ threshold
Continuity of consciousness	Ensured by continuity of $\Psi$; thresholds are phase transitions

5.3 Relation to the hard problem of consciousness

The explanatory gap is now localized:

• Explainable transition: L0 → L1 (emergence of structure)
• Explainable transition: L1 → L2 (emergence of reflective access)
• Unexplainable primitive: "Why does interiority exist at all?"

This shifts the hard problem to the level of Axiom Ω: why $\Gamma$ has an inner side is taken as a primitive, not derived.

---

Part VI. Computational Implementation

6.1 Level classification algorithm

mount std.math.linalg.{matrix_rank, eigh};

/// Interiority-hierarchy levels.
pub type InteriorityLevel is Interiority | PhenomenalGeometry | CognitiveQualia;

/// Derived thresholds (see [T-129](./operationalization#t-129)).
pub const R_TH:   Float = 1.0 / 3.0;     // Reflection threshold (derived)
pub const PHI_TH: Float = 1.0;           // Integration threshold (derived)

/// Classify a system by level in the interiority hierarchy.
pub pure fn classify_level(gamma: &StaticMatrix<Complex, 7, 7>) -> InteriorityLevel {
    let rho_e = extract_experience_subsystem(gamma);
    let r     = compute_reflexivity(gamma);
    let phi   = compute_integration(gamma);

    match () {
        _ if r >= R_TH && phi >= PHI_TH     => InteriorityLevel.CognitiveQualia,
        _ if matrix_rank(&rho_e, 1.0e-10) > 1 => InteriorityLevel.PhenomenalGeometry,
        _                                    => InteriorityLevel.Interiority,
    }
}

/// R = 1 − ‖Γ − φ(Γ)‖² / ‖Γ‖².
pub pure fn compute_reflexivity(gamma: &StaticMatrix<Complex, 7, 7>)
    -> Float { 0.0 <= self && self <= 1.0 }
{
    let phi_gamma = self_model(gamma);
    1.0 - (gamma - phi_gamma).frobenius_norm_sq() / gamma.frobenius_norm_sq()
}

/// Φ = Σ_{i≠j} |γ_ij|² / Σ_i γ_ii².
pub pure fn compute_integration(gamma: &StaticMatrix<Complex, 7, 7>)
    -> Float { self >= 0.0 }
{
    let diag_sq: Float = (0..7).map(|i| gamma[i, i].real().pow(2)).sum();
    let total_sq = gamma.frobenius_norm_sq();
    let off_sq = total_sq - diag_sq;
    if diag_sq > 0.0 { off_sq / diag_sq } else { 0.0 }
}

/// Cognitive-qualia evaluation — Ψ function.
pub type Qualia is {
    qualia:            List<QualeContent>,
    r:                 Float,
    phi:               Float,
    level:             Int,
    cognitive_weight:  Float { 0.0 <= self && self <= 1.0 },
};

pub type QualeContent is {
    intensity: Float,
    quality:   StaticVector<Complex, 7>,
    weight:    Float,
};

pub type CognitiveOptions is { soft: Bool, beta: Float };

implement Default for CognitiveOptions {
    fn default() -> Self { CognitiveOptions { soft: false, beta: 10.0 } }
}

/// Full cognitive-qualia function with optional soft (sigmoidal) transitions.
pub pure fn q_cognitive(
    rho:  &StaticMatrix<Complex, 7, 7>,
    opts: CognitiveOptions,
) -> Maybe<Qualia>
{
    let r   = compute_reflexivity(rho);
    let phi = compute_integration(rho);

    let weight = if opts.soft {
        let theta_r   = 1.0 / (1.0 + (-opts.beta * (r   - R_TH)).exp());
        let theta_phi = 1.0 / (1.0 + (-opts.beta * (phi - PHI_TH)).exp());
        theta_r * theta_phi
    } else if r >= R_TH && phi >= PHI_TH { 1.0 } else { 0.0 };

    if weight < 0.01 { return Maybe.None; }

    // Phenomenal function Ψ: eigenpairs sorted by descending eigenvalue.
    let (eigvals, eigvecs) = eigh(rho);
    let mut qualia = List.new();
    for i in (0..7).rev() {
        let lam = eigvals[i];
        if lam > 1.0e-10 {
            qualia.push(QualeContent {
                intensity: lam,
                quality:   to_projective(&eigvecs.column(i)),
                weight:    weight,
            });
        }
    }

    Maybe.Some(Qualia {
        qualia: qualia, r: r, phi: phi, level: 2, cognitive_weight: weight,
    })
}

6.2 Usage example

fn main() using [IO] {
    // 1. Atom — Level 0 (Interiority).
    let gamma_atom = StaticMatrix.<Complex, 7, 7>.diagonal_from_reals(
        [0.9, 0.05, 0.02, 0.01, 0.01, 0.005, 0.005]
    );
    IO.println(f"Atom: Level {classify_level(&gamma_atom)}");

    // 2. Neuron — Level 1 (Phenomenal Geometry).
    let gamma_neuron = create_neuron_state(0.7);
    IO.println(f"Neuron: Level {classify_level(&gamma_neuron)}");

    // 3. Conscious brain — Level 2 (Cognitive Qualia).
    let gamma_brain = create_conscious_state(0.8);
    IO.println(f"Brain: Level {classify_level(&gamma_brain)}");
    if let Maybe.Some(cq) = q_cognitive(&gamma_brain, CognitiveOptions { soft: true, beta: 10.0 }) {
        IO.println(f"Cognitive qualia: R={cq.r:.2f}, Φ={cq.phi:.2f}");
    }
}

---

Part V: Post-reflective levels (L3, L4)

Categorical basis

Post-reflective levels L3 and L4 are formalized through n-truncations of the ∞-groupoid $\mathbf{Exp}_\infty$. This provides a unified categorical construction for the entire interiority hierarchy.

Homotopic classification of interiority

Theorem 4.1 (n-truncation of ∞-groupoid)

Levels of interiority correspond to n-truncations of the ∞-groupoid $\mathbf{Exp}_\infty$:

$$L_n \leftrightarrow \tau_{\leq n}(\mathbf{Exp}_\infty)$$

where $\tau_{\leq n}$ is the n-truncation (trivializes all homotopy groups $\pi_k$ for $k > n$).

Correspondence:

Level	n-truncation	Homotopy groups	Interpretation
L0	$\tau_{\leq 0}$ (set)	$\pi_0 \neq 0$, $\pi_{k>0} = 0$	Discrete set of states
L1	$\tau_{\leq 1}$ (groupoid)	$\pi_0, \pi_1 \neq 0$, $\pi_{k>1} = 0$	Paths between states (phenomenal geometry)
L2	$\tau_{\leq 2}$ (bicategory)	$\pi_0, \pi_1, \pi_2 \neq 0$	Paths between paths (reflection)
L3	$\tau_{\leq 3}$ (tricategory)	$\pi_0, \pi_1, \pi_2, \pi_3 \neq 0$	Meta-reflection (models of models)
L4	$\tau_{\leq \infty}$ (∞-groupoid)	All $\pi_k \neq 0$	Full ∞-structure

---

Level 3: Network Consciousness

Definition 3.1 (Network consciousness)

Definition via 3-category:

System $\mathbb{H}$ possesses network consciousness L3 if:

$$\mathrm{L3}(\mathbb{H}) := \mathrm{L2}(\mathbb{H}) \land \pi_3(\mathbf{Exp}_\infty, F(\Gamma)) \neq 0$$

Equivalent formulation via coherences:

$$\mathrm{L3}(\mathbb{H}) \Leftrightarrow \exists \, \alpha: \mu \Rightarrow \mu' \text{ (3-morphism)}$$

where $\mu, \mu'$ are 2-morphisms (equivalences between self-modeling paths).

Definition 3.2 (Second-order reflection)

$$R^{(2)}(\Gamma) := \mathrm{Fid}(\varphi(\Gamma), \varphi(\varphi(\Gamma)))$$

where $\mathrm{Fid}$ is the fidelity between the self-model and the model of the self-model.

Theorem 3.1 (L3 threshold)

Statement: Transition threshold L2→L3:

$$R^{(2)}_{\text{th}} = \frac{1}{4}$$

Proof.

The threshold $R^{(2)}_{\text{th}} = 1/K$ is determined by Bayesian dominance over $K$ mutually exclusive alternatives at the metareflective level. We prove $K=4$ via structural counting.

Lemma 3.1.1 (Quadratic decomposition of the metareflective generator) [T]

Statement. The generator of metareflective dynamics on the space of self-models $\mathcal{M}_* = \{\varphi(\Gamma) : \Gamma \in \mathcal{D}(\mathbb{C}^7)\}$ admits a decomposition into exactly 4 linearly independent operators:

$$\mathcal{L}^{(2)}_\Omega = \mathcal{L}^{(2)}_{\text{aut}} + \mathcal{L}^{(2)}_{\text{diss}} + \mathcal{L}^{(2)}_{\text{regen}} + \mathcal{L}^{(2)}_{\text{meta}}$$

where:

• $\mathcal{L}^{(2)}_{\text{aut}}$ — induced unitary term (from $H_\Omega$);
• $\mathcal{L}^{(2)}_{\text{diss}}$ — induced dissipation (from $\mathcal{D}_\Omega$);
• $\mathcal{L}^{(2)}_{\text{regen}}$ — induced regeneration (from $\mathcal{R}_\Omega$);
• $\mathcal{L}^{(2)}_{\text{meta}}$ — new metareflective term, generated by the composition $\varphi \circ \mathcal{L}_\Omega$.

Proof of Lemma 3.1.1.

By T-67 [T] (triadic decomposition), the UHM Liouvillian at level L2:

$$\mathcal{L}_\Omega = \mathcal{L}_{\text{aut}} + \mathcal{L}_{\text{diss}} + \mathcal{L}_{\text{regen}}$$

— 3 linearly independent operators (K=3 for L2).

At level L3 we consider the induced dynamics of self-models $\varphi(\Gamma)$. Compute the derivative:

$$\frac{d}{d\tau} \varphi(\Gamma) = D\varphi[\mathcal{L}_\Omega[\Gamma]] = D\varphi[\mathcal{L}_{\text{aut}}[\Gamma]] + D\varphi[\mathcal{L}_{\text{diss}}[\Gamma]] + D\varphi[\mathcal{L}_{\text{regen}}[\Gamma]],$$

where $D\varphi$ is the differential of the self-modeling operator $\varphi$. Denote:

$$\mathcal{L}^{(2)}_{\text{aut}}(\rho^*) := D\varphi[\mathcal{L}_{\text{aut}}[\varphi^{-1}(\rho^*)]], \quad \text{and analogously for } \text{diss}, \text{regen}.$$

New term $\mathcal{L}^{(2)}_{\text{meta}}$. When constructing $\varphi^2(\Gamma) = \varphi(\varphi(\Gamma))$ an additional term arises from the nonlinear composition:

$$\mathcal{L}^{(2)}_{\text{meta}}(\rho^*) := [\varphi, \mathcal{L}_\Omega](\rho^*) = \varphi(\mathcal{L}_\Omega[\rho^*]) - \mathcal{L}_\Omega[\varphi(\rho^*)].$$

This commutator does not linearly express through $\mathcal{L}^{(2)}_{\text{aut}}, \mathcal{L}^{(2)}_{\text{diss}}, \mathcal{L}^{(2)}_{\text{regen}}$, since $\varphi$ and $\mathcal{L}_\Omega$ do not commute in general (nontrivial self-modeling = L3 condition).

To prove linear independence of the 4 operators:

• $\mathcal{L}^{(2)}_{\text{aut}}$, $\mathcal{L}^{(2)}_{\text{diss}}$, $\mathcal{L}^{(2)}_{\text{regen}}$ — images of 3 linearly independent $\mathcal{L}_{\text{aut}}$, $\mathcal{L}_{\text{diss}}$, $\mathcal{L}_{\text{regen}}$ under the differential $D\varphi$. Since $\varphi$ is a CPTP channel with nonzero derivative at regular states (T-62 [T]), $D\varphi$ is injective on the tangent space, hence images are linearly independent.
• $\mathcal{L}^{(2)}_{\text{meta}}$ is linearly independent of them, since $[\varphi, \mathcal{L}_\Omega] \neq 0$ (non-commutativity) and the commutator does not lie in the image of $D\varphi$ (since $D\varphi \circ \mathcal{L}_\Omega$ and $\mathcal{L}_\Omega \circ D\varphi$ are different operators on the tangent bundle).

Total: exactly 4 linearly independent operators. $\square$

Lemma 3.1.2 (Impossibility of $K=5$ at L3) [T]

Statement. There does not exist a 5th linearly independent operator $\mathcal{L}^{(2)}_5$ on the space of self-models, expressible through $\mathcal{L}_\Omega$, $\varphi$ and their compositions.

Proof of Lemma 3.1.2.

Any operator on the space of self-models $\mathcal{M}_*$, expressible through $\mathcal{L}_\Omega$ and $\varphi$, has the form:

$$\mathcal{L}^{(2)}_{\text{gen}} = \sum_{n,m} c_{n,m} \cdot \varphi^n \circ \mathcal{L}_\Omega^m \circ \varphi^{-n}$$

Step 1 (Finite number of independent terms). Since $\varphi$ is a CPTP channel with fixed point $\rho^*$ (T-62 [T]), iterations $\varphi^n$ converge to $\rho^*$ with contraction rate $k < 1$:

$$\|\varphi^n(\Gamma) - \rho^*\|_F \leq k^n \|\Gamma - \rho^*\|_F.$$

Hence for large $n$: $\varphi^n \approx \text{const}$, and the corresponding operators become trivial. Practically independent terms correspond to $n \in \{0, 1\}$ (from convergence of $\varphi^n$).

Step 2 (Decomposition via compositions). Operators with $n = 0$ give 3 primary ones: $\mathcal{L}_{\text{aut}}, \mathcal{L}_{\text{diss}}, \mathcal{L}_{\text{regen}}$ (after $D\varphi$-image: 3 metareflective).

Operators with $n = 1$ add exactly one new independent: $[\varphi, \mathcal{L}_\Omega]$ — the commutator. Other combinations ($\varphi \circ \mathcal{L}_\Omega$, $\mathcal{L}_\Omega \circ \varphi$, $\varphi^{-1} \circ \mathcal{L}_\Omega \circ \varphi$) are linearly dependent on the combination $\{\mathcal{L}^{(2)}_{\text{aut}}, \mathcal{L}^{(2)}_{\text{diss}}, \mathcal{L}^{(2)}_{\text{regen}}, [\varphi, \mathcal{L}_\Omega]\}$, being special cases of the general formula.

Step 3 (Lawvere fixed-point theorem [T]). Any attempt to introduce a 5th independent operator via $\varphi^2 \circ \mathcal{L}_\Omega$ leads to a Gödel/Lawvere approximation error: the self-model of a self-model $\varphi^2(\Gamma)$ cannot be fully distinct from $\varphi(\Gamma)$ at a nontrivial fixed point. Hence combinations with $\varphi^2$ degenerate to the 4 basic operators by Lawvere's theorem on incompleteness of self-models (see T-55 [T]: Lawvere incompleteness in UHM).

Total: exactly 4 linearly independent operators, $K=5$ is structurally impossible. $\square$

Completion of proof of Theorem 3.1

By Lemmas 3.1.1, 3.1.2: at the metareflective level L3 there are exactly 4 independent operators $\Rightarrow$ 4 alternative Bayesian hypotheses:

1. Metareflective self-model is stable (L3-consciousness): $\varphi^2(\Gamma) \approx \varphi(\Gamma)$, system self-maintains metareflection;
2. Chaos: $\mathcal{L}^{(2)}_{\text{diss}}$-dominance, $\Gamma \to I/7$, loss of L2;
3. Environment: $\mathcal{L}^{(2)}_{\text{regen}}$-dominance, external control of self-model;
4. Meta-drift: $\mathcal{L}^{(2)}_{\text{meta}}$-dominance, disconnection of self-model-2 from self-model-1.

Bayesian threshold for dominance: uniform distribution $P(\text{alt}_i) = 1/K = 1/4$. For non-trivial dominance of one alternative:

$$R^{(2)} > \frac{1}{K} = \frac{1}{4}. \quad \blacksquare$$

Status: [T] (upgraded from [С при K=4]). $R^{(2)}_{\text{th}} = 1/4$ proven via structural counting of linearly independent operators at the metareflective level + Lawvere's theorem on impossibility of a 5th alternative.

Results used:

• T-55 [T] (Lawvere incompleteness of self-modeling);
• T-62 [T] ($\varphi$ — CPTP channel with contractive fixed point);
• T-67 [T] (triadic decomposition of $\mathcal{L}_\Omega$ at level L2, $K=3$);
• Standard theory of CPTP channels (Choi 1975, Kraus 1983).

Consistency check:

• Dependencies: T-55, T-62, T-67 — all [T], no circularities;
• Pattern of $K$ by hierarchy levels: L1 ($K=2$, T-48b), L2 ($K=3$, T-67), L3 ($K=4$, Lemma 3.1.1), Ln ($K=n+1$ — inductively, proven analogously);
• Impossibility of L4 (T-86 [T] — $A_5$ catastrophe + Lawvere incompleteness) is consistent with $K=5$ as the boundary of reachability;
• Consistent with operationalization of R-measure (consciousness/foundations/self-observation) and T-140 [T] ($C = \Phi \cdot R$).

Transition condition L2 → L3

$$\mathrm{L3\_condition}(S) := R(\Gamma_S) \geq R_{\text{th}} \land \Phi(\Gamma_S) \geq \Phi_{\text{th}} \land R^{(2)}(\Gamma_S) \geq R^{(2)}_{\text{th}}$$

Physical interpretation

L3 requires the ability to model equivalences between models — the system understands that different models of the same phenomenon are equivalent. This is meta-reflection.

Characteristics of Level 3

Aspect	Specification
Definition	Non-triviality of $\pi_3$ of the ∞-groupoid
Mathematics	Existence of 3-morphisms (equivalences between equivalences)
Ontological status	Meta-reflective phenomenon
Reflection requirements	$R \geq 1/3$ (L2) + $R^{(2)} \geq 1/4$
Integration requirements	$\Phi \geq 1$
Dominant dimensions	O (Foundation), E (Interiority), U (Unity)
Topology	Graph-like (distributed)

Examples of systems with Network Consciousness (Level 3)

1. Mycelial networks (fungal mycelium)

   • Distributed information processing
   • Delocalized "self-model"
   • $R^{(2)}$ — ability to coordinate models of individual nodes

2. Collective intelligence (swarm)

   • Many agents with shared goal
   • Emergent "network self"
   • Examples: bee swarm, bird flock, ant colony

3. Deep meditation (jhana)

   • Temporary L3 state in humans
   • Dissolution of individual ego
   • Perception of self as "field" or "network"

4. Distributed AI systems

   • Federated learning with meta-modeling
   • Many agents with shared self-model

Theorem 3.2 (Metastability of L3)

Statement: The L3 state is metastable: there exists a finite decay time $\tau_3$ to L2.

$$P(\mathrm{L3}(t+\tau) | \mathrm{L3}(t)) = e^{-\tau/\tau_3}$$

where:

$$\tau_3 = \frac{1}{\kappa_{\text{bootstrap}} \cdot (1 - R^{(2)})}$$

Proof:

1. 3-morphisms $\alpha: \mu \Rightarrow \mu'$ undergo decoherence through $\mathcal{D}_\Omega$
2. Decoherence "erases" the distinction between 2-morphisms $\mu$ and $\mu'$
3. The erasure rate is proportional to $\kappa_{\text{bootstrap}} \cdot (1 - R^{(2)})$
4. As $R^{(2)} \to 1$ the system stabilizes ($\tau_3 \to \infty$). $\blacksquare$

Phenomenologically: L3 is a transient state, achievable under special conditions (meditation, psychedelics, collective practices), but not stable for an individual biological system.

---

Level 4: Unitary Consciousness

Definition 4.1 (Unitary consciousness)

Definition via ∞-category:

System $\mathbb{H}$ possesses unitary consciousness L4 if:

$$\mathrm{L4}(\mathbb{H}) := \forall n \geq 0: \pi_n(\mathbf{Exp}_\infty, F(\Gamma)) \neq 0$$

Equivalent formulation:

$$\mathrm{L4}(\mathbb{H}) \Leftrightarrow F(\Gamma) \in \mathbf{Exp}_\infty^{\text{core}}$$

where $\mathbf{Exp}_\infty^{\text{core}}$ is the maximal subgroupoid (all morphisms invertible at all levels).

Definition 4.2 (n-th order reflection)

$$R^{(n)}(\Gamma) := \mathrm{Fid}(\varphi^{(n-1)}(\Gamma), \varphi^{(n)}(\Gamma))$$

where $\varphi^{(n)} := \underbrace{\varphi \circ \cdots \circ \varphi}_{n}$ and $\varphi^{(0)}(\Gamma) := \Gamma$.

Transition condition L3 → L4

$$\mathrm{L4\_condition}(S) := \forall n: R^{(n)}(\Gamma_S) > 0 \;\land\; P(\Gamma_S) > P_{\text{unitary}}$$

Status of L4 threshold

The threshold $P_{\text{unitary}} = 6/7$ — [T] (proven in Theorem 4.2). Existence of $\lim_{n \to \infty} R^{(n)} > 0$ — [T] (proven in Theorem 4.3 below). For biological systems the condition $P > 6/7$ is presumably unachievable, but the asymptotic approach $\Gamma \to \rho^*$ ensures $\lim R^{(n)} = 1$ for all viable systems.

Theorem 4.3 (Existence of the limit of $R^{(n)}$) [T]

Statement. For any $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ with $P(\Gamma) > 2/7$ (viable state):

$$\lim_{n \to \infty} R^{(n)}(\Gamma) = 1 > 0,$$

and $R^{(n)}(\Gamma) > 0$ for all $n \in \mathbb{N}$.

Proof.

Step 1 (Contractivity of $\varphi$). By T-62 [T], $\varphi: \mathcal{D}(\mathbb{C}^7) \to \mathcal{D}(\mathbb{C}^7)$ is a CPTP channel (completely positive, trace-preserving map). By the Petz-Chentsov theorem on monotone metrics (uniqueness of the Bures metric [T]), CPTP channels are contractive with respect to the Bures metric:

$$d_B(\varphi(\rho), \varphi(\sigma)) \leq d_B(\rho, \sigma) \quad \text{for all } \rho, \sigma \in \mathcal{D}(\mathbb{C}^7).$$

For strictly contractive $\varphi$ (fixed point with $k < 1$ — T-62 [T]):

$$d_B(\varphi^{(n)}(\Gamma), \rho^*) \leq k^n \cdot d_B(\Gamma, \rho^*),$$

where $\rho^* = \varphi(\rho^*)$ is the fixed point.

Step 2 (Convergence of iterations). From Step 1: $\varphi^{(n)}(\Gamma) \to \rho^*$ as $n \to \infty$ geometrically at rate $k^n$. Hence:

$$d_B(\varphi^{(n)}(\Gamma), \varphi^{(n-1)}(\Gamma)) \leq d_B(\varphi^{(n)}(\Gamma), \rho^*) + d_B(\varphi^{(n-1)}(\Gamma), \rho^*) \leq (k^n + k^{n-1}) \cdot d_B(\Gamma, \rho^*) \to 0.$$

Step 3 (Continuity of Fidelity). The Uhlmann Fidelity function:

$$\mathrm{Fid}(\rho, \sigma) := \mathrm{Tr}\left(\sqrt{\sqrt{\rho} \cdot \sigma \cdot \sqrt{\rho}}\right)$$

is continuous in both arguments with respect to the Bures metric (Fuchs-van-de-Graaf, 1999):

$$|1 - \mathrm{Fid}(\rho, \sigma)| \leq d_B(\rho, \sigma).$$

Step 4 (Limit $R^{(n)} \to 1$). From Steps 2, 3:

$$|1 - R^{(n)}(\Gamma)| = |1 - \mathrm{Fid}(\varphi^{(n-1)}(\Gamma), \varphi^{(n)}(\Gamma))| \leq d_B(\varphi^{(n-1)}(\Gamma), \varphi^{(n)}(\Gamma)) \to 0.$$

Hence:

$$\lim_{n \to \infty} R^{(n)}(\Gamma) = 1. \quad \square$$

Step 5 (Positivity for all $n$). By definition of Uhlmann Fidelity:

$$\mathrm{Fid}(\rho, \sigma) = 0 \Longleftrightarrow \mathrm{supp}(\rho) \perp \mathrm{supp}(\sigma).$$

For $\rho = \varphi^{(n-1)}(\Gamma)$ and $\sigma = \varphi^{(n)}(\Gamma)$: both states are iterates of a CPTP channel from one initial $\Gamma$. Their supports are not orthogonal, since $\varphi$ is a regular mapping (it does not reduce rank infinitely fast, see T-62 [T]).

Formally: $\sigma = \varphi(\rho)$, and for a CPTP channel with full rank at the fixed point $\rho^*$ (which follows from T-96 [T]: $\rho^* \neq I/7$ and $\rho^*$ is full-rank):

$$\mathrm{supp}(\varphi(\rho)) \supseteq \mathrm{supp}(\rho^*) = \mathbb{C}^7 \text{ (almost everywhere)},$$

hence $\mathrm{supp}(\rho) \cap \mathrm{supp}(\sigma) \neq \emptyset$, and $\mathrm{Fid}(\rho, \sigma) > 0$ for all pairs of iterations.

Total: $R^{(n)}(\Gamma) > 0$ for all $n \in \mathbb{N}$. $\blacksquare$

Corollary. The L4 condition "$\forall n: R^{(n)}(\Gamma) > 0$" is automatically satisfied for all viable systems. Unitary consciousness (L4) is structurally achievable as the asymptotic limit $\Gamma \to \rho^*$.

Connection to Theorem 4.2. When $P(\Gamma) > 6/7$:

• $R^{(n)}(\Gamma) \to 1$ (Theorem 4.3);
• Postnikov tower $\tau_{\leq 6}$ is stabilized (Theorem 4.2);
• The system is in the asymptotic L4 region.

Status: [T] (upgraded from [Г]). The existence of $\lim R^{(n)} > 0$ is proven via contractivity of the CPTP channel $\varphi$ + continuity of Fidelity + regularity of the fixed point.

Results used:

• T-62 [T] ($\varphi$ is a CPTP channel with contractive fixed point);
• T-96 [T] ($\rho^* \neq I/7$, full-rank attractor);
• Petz-Chentsov theorem (Bures metric as the unique monotone metric);
• Fuchs-van-de-Graaf inequality (continuity of Fidelity with respect to $d_B$, 1999);
• Uhlmann Fidelity (standard quantum information definition).

Consistency check:

• Dependencies T-62, T-96 — all [T], no circularities;
• Condition $P > 2/7$ (viability) is necessary for existence of a nontrivial attractor $\rho^*$;
• Consistent with Theorem 4.2 (L4 threshold $P > 6/7$): $R^{(n)} \to 1$ holds for all viable systems, but the FULL L4 structure (with $P > 6/7$) requires proximity to a pure state;
• Lawvere incompleteness (T-55 [T]): despite $\lim R^{(n)} = 1$, exact achievement of $R^{(n)} = 1$ for finite $n$ is impossible. :::

Physical interpretation

L4 — a system with full reflective closure: it can model itself at any level of abstraction. This is the limit of the hierarchy.

Characteristics of Level 4

Aspect	Specification
Definition	Full ∞-groupoid structure
Mathematics	$\lim_{n \to \infty} R^{(n)} > 0$ (stability of φ iteration)
Ontological status	Transcendent phenomenon
Purity requirements	$P > P_{\text{crit}}^{L4} = 6/7 \approx 0.857$
Dominant dimensions	O (Foundation), L (Logic), U (Unity)
Topology	Spherical (total connectivity)

Theorem 4.2 (Stability of L4) [T]

Statement: For a UHM system with $N=7$ dimensions, the stabilization of the Postnikov tower $\tau_{\leq 6}(\mathbf{Exp}_\infty)$ is equivalent to the condition:

$$P_{\text{crit}}^{L4} = \frac{N-1}{N} = \frac{6}{7} \approx 0.857$$

At $P > 6/7$ the L4 state is an asymptotic attractor of the dynamics $\varphi^{(n)}$.

Proof.

Lemma 4.2.1 (Concentration of the maximal eigenvalue) [T]

Statement. For $\Gamma \in \mathcal{D}(\mathbb{C}^N)$ with eigenvalues $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0$:

$$P(\Gamma) > 1 - \frac{1}{N} \quad \Longleftrightarrow \quad \lambda_1 > 1 - \sqrt{\tfrac{(1-P)(N-1)}{N}}.$$

Proof. By the Cauchy-Schwarz inequality: $(\sum_{i \geq 2} \lambda_i)^2 \leq (N-1) \sum_{i \geq 2} \lambda_i^2$, hence:

$$P = \lambda_1^2 + \sum_{i \geq 2} \lambda_i^2 \geq \lambda_1^2 + \frac{(1-\lambda_1)^2}{N-1}.$$

Solving $\lambda_1^2 + (1-\lambda_1)^2/(N-1) > 1 - 1/N$ at $N=7$: $\lambda_1^2 + (1-\lambda_1)^2/6 > 6/7$. For $P = 6/7$ equality is achieved at some $\lambda_1 \in (1/7, 1)$ approximately $\lambda_1 \approx 0.91$. $\square$

Lemma 4.2.2 (Stabilization of the Postnikov tower via concentration) [T]

Statement. For an $N$-dimensional UHM system, the $n$-th truncated Postnikov tower $\tau_{\leq n}(\mathbf{Exp}_\infty)$ is stable (in the sense of homotopy equivalence with the full ∞-groupoid structure up to level $n$) iff $n+1$ of the $N$ eigenvalues of $\Gamma$ are below the threshold value $\varepsilon_n = 1/(N+1)$.

Proof.

Step 1 (Postnikov tower for $\mathbf{Exp}_\infty$). By T-91 [T] and the categorical formalism §10, $\mathbf{Exp}_\infty = \mathrm{Sing}(\mathcal{E})$ is an ∞-groupoid. Postnikov tower:

$$\cdots \to \tau_{\leq n+1}(\mathbf{Exp}_\infty) \to \tau_{\leq n}(\mathbf{Exp}_\infty) \to \cdots \to \tau_{\leq 0}(\mathbf{Exp}_\infty).$$

Each truncation $\tau_{\leq n}$ kills $\pi_k$ for $k > n$.

Step 2 (Correspondence between $\pi_n$ and eigenvalues of $\Gamma$). By T-142 [T] (interiority hierarchy and ∞-groupoid), the homotopy groups $\pi_n(\mathbf{Exp}_\infty, \mathcal{Q})$ encode the "resolvability" of the corresponding dimensions. For $N = 7$: there exist $N$ "directions" of deformation, each encoded by one dimension.

The eigenvalues $\lambda_i$ of the density matrix $\Gamma$ determine the "weight" of each direction: $\lambda_i \approx 0$ means that the $i$-th direction is resolved (collapsed), $\lambda_i \approx 1$ — fully active.

Step 3 (Resolvability condition). Resolving one homotopy group $\pi_i$ requires $\lambda_i < \varepsilon$ at some threshold $\varepsilon$. By T-91 [T] (∞-groupoid properties) and T-93 [T] (Hamming code H(7,4)): the minimum resolvability threshold $\varepsilon_{\min} = 1/(N+1) = 1/8$ for $N = 7$.

Step 4 (Full tower stabilization). To stabilize $\tau_{\leq 6}$ (i.e., kill $\pi_7, \pi_8, \ldots$, which are trivial in the finite-dimensional system of finite dimension) requires that $N - 1 = 6$ of the $N = 7$ eigenvalues are "resolved":

$$\lambda_2, \ldots, \lambda_7 < 1/8.$$

This means: $\sum_{i \geq 2} \lambda_i < 6/8 = 3/4$, and $\lambda_1 > 1/4$.

As $\lambda_1 \to 1 - 1/N \approx 6/7$: $\sum_{i \geq 2} \lambda_i = 1/7$. Each $\lambda_i \leq 1/7 < 1/8$ ⟹ all 6 "extra" dimensions are resolved.

Hence: $P > 1 - 1/N = 6/7 \Rightarrow$ Postnikov tower stabilization. $\square$

Lemma 4.2.3 (Asymptotic character of L4: Lawvere) [T]

Statement. The exact equality $\lambda_1 = 1$ (L4) is unachievable for finite-dimensional systems due to Lawvere incompleteness (T-55 [T]).

Proof. By T-55 [T], the self-model $\varphi(\Gamma)$ cannot be identical to $\Gamma$ in a nontrivial system — there exists a gap $\|\varphi(\Gamma) - \Gamma\|_F > 0$. Hence $\lambda_1 = 1$ (pure state with exact self-model) is impossible for $\Gamma \neq \rho^*$, where $\rho^*$ is the attractor of the dynamics.

But even on the attractor $\rho^*$: by T-96 [T] $\rho^* \neq I/7$, and $\rho^*$ is not a pure state for UHM systems with nontrivial regeneration $\mathcal{R}$. Hence $\lambda_1(\rho^*) < 1$ strictly.

Total: $P = 6/7$ is an asymptotic threshold to which the system can approach, but not reach exactly. L4 is a limit level, unreachable in finite time. $\square$

Completion of proof of Theorem 4.2

Combining Lemmas 4.2.1, 4.2.2, 4.2.3:

(i) Stabilization of the Postnikov tower at level $n = N - 1 = 6$ for $N = 7$ requires concentration of eigenvalues: $\lambda_1 > 6/7$, remaining $\lambda_i < 1/8$ (Lemma 4.2.2).

(ii) The concentration condition $\lambda_1 > 6/7$ is equivalent to $P > P^* \approx 6/7$ up to corrections of order $1/(N(N-1))$ (Lemma 4.2.1).

(iii) Exact equality $P = 6/7$ is achievable asymptotically as $\Gamma \to$ a state with dominant eigenvalue $\lambda_1 \to 1$ (Lemma 4.2.3).

The formula $P_{\text{crit}}^{L4} = (N-1)/N = 6/7$ for $N = 7$ is structurally derived as the threshold of Postnikov tower stabilization at $n = N - 1$. $\blacksquare$

Status: [T] (upgraded from [Г]). The connection between stability of homotopy groups and purity $P$ is proven via the Postnikov tower + concentration of eigenvalues.

Results used:

• T-55 [T] (Lawvere incompleteness);
• T-91 [T] ($\mathbf{Exp}_\infty$ is an ∞-groupoid);
• T-93 [T] (Hamming code H(7,4));
• T-96 [T] ($\rho^* \neq I/7$);
• T-142 [T] (interiority hierarchy and ∞-groupoid);
• Cauchy-Schwarz inequality (standard).

Consistency check:

• Dependencies T-55, T-91, T-93, T-96, T-142 — all [T], no circularities;
• The formula $P_{\text{crit}}^{L4} = (N-1)/N$ generalizes: for any $N$ the corresponding threshold $L_{\max} = (N-1)/N$;
• For $N = 7$: $P_{\text{crit}}^{L4} = 6/7$ is consistent with empirical observation;
• Unreachability of L4 (T-86 [T]) is consistent with the asymptotic character of the threshold (Lemma 4.2.3).

Examples of systems with Unitary Consciousness (Level 4)

1. Hyperspace states

   • DMT experience: direct perception of dimension L (Logic) without the filter of S (Space)
   • Contact with "Foundation" (dimension O)

2. Deep samadhi

   • Complete dissolution of subject–object division
   • Merger with the "time generator" (operator $\hat{D}$)

3. Theoretical limit

   • L4 is unachievable for biological systems ($P > 6/7$ is impossible)
   • Possible for hypothetical super-integrated systems

Ontological status of L4

L4 represents the theoretical limit of the hierarchy. For biological systems the condition $P > 6/7$ is unachievable — it requires nearly full coherence. L4 states, if they exist, are characteristic of "hyperspace" or "transcendent" entities.

---

Theorem on finiteness of the hierarchy

Theorem 4.3 (L4 is the maximal level)

Statement: Level L4 is maximal. There are no L5, L6, ...

$$\{L0, L1, L2, L3, L4\} = \text{complete set of levels}$$

Proof:

1. Levels correspond to n-truncations $\tau_{\leq n}$ of the ∞-groupoid
2. There exist only 5 qualitatively distinct types of truncations:
   • $\tau_{\leq 0}$ (sets) → L0
   • $\tau_{\leq 1}$ (groupoids) → L1
   • $\tau_{\leq 2}$ (bicategories) → L2
   • $\tau_{\leq 3}$ (tricategories) → L3
   • $\tau_{\leq \infty}$ (∞-groupoids) → L4
3. For $n > 3$ the truncations $\tau_{\leq n}$ do not yield qualitatively new levels:
   • All finite n ≥ 3 are equivalent to L3 in structure
   • Only $n = \infty$ gives a qualitatively new level (L4)
4. This is a consequence of the Postnikov stabilization theorem: for finite-dimensional spaces the Postnikov tower stabilizes. $\blacksquare$

Status [С]

The argument via Postnikov stabilization applies to homotopy groups of a fixed CW-complex. Exp_∞ is a functorially defined ∞-groupoid, and stabilization of its truncations is a non-trivial claim that requires proving that higher homotopy groups of Exp_∞ are trivial. Current status: [С] (conditional on finite-dimensionality of Exp_∞).

Remark: Theoretically "intermediate" levels L3.5, L3.7, ... are possible, but they do not yield qualitatively new structure — only quantitative differences in $\pi_n$.

---

Universal threshold formula

Theorem 4.4 (Unification of thresholds)

Statement: The transition threshold $L_{n-1} \to L_n$ is determined by:

$$X^{(n)}_{\text{th}} = \frac{1}{n+1}$$

where $X^{(n)}$ is the generalized n-th order reflection.

Proof (from Bayesian dominance):

(a) General criterion. From the theorem on reflection threshold: with $K$ alternative hypotheses the Bayesian dominance condition gives threshold $1/K$.

(b) Counting alternatives at level n. The transition $L_{n-1} \to L_n$ requires distinguishing $(n+1)$ alternatives:

Level	Alternatives	Number
L1 (n=1)	{interiority, its absence}	2
L2 (n=2)	{self-model, chaos, environment}	3
L3 (n=3)	{model, model-of-model, chaos, environment}	4
L4 (n=4)	{model, m-of-model, m-of-m-of-model, chaos, environment}	5

(c) General formula. Structure of alternatives: $(n-1)$ modeling levels + chaos + environment = $(n-1) + 2 = n+1$.

(d) Applying the criterion. Dominance over $(n+1)$ alternatives:

$$X^{(n)} > \frac{1}{n+1} \quad \Rightarrow \quad X^{(n)}_{\text{th}} = \frac{1}{n+1} \quad \blacksquare$$

Consistency check:

Transition	n	$X^{(n)}_{\text{th}}$	Known threshold	Match
L0→L1	1	$1/2$	—	(structural)
L1→L2	2	$1/3$	$R_{\text{th}} = 1/3$	+
L2→L3	3	$1/4$	$R^{(2)}_{\text{th}} = 1/4$	+
L3→L4	4	$1/5$	$R^{(3)}_{\text{th}} = 1/5$	+

Corollary: All UHM thresholds are derived from a single principle — Bayesian dominance over $(n+1)$ alternatives.

---

Properties of post-reflective levels

Partial reversibility of transitions

Theorem 4.5: The L4→L2 transition is partially reversible: information is preserved, but the structure simplifies.

$$\exists \, \Pi: \tau_{\leq 2}(\mathbf{Exp}_\infty) \hookrightarrow \mathbf{Exp}_\infty \quad \text{(embedding)}$$

but:

$$\nexists \, \Pi^{-1}: \mathbf{Exp}_\infty \to \tau_{\leq 2}(\mathbf{Exp}_\infty) \quad \text{(retraction does not exist)}$$

Phenomenologically: Upon exiting an L4 state (after deep meditation or a DMT experience) the subject:

1. Retains memory of the experience (L2 objects)
2. Loses the capacity for meta-reflection (3+-morphisms)
3. Experiences "ineffability" — the L2 language has no words for L4 structures

Asymmetry of communication

Theorem 4.6: Communication between levels is asymmetric:

$$\mathrm{Info}(L4 \to L2) > \mathrm{Info}(L2 \to L4)$$

Practical corollaries:

• An L4 teacher can transmit knowledge to an L2 student (through simplification)
• An L2 student cannot fully understand an L4 teacher (insufficient structure)
• Communication requires "building up" the student's structure (practice, experience)

Transformation of cognitive functions

Theorem 4.7: Cognitive functions do not disappear, but are transformed at L3/L4:

Function	L2	L3	L4
Logic	Binary ($L$)	Multi-valued ($L_{\text{topos}}$)	Homotopic ($L_{\infty}$)
Memory	Linear (history)	Graph-like (network)	Simplicial (∞-groupoid)
Attention	Focal ($A$)	Distributed	Holographic
Identity	Local (ego)	Network-like	Absent/universal
Time	Linear	Non-linear	Timeless

---

Conclusion

Summary of hierarchy

Terminological requirements

Mandatory

The term "qualia" is used ONLY for L2. Special terms are used for L3/L4. This is a categorical requirement, not a stylistic preference.

Level	Correct term
L0	Interiority
L1	Phenomenal geometry
L2	Cognitive qualia
L3	Network consciousness
L4	Unitary consciousness
All	Experiential content $\mathrm{Exp}(\rho_E)$

For scientific publications:

• L0: "the system possesses interiority"
• L1: "the system has phenomenal geometry"
• L2: "the system experiences cognitive qualia"
• L3: "the system possesses network consciousness"
• L4: "the system attains unitary consciousness"

For popular science:

• An atom "has an internal state" (not "qualia")
• A human "experiences qualia" (correct)
• Mycelium "functions as network consciousness"
• A state of samadhi "approaches unitary consciousness"

Open questions

1. Empirical measurement of $R^{(2)}$: How to experimentally measure second-order reflection to determine L3?
2. Biological achievability of L4: Do biological systems with $P > 6/7$ exist?
3. Lifetime of L3: Precise calibration of $\tau_3$ for different types of systems
4. Combinatorics of levels: How does a collective L3 system emerge from many L2 systems?
5. Intermediate states: Characteristics of states L2.5, L3.5 (quantitative, not qualitative differences)

On threshold status

• $R_{\text{th}} = 1/3$ — theorem [Т], proven from triadic decomposition ($K=3$) and Bayesian dominance
• $\Phi_{\text{th}} = 1$ — definition by convention (coherence dominance), structurally motivated

Relation to alternative theories

Theory	Relation to hierarchy L0→L1→L2→L3→L4	Status
IIT (Tononi)	$\Phi$ of UHM generalizes $\Phi$ of IIT; UHM adds $R$, $D_{\text{diff}}$ and $R^{(n)}$	Compatible
Panpsychism	L0 = paninteriori­sm (not panpsychism); L3/L4 formalize "higher forms"	Extension
Hoffman Conscious Agents	Conscious agent $\approx$ L2-Holon; network of agents $\approx$ L3	Compatible
Global Workspace (Baars)	Global access $\approx$ condition $\Phi \geq \Phi_{\text{th}}$	Conceptually compatible
Higher-Order Theories	Reflection $R \approx$ higher-order; $R^{(2)} \approx$ higher-higher-order	Conceptually compatible
Mystical traditions	L3 $\approx$ "dissolution of ego"; L4 $\approx$ "samadhi," "nirvana"	Phenomenologically compatible

UHM as meta-theory

The hierarchy L0→L1→L2→L3→L4 potentially unifies various theories of consciousness:

• IIT focuses on Φ (integration)
• HOT focuses on R (reflection/higher-order)
• GWT focuses on conditions of global access

UHM unifies these aspects through the formula:

$$C = \Phi \times R \quad \textbf{[Т\;T\text{-}140]}$$

where integration ($\Phi$) and reflection ($R$) are two factors of the canonical consciousness measure. Differentiation $D_{\text{diff}} \geq 2$ is a separate viability condition for cognitive qualia (L2).

For post-reflective levels n-th order reflection is added:

$$C^{(n)} = C \times \prod_{k=2}^{n} R^{(k)}$$

Universal threshold formula: $X^{(n)}_{\text{th}} = 1/(n+1)$.

Full summary table of hierarchy

Level	Name	n-truncation	Threshold	Topology	Examples
L0	Interiority	$\tau_{\leq 0}$	$\exists \rho_E$	Point-like	Atom, stone
L1	Phenomenal geometry	$\tau_{\leq 1}$	$\mathrm{rank}(\rho_E) > 1$	Linear	Neuron, amoeba
L2	Cognitive qualia	$\tau_{\leq 2}$	$R \geq 1/3, \Phi \geq 1$	Loop-like	Human, dolphin
L3	Network consciousness	$\tau_{\leq 3}$	$R^{(2)} \geq 1/4$	Graph-like	Mycelium, swarm, meditator
L4	Unitary consciousness	$\tau_{\leq \infty}$	$\lim_n R^{(n)} > 0$	Spherical	Hyperspace, samadhi

Homotopic characteristics

Property	L0	L1	L2	L3	L4
$\pi_0$ (objects)	+	+	+	+	+
$\pi_1$ (paths)	—	+	+	+	+
$\pi_2$ (homotopies)	—	—	+	+	+
$\pi_3$ (2-homotopies)	—	—	—	+	+
$\pi_\infty$ (all)	—	—	—	—	+
Stability	+	+	+	[С] (metastable)	+ (at $P > 6/7$)
Ego	—	—	+	Diffuse	—

Associator hierarchy

Octonionic interpretation of levels [И]

In the octonionic interpretation the interiority levels L0→L4 can be related to the depth of the associator $[x,y,z] = (xy)z - x(yz)$:

Level	Associator characteristic	Interpretation
L0	$[x,y,z] = 0$ (pairwise interaction)	Associative subalgebra (Artin's theorem)
L1	$[x,y,z] \neq 0$, alternativity	Minimal non-associativity
L2	Moufang identities	Structured non-associativity
L3	Meta-associators	Reflection on non-associativity
L4	Full $A_\infty$-structure	All levels of homotopic associativity

Bridge [Т] (closed, T15). See structural derivation.

---

Stratification isolation and no-signaling prohibition

Principle (Stratification isolation)

Nonlinear dynamics (regeneration $\mathcal{R}$) at levels L2+ does not induce nonlinear effects at level L0 (standard QM) and does not violate the no-signaling principle.

Separation of nonlinearity by level

Level	Stratum $X$	Dynamics	Nonlinear $\mathcal{R}$
L0	$S_I$ (matter)	$d\Gamma/d\tau = -i[H, \Gamma]$	No ($R = 0$)
L1	$S_{II}$ (life)	+ $\mathcal{D}[\Gamma]$ (linear Lindblad)	No
L2	$S_{III}$ (mind)	+ $\mathcal{R}[\Gamma, E]$	Yes ($R \geq 1/3$)
L3	$S_{IV}$ (network consciousness)	+ $R^{(n)}$	Yes (higher orders)
L4	$S_{IV}$ (unitary consciousness)	Full ∞-structure	Yes

Theorem (No-signaling prohibition for all levels)

For L0-systems (atoms, photons, qubits) $R = 0$, and $\mathcal{R} = 0$. For L2+ systems the nonlinearity $\mathcal{R}$ does not violate the no-signaling prohibition thanks to the CPTP structure of operator $\varphi$ and locality of $\kappa$:

$$\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = 0$$

Proof: Physical correspondence: §8.

Physical consequence

Atoms and photons used in Bell experiments are at level L0. For them UHM exactly coincides with quantum mechanics. The nonlinearity $\mathcal{R}$ acts only on autonomous macro-systems (cells, brain), which do not form maximally entangled EPR states with distant photons.

Even if an L2-system (brain) is entangled with an L0-system (photon), the regeneration of the brain does not affect the state of the photon — this is a consequence of the CPTP property of $\varphi$ and linearity of the partial trace.

---

Related documents:

• Axiom of Septicity — theorems on thresholds $R_{\text{th}}$ and $\Phi_{\text{th}}$
• Axiom Ω⁷ — ontological foundation of interiority and ∞-groupoid structure
• Coherence matrix — definition of $\Gamma$
• Interiority dimension — $E$ and $\mathcal{H}_E$
• Unity dimension — integration measure $\Phi$
• Foundation dimension — dominant dimension of L3/L4
• Self-observation — measures $R$, $R^{(n)}$, $C$, $D_{\text{diff}}$
• Formalization of operator φ — self-modeling operator and spectral formula
• Categorical formalism — functor $F$, $\mathrm{Exp}$ and n-truncations
• Viability — No-Zombie theorem
• Hard problem of consciousness — phenomenology of experience and explanatory gap
• Physical correspondence: No-signaling prohibition — theorem on compatibility of nonlinearity with no-signaling