Viability Measure

This chapter answers one of the most fundamental questions of the theory: what makes a system alive? In ordinary language we intuitively distinguish the living from the dead, the whole from the disintegrated. UHM gives this distinction a precise mathematical expression through purity $P$ — a number that measures how coherent and organised a system is. The reader will learn: how to compute $P$; what the critical threshold $P_{\text{crit}} = 2/7$ means; under what conditions a system is viable and under what conditions it inevitably disintegrates; and how the four conditions of consciousness relate to viability.

Historical precursors

The question "what is life?" from the perspective of physics was first seriously posed by Erwin Schrödinger in the book "What is Life?" (1944). He proposed that living systems maintain low entropy by "feeding on negentropy" from the environment.

Ilya Prigogine (Nobel Prize, 1977) developed this idea in the theory of dissipative structures: systems far from equilibrium can spontaneously form ordered states by consuming free energy.

Integrated Information Theory (IIT, Tononi, 2004) introduced a quantitative threshold of consciousness $\Phi > 0$.

UHM synthesises these ideas: viability is determined by purity $P > 2/7$ (threshold of distinguishability from noise), while full consciousness requires three additional conditions ($R \geq 1/3$, $\Phi \geq 1$, $D \geq 2$). Free energy $\Delta F > 0$ plays the role of Schrödinger's "negentropy" — fuel for regeneration.

Intuitive explanation of purity P

Imagine photographing someone:

• $P = 1$ (pure state) — a perfectly sharp photograph. Every detail is distinguishable. The system is fully determined, nothing random.
• $P \approx 0.5$ — a photograph with slight blur. The main features are visible, but fine details are smeared. The system is alive and functioning, but not ideal.
• $P = 2/7 \approx 0.286$ — the critical threshold. The photograph is so blurred that it is impossible to confidently distinguish a person from background noise. This is the boundary between "distinguishable" and "indistinguishable."
• $P = 1/7 \approx 0.143$ — a fully overexposed film. No information. All measurements are equally probable. Maximum entropy. The system is "dead" (thermal equilibrium).

The threshold $P_{\text{crit}} = 2/7$ is not an arbitrary number. It has been proved to be the unique value at which the signal (the system's structure) is separated from noise (thermal background) in the Bures metric.

Definition of purity

Purity $P$ — a scalar measure of the integrity and viability of the Holon.

$$P = \mathrm{Tr}(\Gamma^2)$$

In the orthonormal basis $\{|A\rangle, |S\rangle, |D\rangle, |L\rangle, |E\rangle, |O\rangle, |U\rangle\}$:

$$P = \sum_{i} \gamma_{ii}^2 + \sum_{i \neq j} |\gamma_{ij}|^2$$

where the first sum is the contribution of diagonal elements, and the second is the contribution of coherences.

Range of values

For a 7-dimensional system:

$$P \in \left[\frac{1}{7}, 1\right] \approx [0.143, 1]$$

$P$	State	Description
$1.0$	Pure	Full coherence: $\Gamma = \vert\psi\rangle\langle\psi\vert$, $\mathrm{rank}(\Gamma) = 1$
$0.5 - 1.0$	Alive	Partial coherence, system is alive and adapts
$2/7 - 0.5$	Stressed	Coherence under threat, regeneration required
$1/7 - 2/7$	Fading	Decoherence exceeds regeneration
$1/7 \approx 0.14$	Minimum	Fully mixed: $\Gamma = I_7/7$, maximum entropy

where $I_7$ is the $7 \times 7$ identity matrix.

Relation to entropy

Von Neumann entropy:

$$S_{vN} = -\mathrm{Tr}(\Gamma \log \Gamma) = -\sum_k \lambda_k \log \lambda_k$$

where $\{\lambda_k\}$ are the eigenvalues of $\Gamma$.

Relation between purity and entropy:

Condition	Purity	Entropy
Pure state	$P = 1$	$S_{vN} = 0$
Maximally mixed	$P = 1/7$	$S_{vN} = \log 7 \approx 1.95$

Monotonic relation

Purity $P$ and entropy $S_{vN}$ are monotonically related: an increase in $P$ corresponds to a decrease in $S_{vN}$ and vice versa. However, the relation is nonlinear.

Critical purity

$$P_{\text{crit}} = \frac{2}{N} = \frac{2}{7} \approx 0.286$$

Unification via the Bures metric

P_crit — a mathematical theorem [Т]; interpretation via PIR [О]

Critical purity $P_{\text{crit}} = 2/N$ is proved from the Bures metric (the unique monotone Riemannian metric by the Chentsov–Petz theorem). PIR [О] provides the ontological interpretation:

$$P > P_{\text{crit}} \Leftrightarrow d_B(\Gamma, I/N) > d_B^{\text{noise}}$$

where $d_B$ is the Bures metric, $d_B^{\text{noise}}$ is the characteristic noise scale.

Physical meaning: A system is viable ⟺ it is informationally distinguishable from background noise in the Bures metric.

See unification of thresholds via PIR.

Theorem on critical purity [Т]

The value $P_{\text{crit}} = 2/N$ is strictly derived from five independent arguments converging to the same value. The structural deviation from chaos must exceed the scale of chaos. At $P = 2/7$ the dominant regime captures ~49% of coherence.

Master definition and table of paths: Axiom of Septicity → Critical purity

Full proof: theorem-purity-critical

Derivation of P_crit = 2/N from the Bures metric

Critical purity is not postulated — it is strictly derived from the requirement of informational distinguishability.

Step 1. Frobenius distance from the maximally mixed state.

$$d_F^2(\Gamma, I/N) = \mathrm{Tr}\left((\Gamma - I/N)^2\right) = \mathrm{Tr}(\Gamma^2) - \frac{2}{N}\mathrm{Tr}(\Gamma) + \frac{1}{N} = P - \frac{1}{N}$$

Step 2. Noise scale. Typical deviation from $I/N$ under thermal fluctuations:

$$\|\delta\Gamma\|_F^2 \sim \frac{1}{N^2} \quad \Rightarrow \quad d_F^{\text{noise}} \sim \frac{1}{N}$$

Step 3. Distinguishability condition. A system is distinguishable from noise if the structural deviation exceeds the noise:

$$d_F^2(\Gamma, I/N) > d_F^{\text{noise}} \quad \Rightarrow \quad P - \frac{1}{N} > \frac{1}{N} \quad \Rightarrow \quad \boxed{P > \frac{2}{N} = \frac{2}{7}}$$

Five independent paths to one threshold [Т]

The value $P_{\text{crit}} = 2/7$ is derived by five independent arguments:

1. Frobenius distinguishability (above): $d_F > d_F^{\text{noise}}$
2. Dominant eigenvalue: at $P = 2/7$ the largest $\lambda_1 \geq 2/7 \approx 0.286$, capturing $\geq 49\%$ of coherence
3. Bayesian distinguishability: posterior likelihood ratio $\Lambda(\Gamma \| I/N) > 1$ at $P > 2/N$
4. Fano channel: coherence contraction with factor $1/3$ preserves structure only at $P > 2/7$
5. Self-observation: minimum reflection $R \geq R_{\text{th}} = 1/3$ requires $P \geq P_{\text{crit}}$

Full proof → | Status: [Т]

Temporal interpretation of P_crit

Theorem (Relation of P_crit to time)

Critical purity is connected to the minimum rate of flow of internal time:

$$P > P_{crit} \Leftrightarrow \frac{d\tau}{d\sigma} > \frac{d\tau}{d\sigma}\bigg|_{min}$$

Viability ($P > 2/7$) means that the Holon continues to exist in time.

At $P \leq 2/7$ the system loses coherence and "smears out" over the state space — time ceases to be well-defined for it.

More →

Viability condition

A Holon $\mathbb{H}$ is viable if and only if:

$$\mathrm{Viable}(\mathbb{H}) := P(\Gamma) > P_{\text{crit}} = \frac{2}{7}$$

Strengthened condition (stable viability):

$$\mathrm{Viable}_{\text{stable}}(\mathbb{H}) := P(\Gamma) > \frac{2}{7} \land \left.\frac{dP}{d\tau}\right|_{\mathcal{R}} + \left.\frac{dP}{d\tau}\right|_{\mathcal{D}} \geq 0$$

The system is not only above the threshold, but also does not lose purity (balance of regeneration and dissipation).

Stochastic extension (T-145 [Т]): under stochastic perturbations $h_{\text{ext}}$ with $\mathbb{E}[\|h_{\text{ext}}\|^2] \leq \sigma_h^2$, the probability of maintaining full viability $V_{\text{full}}$ is exponentially close to 1: $\mathbb{P}[\Gamma \in V_{\text{full}}] \geq 1 - \exp(-r_{\text{stab}}^2/(2\sigma_h^2))$.

Full axiomaticity of the viability condition [Т]

Both terms in the strengthened condition are fully determined by the axioms:

• $dP/d\tau|_\mathcal{D}$ — from $\mathcal{L}_\Omega$ (A1, classifier Ω)
• $dP/d\tau|_\mathcal{R}$ — from the form of ℛ, derived from axioms A1–A5 + standard thermodynamics [Т]

The viability condition is not a phenomenological criterion, but a strict consequence of the axiomatics.

Viability domain

Minimal viability

$$\mathcal{V}_P := \left\{\Gamma \in \mathcal{D}(\mathcal{H}) : P(\Gamma) > \frac{2}{7}\right\}$$

where $\mathcal{D}(\mathcal{H})$ is the space of density matrices on $\mathcal{H}$.

Topological properties:

Property	Description
$\mathcal{V}_P \subset \mathcal{D}(\mathcal{H})$	Open subset
$\partial\mathcal{V}_P = \{\Gamma : P(\Gamma) = 2/7\}$	Boundary (critical surface)
$\mathrm{int}(\mathcal{V}_P) = \mathcal{V}_P$	Interior coincides with the set

Full viability

Minimal viability $\mathcal{V}_P$ is a necessary, but insufficient condition for a fully functional Holon. Full viability is defined via the stress tensor:

tip

Non-emptiness of $\mathcal{V}_{\mathrm{full}}$ [T-124]

The set of full viability is non-empty: the existence of $\Gamma$ with $P \in (2/7, 3/7]$, $\Phi \geq 1$, $\forall k: \sigma_k < 1$ is constructively proved. Goldilocks zone: $P \in (2/7, 3/7]$ — the optimal range for consciousness. See T-124 [Т].

$$\mathcal{V}_{\mathrm{full}} := \left\{\Gamma \in \mathcal{D}(\mathcal{H}) : \|\sigma_{\mathrm{sys}}(\Gamma)\|_\infty < 1\right\}$$

where each of the 7 components $\sigma_i$ controls a separate condition (purity, structure, dynamics, logic, differentiation, regeneration, integration). See Theorem 10.1 / T-92.

Theorem (Embedding of viability domains) [Т]

$$\mathcal{V}_{\mathrm{full}} \subsetneq \mathcal{V}_P$$

Proof. ($\subseteq$): $\|\sigma_{\mathrm{sys}}\|_\infty < 1$ implies, in particular, $\sigma_A < 1$, which via $\sigma_A = 1 - \gamma_{AA}/P$ and the conditions on all 7 components guarantees $P > 2/7$ (nonzero coherences increase $P$). ($\subsetneq$): Counterexample — the pure state $|1\rangle\langle 1| \in \mathcal{V}_P$ ($P = 1$), but $\sigma_U = 1 - \Phi/\Phi_{\mathrm{th}} = 1$ (zero integration $\Phi = 0$), therefore $\|\sigma_{\mathrm{sys}}\|_\infty \geq 1$ and $|1\rangle\langle 1| \notin \mathcal{V}_{\mathrm{full}}$. $\blacksquare$

Viability stratification

The notation $\mathcal{V}$ without an index further in the theory denotes minimal viability $\mathcal{V}_P = \{P > 2/7\}$. For results requiring all 7 conditions simultaneously (e.g., Theorem 10.1), $\mathcal{V}_{\mathrm{full}}$ is used.

Invariance and positivity preservation

Theorem (Invariance of the viability domain)

The viability domain $\mathcal{V}$ is invariant under the full evolution of the Holon given sufficient regeneration:

$$\Gamma(0) \in \mathcal{V} \land \mathcal{R}[\Gamma] \geq \mathcal{D}[\Gamma] \implies \Gamma(\tau) \in \mathcal{V} \quad \forall \tau > 0$$

The proof relies on the positivity preservation theorem:

Despite the nonlinearity of the regenerative term $\mathcal{R}[\Gamma, E]$, the full evolution equation preserves positivity $\Gamma \geq 0$ and normalisation $\mathrm{Tr}(\Gamma) = 1$. This is guaranteed by the interpolation formulation:

$$\Gamma(\tau + \Delta\tau) = (1 - \alpha) \cdot \mathcal{E}[\Gamma(\tau)] + \alpha \cdot \varphi(\Gamma(\tau))$$

where:

• $\mathcal{E}$ — CPTP dissipation channel
• $\varphi(\Gamma)$ — CPTP self-modeling channel, canonical form $\varphi_{\text{coh}}$
• $\alpha = \kappa \cdot \Delta\tau < 1$ — correctness condition

Why this matters for viability:

Property	Guarantee	Consequence
$\Gamma \geq 0$	Interpolation of CPTP channels	State remains physical
$\mathrm{Tr}(\Gamma) = 1$	Linearity of trace	Normalisation preserved
$P(\Gamma) \leq 1$	CPTP channel does not increase purity above 1	Purity bounded above
$P(\Gamma) \geq 1/7$	Convexity	Purity bounded below

The fixed point of canonical $\varphi_{\text{coh}}$ has $P(\Gamma^*) = P_{\text{crit}} = 2/7$ [Т] — this is a mixed state, not pure (see operator φ). Under active regeneration ($\mathcal{R} \geq \mathcal{D}$) the attractor maintains $P > P_{\text{crit}} = 2/7$.

More on positivity preservation →

Purity dynamics

The derivative of purity with respect to time (see evolution):

$$\frac{dP}{d\tau} = 2 \cdot \mathrm{Tr}\left(\Gamma \cdot \frac{d\Gamma}{d\tau}\right)$$

Contributions of the evolution equation components:

Component	Contribution to $\frac{dP}{d\tau}$	Description
Unitary $-i[H, \Gamma]$	$= 0$	Preserves purity
Dissipation $\mathcal{D}[\Gamma]$	$\leq 0$	Decreases purity
Regeneration $\mathcal{R}[\Gamma, E]$	$\geq 0$ (when $\Delta F > 0$)	Can increase purity

Death condition

Theorem (Irreversibility of decoherence) [Т]

A Holon irreversibly loses viability if two conditions are simultaneously satisfied:

$$P(\Gamma) < P_{\text{crit}} = \frac{2}{7} \quad \land \quad \frac{dP}{d\tau} < 0$$

Under these conditions $\Gamma(\tau) \to I/7$ exponentially fast:

$$\|\Gamma(\tau) - I/7\|_F \leq \|\Gamma(0) - I/7\|_F \cdot e^{-\Delta(\mathcal{L}_0)\tau}$$

where $\Delta(\mathcal{L}_0) > 0$ is the spectral gap of the Liouvillian [T-39a].

Proof. At $P < P_{\text{crit}}$: the regeneration gate $g_V(P) = 0$ (V-preservation gate), therefore $\mathcal{R} = 0$. Dynamics is governed only by the linear part $\mathcal{L}_0 = -i[H,\cdot] + \mathcal{D}$. By primitivity of $\mathcal{L}_0$ [T-39a]: the unique stationary state is $I/7$, spectral gap $\Delta > 0$. Exponential convergence follows from the spectral theorem for the superoperator. The condition $dP/d\tau < 0$ guarantees that the system does not leave the region $P < P_{\text{crit}}$: a return would require $dP/d\tau > 0$, but $\mathcal{R} = 0$ and $dP/d\tau|_{\mathcal{D}} \leq 0$. $\blacksquare$

Physical interpretation. Below $P_{\text{crit}}$ regeneration is switched off (Landauer's principle: free energy $\Delta F \leq 0$, regeneration is thermodynamically forbidden). The system inevitably decays to thermal equilibrium $I/7$ — "death" in UHM terminology.

Connection to time dilation. By T-53d [Т]: internal time $d\tau_{\text{int}}/dt_{\text{ext}} \propto (P - P_{\text{crit}})^{1/2}$. Near $P_{\text{crit}}$ subjective time slows down infinitely — "death" is not experienced from within.

Primitivity proved [Т]

Primitivity of the linear part $\mathcal{L}_0$ for viable holons has been proved via the Evans–Spohn criterion: atomic operators $L_k = |k\rangle\langle k|$ together with the connectivity condition on the interaction graph $G_H$ (which follows from (AP)+(PH)+(QG)+(V) by the connectivity theorem) guarantee triviality of the commutant $\mathcal{F}(\mathcal{L}_0) = \mathbb{C} \cdot I$. Convergence to $I/7$ under dominant dissipation is guaranteed (T-39a).

Proof →

Phase diagram

Examples

Biological analogies

State	$P$	Biological analogue
Pure	$\approx 1$	Embryonic stem cells
Healthy	$0.5 - 0.8$	Healthy organism
Stressed	$2/7 - 0.5$	Disease, exhaustion
Fading	$< 2/7$	Terminal state
Minimum	$\approx 1/7$	Death (thermal equilibrium)

Psychological analogies

State	$P$	Psychological analogue
High coherence	$> 0.7$	Flow state
Normal	$0.5 - 0.7$	Wakefulness
Stressed	$2/7 - 0.5$	Fatigue, anxiety
Critical	$< 2/7$	Dissociation, psychosis

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Numerical example: viable and non-viable Γ

Example 1. Viable system ($P > 2/7$):

Let $\Gamma$ have eigenvalues $\lambda = (0.35, 0.20, 0.15, 0.10, 0.08, 0.07, 0.05)$. Then:

$$P = \sum_k \lambda_k^2 = 0.35^2 + 0.20^2 + 0.15^2 + 0.10^2 + 0.08^2 + 0.07^2 + 0.05^2 = 0.2118$$

Wait — $0.2118 < 2/7 \approx 0.286$! This system is non-viable — too diffuse. More pronounced "peaks" are required. Take different eigenvalues:

$\lambda = (0.45, 0.20, 0.12, 0.08, 0.06, 0.05, 0.04)$:

$$P = 0.45^2 + 0.20^2 + 0.12^2 + 0.08^2 + 0.06^2 + 0.05^2 + 0.04^2 = 0.2930 > \frac{2}{7}$$

The system is viable: the dominant eigenvalue $\lambda_1 = 0.45$ captures enough coherence. Life requires concentration — at least one direction must be significantly stronger than the rest.

Example 2. Maximally mixed (dead) system:

$\lambda = (1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7)$, $P = 7 \cdot (1/7)^2 = 1/7 \approx 0.143$ — minimum. All dimensions are equal, no structure.

Four conditions of consciousness

Viability ($P > 2/7$) is a necessary, but insufficient condition for consciousness. Full consciousness requires four conditions to hold simultaneously:

Condition	Formula	Meaning	Threshold	Status
Viability	$P > P_{\text{crit}}$	System is distinguishable from noise	$P_{\text{crit}} = 2/7$	[Т]
Reflection	$R \geq R_{\text{th}}$	System can model itself	$R_{\text{th}} = 1/3$	[Т]
Integration	$\Phi \geq \Phi_{\text{th}}$	Parts of the system are bound into a whole	$\Phi_{\text{th}} = 1$	[Т]
Differentiation	$D \geq D_{\min}$	System distinguishes its states	$D_{\min} = 2$	[Т]

Analogy with an organism

All four conditions can be compared to the hallmarks of a living organism:

• $P > 2/7$ — the organism exists (distinguishable from its environment).
• $R \geq 1/3$ — the organism feels itself (the nervous system forms a self-model).
• $\Phi \geq 1$ — the organs are connected into a unified whole (not a collection of separate cells).
• $D \geq 2$ — the organism distinguishes at least two different states (not locked into one pattern).

Violation of any of the four conditions destroys consciousness: zombie (no $R$), dissociation ($\Phi < 1$), catatonia ($D < 2$), death ($P < 2/7$).

More: Reflection measure R | Integration measure Φ | Interiority hierarchy

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Octonionic norm

note

Relation of purity P to the $\mathbb{O}$ norm [С]

In the octonionic interpretation, purity $P = \mathrm{Tr}(\Gamma^2)$ is related to the norm on $\mathrm{Im}(\mathbb{O})$: the normativity of octonions ($|xy| = |x||y|$) provides a consistent metric on the state space. The viability condition $P > 2/7$ corresponds to the minimum "distinguishability from noise" in the normed space $\mathrm{Im}(\mathbb{O})$. Bridge [Т] (closed, T15). See structural derivation.

Related documents:

• Theorem on critical purity — full proof of $P_{\text{crit}} = 2/N$
• Theorem on emergent time — temporal interpretation of P_crit
• Axiom of Septicity — axiom context
• Coherence matrix — definition of Γ and P
• Evolution — dynamics of Γ(τ) with internal time
• Foundation (O) — source of regeneration and internal clock
• Spacetime — emergent geometry
• Interiority hierarchy — L2 thresholds
• Mathematical apparatus — formal specification
• Γ measurement protocol — AI viability validation (research programme)