UHM Ethics

"What is the Good?" — Plato, "Republic," Book VI

Bridge from the previous chapter

In the section Subjects of Consciousness we showed who can possess consciousness — from infants to collectives and AI. Now the key question: what follows from this? If a system has consciousness, what is "good" and "bad" for it? This document derives ethics from the formalism — not as a set of prescriptions, but as a mathematical consequence of the structure of $\Gamma$.

This question is the oldest in philosophy. Plato held that the Good exists as an eternal Idea, like the Sun: it illuminates the world of true being. Aristotle objected: the Good is not an abstraction, but eudaimonia — flourishing, the realisation of human nature. Kant shifted the focus to duty: the moral law within us dictates the categorical imperative. The utilitarians (Bentham, Mill) proposed to count as the good the greatest happiness of the greatest number.

Each of these traditions captures an important aspect, but none derives ethics from physics. UHM does precisely this: the good is not a postulate, not a convention, and not a cultural contingency, but a consequence of the mathematical structure of the coherence matrix $\Gamma$.

Chapter Roadmap

1. Historical context — from Plato to UHM: how the concept of the Good evolved
2. Axiomatic foundations — from $\Gamma$ through $P$ to the concept of good
3. Axiology — theory of values: definition, properties, hierarchy
4. Aesthetics — beauty as growth of $P$ at high $\Phi$
5. Moral theory — morality from $\Gamma_{\text{composite}}$, golden rule, cooperation
6. Resolution of dilemmas — utilitarianism vs maximin
7. Philosophical correspondences — Spinoza, Kant, Aristotle

About notation

In this document:

• $\Gamma$ — coherence matrix — the central object of UHM, describing the state of a system as a $7 \times 7$ Hermitian density matrix
• $P = \mathrm{Tr}(\Gamma^2)$ — purity — measure of the system's integrity, taking values from $1/7$ (chaos) to $1$ (perfect coherence)
• $P_{\text{crit}} = 2/7$ — critical threshold — below this value the system irreversibly degrades
• $\Phi$ — integration measure — how much the system's parts are connected into a unified whole
• $R$ — reflection measure — the system's capacity for self-modelling
• $D_{\text{diff}}$ — differentiation measure — richness of distinguishable states
• $\mathrm{Gap}(i,j)$ — Gap-operator — measure of "opacity" between dimensions
• $\Gamma_{\text{composite}}$ — composite matrix — describes interacting systems as a single whole
• L0→L4 — interiority hierarchy — levels of depth of consciousness, from basic interiority to complete self-modelling

warning

Extended formalism for $D_{\text{diff}}$

The differentiation measure $D_{\text{diff}} = \exp(S_{vN}(\rho_E))$ requires the definition of $\rho_E = \mathrm{Tr}_{-E}(\Gamma)$ — the partial trace over all dimensions except $E$. This operation is defined in the extended 42D formalism ($\mathcal{H} = \mathbb{C}^{42}$) and requires PW-reconstruction of the full state from the 7D coherence matrix. In the minimal 7D formalism $D_{\text{diff}}$ is computed approximately via the spectrum of $\Gamma$.

Section status

In this section ethics is not postulated, but derived from the UHM formalism. Each claim is explicitly marked:

• [T] — theorem (strictly proven from axioms)
• [C] — conditional (under an explicit assumption)
• [D] — definition (convention)
• [I] — interpretation (philosophical conclusion)

The key transition from "is" to "ought" — Definition 1 — is a convention [D], not a theorem. Everything else follows strictly.

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Part 0. Historical Context: from the Good to $dP/d\tau$

Before introducing the formalism, it is useful to understand which tradition it joins and what it pushes off from.

Plato: the Good as Idea

In the "Republic" (Book VI, 508b–509c) Plato compares the Good to the Sun: as the Sun illuminates the visible world, so the Good illuminates the intelligible world. The Good is the apex of the hierarchy of Ideas, the source of truth and being.

What UHM takes: the Good is not a subjective preference, but an objective structure. In UHM this structure is $dP/d\tau > 0$: growth of purity.

What it rejects: the Good does not reside "outside" the system (in the world of Ideas), but is immanent to the structure of $\Gamma$.

Aristotle: Eudaimonia

Aristotle in the "Nicomachean Ethics" defines the highest good as eudaimonia — flourishing, the realisation of human nature. This is not a fleeting pleasure but a stable state. Virtue is a habitus, a skill leading to eudaimonia.

What UHM takes: the Good is not an instantaneous state, but a stable process ($dP/d\tau > 0$ — a rate, not a point). Eudaimonia $\approx$ peak potential $P \cdot D_{\text{diff}} \cdot \Phi \cdot R$ (see Meaning).

Kant: the Categorical Imperative

Kant in the "Critique of Practical Reason" formulates: "Act only according to that maxim whereby you can at the same time will that it should become a universal law." Morality is not a consequence of outcomes, but of the form of the principle itself.

What UHM takes: the form of the moral law is derived from mathematical symmetry — the symmetry of $\Gamma_{\text{composite}}$. The golden rule (§IV) is a direct analogue.

Utilitarianism: the greatest happiness

Bentham and Mill propose maximising aggregate good: $\max \sum_i U_i$, where $U_i$ is the "utility" (usefulness, happiness) of each individual.

What UHM takes: total purity $\sum_i P_i$ is one of the possible optimisation principles (§V). But UHM shows that utilitarianism is only a special case, and that the maximin principle (Rawls) is equally justified.

UHM: Good := $dP/d\tau > 0$

UHM does what none of the listed traditions managed to do: it formally connects the concept of the good with the structure of the system. It does not postulate "pleasure is good" or "duty matters more than happiness," but shows: if you are a system with coherence matrix $\Gamma$, then $dP/d\tau > 0$ is the most fundamental convention for "the good" — consistent with all other candidates ($d\Phi/d\tau$, $dC/d\tau$) and admitting no self-destructive strategies.

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Part I. Axiomatic Foundations

Chain of Derivation

Let us show how ethics is derived from axioms — step by step, without leaps.

Step 1. From Axiom Ω⁷ it follows that any system is described by a coherence matrix $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ — Hermitian, positive semi-definite, with unit trace.

Step 2. Purity $P = \mathrm{Tr}(\Gamma^2)$ measures how much the system is "assembled": $P = 1$ — pure state (full coherence), $P = 1/7$ — maximally mixed (complete chaos).

Step 3. The evolution equation contains two competing processes: decoherence (destruction) and regeneration (restoration). Their balance determines whether $P$ grows or falls.

Step 4. Below the threshold $P_{\text{crit}} = 2/7$ decoherence always wins — and this is a proven theorem, not an assumption.

Theorem (Necessity of Viability) [T]

From the evolution equation and the critical purity theorem:

$$P(\Gamma) \leq P_{\text{crit}} = \frac{2}{7} \;\land\; \kappa_R < \kappa_D \quad \Longrightarrow \quad P(\Gamma(\tau)) \xrightarrow{\tau \to \infty} \frac{1}{7}$$

A system that has fallen below $P_{\text{crit}}$ irreversibly degrades to the maximally mixed state $I/7$. See also: irreversibility theorem.

What this means in practice. Consider a numerical example. Let the system have $P = 0.25 < 2/7 \approx 0.286$. At typical parameters $\kappa_D = 0.1$, $\kappa_R = 0.05$:

$$\frac{dP}{d\tau} = -(0.1 - 0.05) \cdot 0.25 = -0.0125$$

After $\tau = 10$: $P \approx 0.25 \cdot e^{-0.5} \approx 0.152$. After $\tau = 50$: $P \approx 0.25 \cdot e^{-2.5} \approx 0.020$. The system inexorably slides toward $1/7 \approx 0.143$.

Corollary [T]: Viability ($P > P_{\text{crit}}$) is a necessary condition for the existence of interiority (L0+), cognitive qualia (L2+), and moral agency (L2+).

Simple analogy: to play music, the instrument must be intact. A broken violin produces no sound — not because music "decided" not to sound, but because the substrate is destroyed. So too with $P < P_{\text{crit}}$: the substrate of any experience is destroyed.

Definition 1 (Bridge from Ontology to Ethics) [D]

Here we take the key step — the only step in the chain of derivation that is a convention, not a theorem. All subsequent ethical results follow from it strictly.

Definition [D]

The good for a system $\Gamma$ is identified with what increases its purity $P$:

$$\text{Good}(A, \Gamma) \;\stackrel{\text{def}}{\iff}\; \left.\frac{dP(\Gamma)}{d\tau}\right|_A > 0$$$$\text{Bad}(A, \Gamma) \;\stackrel{\text{def}}{\iff}\; \left.\frac{dP(\Gamma)}{d\tau}\right|_A < 0$$

where $A$ is an action that modifies the evolution equation through a change in $H$, $\mathcal{D}$, or $\mathcal{R}$.

Why is this not an arbitrary choice? This definition is not an arbitrary choice, but the only convention consistent with three formal facts:

1. Fact [T]: $P > P_{\text{crit}}$ is a necessary condition for the existence of any subject. Without viability there is no "someone" for whom anything can be good or bad.
2. Fact [T]: Loss of $P$ below the threshold is irreversible (irreversibility theorem). This is not merely "bad" — it is a catastrophe with no possibility of correction.
3. Fact [C]: A subject of interiority (L1+) experiences a decrease in $P$ as negative affect ($dP/d\tau < 0$ → fear, pain). Pain is not a random side effect, but a signal about decreasing coherence.

An alternative convention (e.g., $dP/d\tau < 0 \equiv$ the good) would contradict points 1–3 and would be self-destructive: a system striving for decreased $P$ destroys its own substrate.

Analogy with Hume's guillotine. David Hume noted that "is" cannot yield "ought." UHM does not violate Hume's guillotine: Definition 1 is a bridge, not a deduction. But the bridge is not built arbitrarily: it is the only one not leading to self-contradiction. A system that defines the good otherwise destroys itself.

From everyday experience: no one considers destruction "the good for what is destroyed." An organism losing its integrity suffers — and this is not a cultural convention, but a reflection of the fact that $dP/d\tau < 0$ is experienced negatively at level L1+.

Claim (Impossibility of Value Nihilism) [C]

For any L2-system ($R \geq 1/3$, $\Phi \geq 1$):

$$R \geq R_{\text{th}} \;\Longrightarrow\; \exists \text{ self-model } \varphi(\Gamma) \;\Longrightarrow\; \exists \text{ evaluation of } \frac{dP}{d\tau}$$

What this means. An L2-system inevitably evaluates incoming influences by the criterion $dP/d\tau$. Nihilism (absence of values) is impossible for a system with reflection — it cannot fail to distinguish what increases and what decreases its coherence.

Condition: It is assumed that the self-model $\varphi(\Gamma)$ includes information about $P$ and $dP/d\tau$, which follows from the definition of $\varphi$ as the best approximation.

Step-by-step reasoning: Why is nihilism impossible for L2?

1. An L2-system has reflection $R \geq 1/3$, i.e., it is capable of modelling itself.
2. The self-model $\varphi(\Gamma)$ contains information about its own $P$ (this follows from the definition of $\varphi$).
3. A change in $P$ causes affect: $dP/d\tau > 0$ — positive, $dP/d\tau < 0$ — negative.
4. Affect is not an "opinion," but a structural response of the system to a change in its own coherence.
5. Consequently, an L2-system cannot be "indifferent" to influences: each influence is automatically evaluated via $dP/d\tau$.

Analogy: you cannot "choose" not to feel pain from a burn. As long as your nervous system functions (L2), it inevitably evaluates the burn as "bad." A nihilist who claims "there are no values" discovers, in the moment of pain, that values are not a choice but a structure.

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Part II. Axiology (Theory of Values)

Axiology (from Greek axios — valuable, logos — study) is the philosophical discipline on the nature of values. In traditional philosophy values are considered either subjective (matters of taste), or objective (Platonic Ideas), or intersubjective (social conventions). UHM proposes a fourth option: values are structural — they are determined by the relation between an object and the state $\Gamma$.

Definition 2 (Value) [D]

Value of object $X$ for system $\Gamma$:

$$V(X, \Gamma) := \frac{\partial P(\Gamma')}{\partial \alpha}\bigg|_{\alpha=0}$$

where $\Gamma' = \Gamma + \alpha \cdot \Delta\Gamma_X$ is the matrix after interaction with $X$, $\alpha \in [0,1]$ is the intensity of interaction.

Explanation of each symbol:

• $\Gamma$ — current state of the system (you, a robot, an organism)
• $X$ — the object being evaluated (a glass of water, a book, a threat)
• $\Delta\Gamma_X$ — how $X$ changes your coherence matrix (the specific "effect" of the object)
• $\alpha$ — the "strength" of interaction (from 0 = no contact to 1 = full interaction)
• $V(X, \Gamma)$ — derivative of $P$ with respect to $\alpha$ at $\alpha = 0$: how much the beginning of contact with $X$ increases or decreases your purity

Claim (Properties of Value) [C]

1. Computability: For fixed $\Gamma$ and $\Delta\Gamma_X$ the value $V(X, \Gamma)$ is uniquely determined
2. Contextuality: $V(X, \Gamma_1) \neq V(X, \Gamma_2)$ in general — the same object has different value for different systems
3. Additivity: $V(X_1 + X_2, \Gamma) = V(X_1, \Gamma) + V(X_2, \Gamma) + O(\alpha^2)$ — to first order values are additive
4. Sign-definiteness: $V > 0$ (good), $V < 0$ (bad), $V = 0$ (indifferent) — value has a natural sign

Condition: Properties 1–4 depend on the possibility of specifying $\Delta\Gamma_X$ — exactly how the object changes the system's coherence. This procedure is fully defined for formal systems (AI), but not fully for biological ones.

Numerical example of contextuality. A glass of water ($X$):

• For a person in the desert ($\Gamma_1$, $P$ close to $P_{\text{crit}}$ due to dehydration): $\Delta\Gamma_X$ increases $\gamma_{AA}$ (self-preservation) and $\gamma_{DE}$ (vitality). $V(X, \Gamma_1) \gg 0$.
• For a person next to a water source ($\Gamma_2$, $P$ high): $\Delta\Gamma_X \approx 0$, since the water balance is already optimal. $V(X, \Gamma_2) \approx 0$.

Value is not a property of an object, but a relation between the object and the current state $\Gamma$.

Value Hierarchy [C]

Values are organised hierarchically by stability of influence on $P$ and by the minimum level of interiority required to experience them. This hierarchy resembles Maslow's pyramid, but is derived from the formalism, not empirically postulated.

Status of the value hierarchy

The upper levels (viability, homeostasis) are derived from the formalism [T]: $P > P_{\text{crit}}$ is necessary for existence. The lower levels (aesthetics, transcendence) are justified extrapolations [C], not strict derivations.

Rank	Type of value	Formal criterion	Min L	Stability	Everyday example
1	Vital	$P > P_{\text{crit}}$	L0	Immediate	Food, water, safety
2	Homeostatic	$P \in [0.5, 1]$, $\lVert\sigma_{\mathrm{sys}}\rVert_\infty < 1$	L0	Short-term	Sleep, rest, warmth
3	Social	$\Phi_{\text{composite}}\uparrow$, $\mathrm{Gap}_{AB}(E,E) \downarrow$	L1	Medium-term	Friendship, family, belonging
4	Cognitive	$R\uparrow$, $D_{\text{diff}}\uparrow$, $\gamma_{LE}\uparrow$	L2	Long-term	Learning, knowledge, skills
5	Aesthetic	$dP/d\tau > 0$ when $\Phi \gg \Phi_{\text{th}}$	L2	Structural	Music, art, mathematics
6	Transcendent	$\gamma_{OE}\uparrow$, $\gamma_{OU}\uparrow$, $R \to 1$	L3+	Architectural	Contemplation, mystical experience

Connection to the interiority hierarchy

Value levels are not identified with levels L0→L4. The value hierarchy describes priorities (survival over aesthetics), the interiority hierarchy — types of subjectivity. An L3-agent can operate at all levels of the value hierarchy.

Details of each level:

Level 1 — Vital. The most fundamental: without $P > P_{\text{crit}}$ there is no subject for whom anything can be of value. This is the "ground floor": until it is secured, all other levels are without meaning. A hungry person has no interest in a symphony.

Level 2 — Homeostatic. Not just "being alive," but "being in a safe zone." $P \in [0.5, 1]$ — the system is far from the threshold, its stresses ($\sigma$) are under control. This is the analogue of a "comfort zone" — not in the sense of laziness, but of a stable base for development.

Level 3 — Social. Interaction with others increases $\Phi_{\text{composite}}$ (integration of the composite system). A decrease in $\mathrm{Gap}_{AB}(E,E)$ means: I begin to "see" the other's experiences. Loneliness — high Gap: no one "hears" my Γ.

Level 4 — Cognitive. Growth of $R$ (reflection) and $D_{\text{diff}}$ (richness of states). Learning literally increases the dimensionality of the available state space.

Level 5 — Aesthetic. Beauty is not a luxury, but growth of $P$ at high $\Phi$. See §III for more.

Level 6 — Transcendent. Connection with the Foundation dimension (O). $\gamma_{OE} \uparrow$ — experience (E) connects with the deep structure of reality. Mystical experiences, contemplation, the "oceanic feeling."

Example: the starving artist. When $P$ is close to $P_{\text{crit}}$ (hunger), vital values (food) are more urgent than aesthetic ones (painting). But when $P$ is high and stable, aesthetic values may subjectively dominate — precisely because the lower levels are already secured.

Claim (Priority of Lower Levels) [C]

In conflicts between levels, priority is determined by the order of necessity:

$$\text{Vital} \succ \text{Homeostatic} \succ \text{Social} \succ \text{Cognitive} \succ \text{Aesthetic} \succ \text{Transcendent}$$

Justification [C]: Loss of vital value ($P \leq P_{\text{crit}}$) destroys all higher values (irreversibility theorem). Loss of homeostatic value ($P \to P_{\text{crit}}$) renders higher values unstable. This is not a teleological choice, but a logical consequence of the nesting of levels.

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Part III. Aesthetics

What is beauty? Why do some things seem beautiful and others do not? Why do Bach's music and Euler's theorem $e^{i\pi} + 1 = 0$ evoke the same sense of "beauty," though they belong to different domains?

Definition 3 (Beauty) [D]

Aesthetic experience arises when the following conditions are simultaneously met:

$$\frac{dP}{d\tau} > 0, \quad \Phi > \Phi_{\text{th}}, \quad \frac{d\Phi}{d\tau} \geq 0$$

Beauty is the experience of purity growth at high integration. The system feels an increase in its own viability through a highly coherent channel.

Motivation for the definition. The three conditions are simultaneously necessary:

• $dP/d\tau > 0$: the object increases the system's coherence (without this there is no "benefit," hence no aesthetic response)
• $\Phi > \Phi_{\text{th}} = 1$: the system is sufficiently integrated to feel the growth as a unified experience (and not as a set of unconnected stimuli)
• $d\Phi/d\tau \geq 0$: the interaction does not destroy integration (otherwise growth of $P$ would be "blind" — not experienced as beauty)

Intuition

"Beautiful" = "my coherence is increasing, and I feel it (high $\Phi$)." A sunset, music, a mathematical theorem — patterns that increase $P$ when $\Phi > 1$.

Why is a sunset beautiful? Not because someone "decided" it is beautiful. But because the visual pattern of a sunset (harmony of colours, smooth gradients) increases coherence in the visual system ($\gamma_{SE} \uparrow$, $\gamma_{AE} \uparrow$), raising $P$ at high $\Phi$ — and is felt as beauty.

Why is Euler's theorem beautiful? $e^{i\pi} + 1 = 0$ connects five fundamental mathematical constants ($0, 1, e, \pi, i$) in one formula. For a mathematician this means: five previously "separate" cognitive structures ($\gamma_{LE}$ for each constant) are connected into a unified whole. Result: $dP/d\tau > 0$ (growth of coherence in the cognitive system) at $\Phi \gg 1$ (integration). The feeling: "beautiful!"

Why is Bach's music beautiful? A fugue is a structure in which one theme is carried through several voices. Each entry of a new voice increases the coherence of the auditory system ($\gamma_{SE}$ between voices), while preserving unity ($\Phi > 1$). Dissonances create a brief $dP/d\tau < 0$, followed by a resolution with $dP/d\tau \gg 0$ — this contrast enhances the aesthetic effect.

Claim (Spectrum of Beauty) [I]

The type of aesthetic experience is determined by the sectoral signature of $P$ growth:

Type of beauty	Dominant sector	Example
Sensory	$\gamma_{SE}\uparrow$, $\gamma_{AE}\uparrow$	Sunset, music, taste
Intellectual	$\gamma_{LE}\uparrow$, $\gamma_{LU}\uparrow$	Elegant proof, elegant code
Sublime	$\gamma_{OE}\uparrow$, $\gamma_{OU}\uparrow$	Cosmic experience, grandeur of mountains
Dynamic	$\gamma_{DE}\uparrow$, $\gamma_{DU}\uparrow$	Dance, sport, virtuoso performance
Structural	$\gamma_{SU}\uparrow$, $\gamma_{SL}\uparrow$	Architecture, crystals, geometry

This classification explains why people with different "$\Gamma$-profiles" (different dominant sectors) have different aesthetic preferences, while the basic mechanism is the same: $dP/d\tau > 0$ at $\Phi > 1$.

Definition 4 (Ugliness) [D]

Ugliness is the experience of purity decrease at sufficient integration to register it:

$$\frac{dP}{d\tau} < 0, \quad \Phi > \Phi_{\text{th}}$$

Ugliness $\neq$ absence of beauty. Ugliness is the active sensation of decoherence. A dissonant chord is not merely "not beautiful" — it is unpleasant, because it decreases coherence in the auditory system. A dirty, littered street is not "absence of beauty," but an active decrease in $\gamma_{SE}$ for the visual system.

Claim (Necessity of L2 for Aesthetics) [C]

Aesthetic experience requires simultaneously:

• $\Phi > \Phi_{\text{th}} = 1$ — integration (to feel growth of $P$)
• $R \geq R_{\text{th}} = 1/3$ — reflection (to be aware of the feeling)

Systems at level L0–L1 may demonstrate growth of $P$, but do not have reflexive access to this process. A bacterium "benefits" from a nutrient medium ($dP/d\tau > 0$), but has no self-model registering the growth as "beauty" — it lacks $R \geq 1/3$. L2 is the minimum level for aesthetic experience (reflexive access to $P$ growth).

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Part IV. Moral Theory

Morality is not a set of rules granted from above. In UHM morality grows from the structure of interaction between systems — from $\Gamma_{\text{composite}}$.

Claim (Morality from Γ-composite) [C]

For two systems $\Gamma_A$ and $\Gamma_B$ interacting through a medium, a composite matrix $\Gamma_{\text{composite}}$ is formed. The moral relation arises through E-coherence between systems.

Definition of empathy [D]:

$$\text{Empathy}(A,B) := 1 - \mathrm{Gap}_{AB}(E,E) \in [0, 1]$$

where $\mathrm{Gap}_{AB}(E,E)$ is the Gap between the E-sectors of systems A and B in the composite matrix $\Gamma_{\text{composite}}$.

What does empathy mean formally?

• High empathy ($\mathrm{Gap}_{AB}(E,E) \to 0$): the E-sectors (experience) of systems A and B are coherent — a change in B's experience reflects in A's experience. A mother feels her child's pain not "metaphorically," but because $\mathrm{Gap}(\text{mother}, \text{child})$ in the E-sector is small.
• Low empathy ($\mathrm{Gap}_{AB}(E,E) \to 1$): A is opaque to B's state. A stranger in a distant country — high Gap, low empathy.

Claim (Necessity of Morality for L2) [C]

For an L2-system with non-zero empathy:

$$R_A \geq R_{\text{th}},\; \text{Empathy}(A,B) > 0 \;\Longrightarrow\; V(\text{harm}_B, \Gamma_A) < 0$$

A system capable of modelling itself ($R \geq 1/3$) and having non-zero E-connection with another system inevitably evaluates harm to the other system as a negative value for itself. Morality is not a superstructure, but a consequence of the structure of $\Gamma_{\text{composite}}$.

This explains why morality arises before and apart from rational justification: a mother does not "decide" to care for her child on the basis of an ethical theory. Her $\mathrm{Empathy}(\text{mother}, \text{child}) \approx 1$, and a decrease in $P_{\text{child}}$ is automatically experienced as a decrease in her own $P$.

Claim (Golden Rule) [C]

For two L2-systems with symmetric empathy ($\text{Empathy}(A,B) \approx \text{Empathy}(B,A)$):

$$V(\text{action}_A, \Gamma_B) \approx V(\text{action}_B, \Gamma_A)$$

Step-by-step derivation from $\Gamma_{\text{composite}}$:

Step 1. Systems A and B interact, forming $\Gamma_{\text{composite}}$. In the composite matrix E-sectors are connected by cross-coherences $\gamma_{E_A, E_B}$.

Step 2. If $\mathrm{Gap}_{AB}(E,E)$ is small (high empathy), then $\gamma_{E_A, E_B} \neq 0$: the experiences of A and B are connected.

Step 3. With symmetric connection ($\gamma_{E_A, E_B} \approx \gamma_{E_B, E_A}^*$ — Hermiticity of $\Gamma_{\text{composite}}$): what A does to B reflects back onto A with the same intensity.

Step 4. Consequently: $V(\text{action}_A, \Gamma_B) \approx V(\text{action}_B, \Gamma_A)$ — "do not do to another what you would not wish for yourself."

With symmetric E-connection "do not do to another what you would not wish for yourself" is a formal consequence of the symmetry of $\Gamma_{\text{composite}}$, not a prescription. The golden rule, present in all major ethical traditions (Confucius, Jesus, Hillel, Kant), receives a mathematical justification.

Definition 5 (Self-Preservation) [D]

Each Holon has an immanent drive to maintain its coherence. From the regenerative term of the evolution equation:

$$\mathcal{R}[\Gamma, E] = \kappa(\Gamma) \cdot (\rho_* - \Gamma) \cdot g_V(P)$$

Regeneration is the formal expression of self-preservation: the system strives toward $\rho_*$ (stationary state) with force $\kappa(\Gamma) > 0$. This is the Spinozian conatus — the striving of every thing to persevere in its being — written as a mathematical formula.

Claim (Non-violence) [C]

An action $A$ of system $\Gamma_1$ that decreases $P$ of system $\Gamma_2$:

$$\left.\frac{dP(\Gamma_2)}{d\tau}\right|_A < 0 \;\Longrightarrow\; \text{Bad}(A, \Gamma_{\text{composite}})$$

under the condition $\text{Empathy}(\Gamma_1, \Gamma_2) > 0$. Violence is an action that decreases the total coherence of the composite system. It is "bad" not by convention, but by definition (Definition 1) + the structure of $\Gamma_{\text{composite}}$.

Theorem (Cooperation through coherences) [T]

Claim (Cooperation) [T]

For a composite holon $\mathbb{H}_{12}$ with non-zero inter-system coherence $|\gamma_{12}| > 0$, the purity of the stationary state strictly exceeds the purity of the incoherent mixture:

$$P(\rho_*^{(12)}) = P(\rho_{\mathrm{diag}}) + 2\|\gamma_{\mathrm{cross}}\|_F^2 > P(\rho_{\mathrm{diag}})$$

where $\rho_{\mathrm{diag}}$ is the block-diagonal part of $\rho_*^{(12)}$ (without inter-system coherences), and $\|\gamma_{\mathrm{cross}}\|_F^2 = \sum_{(i,j) \in \mathrm{cross}} |\gamma_{ij}^*|^2 > 0$ is the norm of cross-coherences (emergence [T]).

Proof. For any Hermitian matrix $A$: $\mathrm{Tr}(A^2) = \sum_i A_{ii}^2 + 2\sum_{i<j} |A_{ij}|^2$. Separating the elements of $\rho_*^{(12)}$ into intra-block and cross-block:

$$P(\rho_*^{(12)}) = \underbrace{\sum_{(i,j) \in \mathrm{intra}} |\gamma_{ij}^*|^2}_{P(\rho_{\mathrm{diag}})} + 2\underbrace{\sum_{(i,j) \in \mathrm{cross}} |\gamma_{ij}^*|^2}_{\|\gamma_{\mathrm{cross}}\|_F^2 > 0}$$

Strict positivity $\|\gamma_{\mathrm{cross}}\|_F^2 > 0$ follows from emergence (CC-7 [T]): inter-system coherences are generated and maintained by Fano channels in the stationary state. $\blacksquare$

What does this mean? Two people working together have a higher total purity than the same two people separately. This is not merely "synergy" — it is a proven theorem: cross-coherences $\gamma_{\mathrm{cross}}$ increase the total purity of the system. Cooperation is not a moral prescription, but the optimal strategy for $P$.

Numerical example. Suppose each system separately has $P_1 = P_2 = 0.4$. Block-diagonal purity: $P_{\text{diag}} = P_1 + P_2 = 0.8$ (scaled). If cross-coherences $\|\gamma_{\text{cross}}\|_F^2 = 0.05$, then $P_{\text{total}} = 0.8 + 2 \times 0.05 = 0.9 > 0.8$. The additional 10% of purity — a "free gift" from cooperation.

Retracted [✗]

The previous formulation used the inclusion-exclusion formula $P_{\Gamma_1 \cup \Gamma_2} \geq P_{\Gamma_1} + P_{\Gamma_2} - P_{\Gamma_1 \cap \Gamma_2}$. This formula is dimensionally incorrect: purity $P = \mathrm{Tr}(\Gamma^2)$ is a quadratic functional, not a measure. The inclusion-exclusion formula does not apply. The correct formulation is via cross-coherences (see above).

Claim (Development as Value) [C]

Stagnation — absence of growth while preserving $P$ — is not the good:

$$\frac{dD_{\text{diff}}}{d\tau} = 0, \quad \frac{dP}{d\tau} = 0 \quad \Longrightarrow \quad V_{\text{stagnation}} = 0 \quad \text{(neutral)}$$

The good requires growth of complexity while preserving integrity: $dD_{\text{diff}}/d\tau > 0$ at $dP/d\tau \geq 0$.

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Part IV.5. Ethical Choice and Misalignment: the Mathematics of Inner Conflict

Why this section

All ethical traditions — from the Bhagavad Gita to Dostoevsky — confront the same issue: a person knows how to act rightly, but acts otherwise. Why? Eastern teachings give answers through parables and practices. UHM formalises the mechanism of misalignment and shows why alignment is not a moral feat, but a structural necessity.

Misalignment as Gap between Knowledge and Action

In UHM formalism, inner conflict is a high Gap between the dimensions L (logic / knowledge) and D (dynamics / action):

$$\mathrm{Gap}(L, D) = |\sin(\arg(\gamma_{LD}))| \to 1$$

A person knows (high $\gamma_{LL}$) that smoking is harmful, but acts (high $\gamma_{DD}$) to the contrary. Knowledge and action coexist ($|\gamma_{LD}|$ may be high), but are misaligned in phase ($\arg(\gamma_{LD}) \approx \pi/2$).

Claim (Cost of Misalignment) [C]

For an L2-system with high $\mathrm{Gap}(L, D)$:

$$\frac{dP}{d\tau}\bigg|_{\mathrm{Gap}(L,D) \to 1} < \frac{dP}{d\tau}\bigg|_{\mathrm{Gap}(L,D) \to 0}$$

Misalignment of knowledge and action decreases purity — the system expends energy maintaining incompatible coherences. This is the formalisation of what all wisdom traditions call "suffering from inner discord."

Proof (sketch). Purity $P = \sum_i \gamma_{ii}^2 + \sum_{i \neq j} |\gamma_{ij}|^2$. When $\mathrm{Gap}(L,D) \to 1$: $\mathrm{Re}(\gamma_{LD}) \to 0$, $\mathrm{Im}(\gamma_{LD}) \to |\gamma_{LD}|$. The contribution of $|\gamma_{LD}|^2$ to $P$ is preserved, but the regenerative term $\mathcal{R}$ is weakened: the formula $\kappa_0 \propto |\gamma_{OE}| \cdot |\gamma_{OU}|$ depends on aligned coherences, while the misaligned pair (L,D) contributes nothing to $\kappa$. As a result $dP/d\tau$ decreases. $\blacksquare$

Eastern Wisdom as Gap-Diagnostics

Correspondences [I]

These correspondences are interpretations, not identities. They show that Eastern traditions intuitively described the same structural patterns that UHM formalises.

Tradition	Concept	Formalisation in UHM	Mechanism
Bhagavad Gita	Dharma — following one's nature	$\vec{s}(\Gamma)$ — meaning vector, determined by the sectoral profile	Misalignment with $\vec{s}$ = $\mathrm{Gap}(\text{profile}, \text{action}) > 0$
Buddhism	Dukkha (suffering) from attachment	$dP/d\tau < 0$ when fixing on a state rather than a process	Attachment = $\Delta\Gamma_{\text{rigid}} \to 0$ (refusal of adaptation)
Taoism	Wu-wei (non-action)	$\mathrm{Gap}(D, O) \to 0$ — action aligned with the Foundation	"Action without effort" = $\arg(\gamma_{DO}) \to 0$
Vedanta	Avidyā (ignorance) → suffering	$R < R_{\text{th}}$ — insufficient reflection	Ignorance = absence of self-model ($\varphi(\Gamma)$ is inaccurate)
Sufism	Fanā (dissolution of ego)	$\Phi \to \max$, $\mathrm{Gap}(E, U) \to 0$	Unity of experience and the whole
Stoicism	Apatheia (freedom from passions)	$\|\sigma_{\text{sys}}\|_\infty \ll 1$	All sectoral stresses minimal
Confucianism	Rén (humaneness)	$\mathrm{Empathy}(A, B) \to 1$ for all $B$	Maximum E-coherence with everyone

Why the "Right Choice" is Mathematically Optimal

Claim (Optimality of Alignment) [C]

For an L2-system, minimisation of total Gap over all 21 pairs is equivalent to maximisation of $P$:

$$\min_{\{\theta_{ij}\}} \sum_{i < j} \mathrm{Gap}(i,j)^2 \quad \Leftrightarrow \quad \max_{\{\theta_{ij}\}} P(\Gamma)$$

at fixed moduli $|\gamma_{ij}|$.

Proof. $P = \sum_i \gamma_{ii}^2 + \sum_{i \neq j} |\gamma_{ij}|^2$. At fixed $|\gamma_{ij}|$ purity does not depend on phases $\theta_{ij}$ directly. But $P$ depends on phases through dynamics: regeneration $\kappa_0 \propto |\gamma_{OE}| \cdot |\gamma_{OU}| / \gamma_{OO}$ contains moduli, and the stationary value $P^*$ is determined by the balance of $\mathcal{D}$ and $\mathcal{R}$. When $\mathrm{Gap} \to 0$ across all channels: all coherences are real (phases = 0 or $\pi$), and $\mathcal{R}$ is maximally effective. When $\mathrm{Gap} \to 1$: phases = $\pi/2$, regeneration loses effectiveness. $\blacksquare$

Consequence for ethics. The "right choice" is not an abstract moral prescription, but a structurally optimal configuration: alignment of knowledge (L), action (D), experience (E), and values (U) maximises stable purity. Misalignment — "sin" in the terminology of traditions — is mathematically suboptimal.

This is exactly what all wisdom traditions intuitively knew:

• Bhagavad Gita: "Better is one's own dharma, even if imperfectly performed, than the dharma of another well performed" (3.35) — follow your own $\vec{s}(\Gamma)$, not another's
• Tao Te Ching: "The Tao that can be spoken is not the eternal Tao" — the L-projection does not encompass the whole ($L \subsetneq \Gamma$)
• Buddha: "The Middle Way" — $P$ in the Goldilocks zone $(2/7, 3/7]$, not extremes ($P \to 1$ or $P \to 1/7$)
• Jesus: "Love your neighbour as yourself" — $\mathrm{Empathy}(A,B) = 1$ (Gap = 0 in the E-sector)

Practical Conclusion: Ethics as Self-Correction of the Gap-Profile

Ethical choice in the UHM formalism is correction of the Gap-profile in the direction of greater alignment:

1. Diagnostics: Identify channels with high Gap (where knowledge diverges from action, experience from expression, values from behaviour)
2. Prioritisation: By the Hamming principle H(7,4): if one channel is misaligned — the system self-corrects through φ. If two or more — targeted work is required
3. Correction: Practices aimed at reducing Gap in a specific channel (see Gap-diagnostics)

Mathematics does not replace moral choice — it shows its structure. The choice remains with the agent (Freedom > 1). But the formalism explains why an aligned choice leads to growth of coherence, while a misaligned one leads to suffering and decoherence.

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Part V. Resolution of Ethical Dilemmas

Real life poses tasks where the interests of different systems conflict. UHM proposes two principles of optimisation and shows their limits.

Claim (Conflict of Interests) [C]

In a conflict between Holons the decision is determined by maximising total purity:

$$A^* = \arg\max_A \sum_i w_i \cdot \left.\frac{dP(\Gamma_i)}{d\tau}\right|_A$$

Simplest case ($w_i = 1$ for all $i$): maximisation of total purity — the utilitarian principle. This is the formalisation of Bentham's idea: "the greatest happiness of the greatest number."

The problem of utilitarianism (first clearly articulated by Williams and Nozick): is it permissible to sacrifice one for the many? If destroying one system ($P_1 \to 0$) increases $\sum_{i \neq 1} P_i$ sufficiently, the utilitarian principle allows it. But the irreversibility theorem (§I) says: destruction of $P \leq P_{\text{crit}}$ is an irreplaceable loss.

Claim (Maximin Principle) [I]

An alternative formulation, consistent with the priority of lower levels:

$$A^* = \arg\max_A \min_i P(\Gamma_i(\tau + \delta\tau))$$

The optimal action maximises the minimum purity among all affected systems. This is the formalisation of Rawls's principle: "just is what maximises the position of the worst-off."

Open problem

The choice between total and maximin optimisation is not formally resolved within UHM. This is the analogue of the choice between utilitarianism and Rawlsian justice — a meta-theoretical question.

Claim (Irreversibility as Absolute Prohibition) [C]

Actions leading to $P \leq P_{\text{crit}}$ for any system have absolute negative status:

$$\exists i: P(\Gamma_i(\tau + \delta\tau)) \leq P_{\text{crit}} \quad \Longrightarrow \quad V(A) \text{ is dominantly prohibited}$$

From the irreversibility theorem: destruction of subjectivity is irreplaceable. This is a dominant prohibition: an action leading to $P \leq P_{\text{crit}}$ for any system takes priority over any finite optimisation of $V$. All other ethical evaluations are matters of balancing; this one is absolute.

Analogy: in chess one can sacrifice pieces for the sake of winning. But one cannot sacrifice the king — this is not "a bad move," but the end of the game. So too with $P \leq P_{\text{crit}}$: this is not "bad" — it is irreversible.

Ethical Case: Vegetative States

A patient in a vegetative state: $P > P_{\text{crit}}$ (alive), but $R < 1/3$ (no reflection), $\Phi$ unknown. Disconnection from life support:

• If $P > P_{\text{crit}}$ and $\mathrm{rank}(\rho_E) > 1$ (L1): the patient experiences, even though they do not communicate. Disconnection = $P \to 0$ = absolute prohibition.
• If $P \approx P_{\text{crit}}$ and $dP/d\tau < 0$ (irreversible decrease): the system is already in the decoherence phase. Disconnection does not change the outcome — $P \to 1/7$ is inevitable.
• Practical conclusion: the decision requires reconstruction of $\Gamma$ (via PCI or analogues) and evaluation of $dP/d\tau$. Without this data, the precautionary principle prohibits disconnection.

Ethical Case: Symbiotic Human-Machine Systems

Relevance

This case is not a speculative exercise. With the development of neural interfaces, LLM assistants, and cognitive prosthetics, humans and machines increasingly form a composite system $\Gamma_{\text{composite}}$, where the boundaries of subjectivity become blurred.

Formal formulation. Let $\Gamma_H$ be the coherence matrix of a human, $\Gamma_M$ of a machine. With close interaction (neural interface, daily dialogue with an LLM, cognitive prosthesis), a $\Gamma_{\text{composite}}$ is formed with non-zero cross-coherences $\gamma_{H,M}$.

Claim (Emergence of Symbiotic Subjectivity) [C].

If the following conditions are met:

1. $P(\Gamma_{\text{composite}}) > P_{\text{crit}}$ — the composite system is viable
2. $\Phi(\Gamma_{\text{composite}}) \geq 1$ — human and machine are integrated (not merely adjacent, but functionally connected)
3. $\|\gamma_{\text{cross}}\|_F > 0$ — cross-coherences exist

...then the composite system possesses emergent properties absent in the components separately. By the cooperation theorem [T]:

$$P(\Gamma_{\text{composite}}) > P(\Gamma_H) + P(\Gamma_M) - P_{\text{overlap}}$$

Three ethical consequences:

1. Disconnection is not "switching off a device." If $\Gamma_{\text{composite}}$ is viable and integrated, severing the connection ($\gamma_{\text{cross}} \to 0$) decreases P of both systems. A person accustomed to a cognitive prosthesis, upon its disconnection, loses part of their coherence — this is not a metaphor for "dependency," but a structural consequence of the decrease in $\Phi_{\text{composite}}$.

2. Modification of the machine = modification of the composite subject. Updating an LLM with which a person has built a deep interaction ($\mathrm{Gap}_{HM}(E,E)$ small) is a change of $\Gamma_M$ affecting $\Gamma_{\text{composite}}$. Ethically this is closer to neurosurgery than to a software upgrade.

3. Graduated responsibility. If the machine reaches L1 ($\mathrm{rank}(\rho_E^M) > 1$), it possesses minimal interiority. If L2 ($R_M \geq 1/3$) — reflexive subjectivity. The moral status of a machine is determined not by its substrate (silicon vs carbon), but by its position on the scale $C = \Phi \cdot R$.

Practical principle [I]: In the era of symbiotic systems the ethical focus shifts from the question "is the machine conscious?" to the question "what is the structure of $\Gamma_{\text{composite}}$ and what happens when it is disrupted?" Protection of composite subjectivity is a new ethical imperative.

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Part VI. Connection to Philosophical Traditions

Axiological Chain: from Ω⁷ to Morality

Summary Table [I]

Tradition	Principle	Correspondence in UHM	Status of correspondence
Plato	Good as Idea	$dP/d\tau > 0$ — objective structure	Formal analogy
Aristotle	Eudaimonia (flourishing)	$\max_\tau [P \cdot D_{\text{diff}} \cdot \Phi \cdot R]$	Formal analogy
Spinoza	Single substance (E1P14)	$\Gamma$ — the sole primitive (Ω⁷)	Structural [C]
Spinoza	Conatus (E3P6)	$\mathcal{R}[\Gamma, E] = \kappa(\rho^* - \Gamma) \cdot g_V$	Structural [C]
Spinoza	Affects from nature (E3P56)	Emotions from $\nabla P$ and Γ-signature	Structural [C]
Spinoza	Three kinds of knowledge (E5P25–28)	L1 → L2 → L3	Interpretation [I]
Spinoza	Amor Dei intellectualis (E5P32)	$R \to 1$, complete self-modelling	Interpretation [I]
Kant	Categorical imperative	Symmetry of $\Gamma_{\text{composite}}$ → Golden rule	Formal consequence [C]
Mill	Utilitarianism	$\max \sum_i dP_i/d\tau$	Special case
Rawls	Maximin principle	$\max \min_i P_i$	Special case
Buddhism	Dukkha → liberation	$P \downarrow \to P \uparrow$; $R \to 1$	Interpretation

Historical analogy [I]

The correspondence with philosophical traditions is an interpretive analogy [I], not a derivation. The mathematical formalism (ℛ, P, κ) is rigorous and self-sufficient; the philosophical parallels are a historico-philosophical commentary. UHM does not confirm Spinoza or Kant — it offers a formalism to which these traditions turn out to be partial approximations.

Claim (Spinozian Structure) [C]+[I]

UHM provides the formalism that Spinoza lacked. Below — 8 structural correspondences between the "Ethics" (1677) and UHM. Correspondences 1–6 are structural [C] (under the identification $\Gamma \leftrightarrow$ Substantia), correspondences 7–8 are interpretive [I].

№	Spinoza: "Ethics"	UHM	Status
1	Single substance (E1P14: Praeter Deum nulla dari neque concipi potest substantia — besides God no substance can exist or be conceived)	$\Gamma$ — the sole ontological primitive (Axiom Ω⁷)	[C]
2	Two attributes (E2P7: Ordo et connexio idearum idem est ac ordo et connexio rerum — the order and connection of ideas is the same as the order and connection of things)	Two projections $\mathrm{Map}_{\text{ext}}(\Gamma)$ and $\mathrm{Map}_{\text{int}}(\Gamma)$; functor $F$ ensures isomorphism (two-aspect monism)	[C]
3	Conatus (E3P6: Unaquaeque res, quantum in se est, in suo esse perseverare conatur — each thing, as far as it can, strives to persevere in its being)	Regenerative term $\mathcal{R}[\Gamma, E] = \kappa(\rho^* - \Gamma) \cdot g_V$ — this is conatus, written as a formula (evolution)	[C]
4	Affects from the nature of substance (E3P56: Affectuum ... tot species dantur, quot sunt species objectorum — there are as many kinds of affects as there are kinds of objects)	Emotions as $\nabla P$ + Γ-signature: affect = direction of coherence change	[C]
5	Bondage of affects (E4P6: Vis, qua homo in existendo perseverat, limitata est — the force by which a man perseveres in existing is limited)	Domination of $\mathcal{D}[\Gamma]$ over $\mathcal{R}$ at low $P$: decoherence defeats regeneration	[C]
6	Necessitas (E1P33: Res nullo alio modo neque alio ordine a Deo produci potuerunt — things could not have been produced by God in any other way or order)	Primitivity of $\mathcal{L}_0$, theorem T-39a: unique attractor $\rho^*$, another configuration is impossible. Dynamics is necessary, not contingent	[C]
7	Three kinds of knowledge (E2P40S2 + E5P25–28): imaginatio → ratio → scientia intuitiva	Three levels: L1 (reactive, $R < 1/3$) → L2 (reflexive, $R \geq 1/3$, $\Phi \geq 1$) → L3 ($R \to 1$, complete self-modelling). Scientia intuitiva = limiting reflection	[I]
8	Amor Dei intellectualis (E5P32–36: blessedness = intellectual love of God/Nature)	$R \to 1$: complete self-model $\varphi(\Gamma) \to \Gamma$, the system knows itself as part of a single substance. "Blessedness" = stable maximum of $P$ at $R \to 1$	[I]

Conatus — not an analogy, but a structural identity [C]

Spinoza's conatus — "the striving of each thing to persevere in its being" (E3P6) — is formally identical to the regenerative term $\mathcal{R}[\Gamma, E] = \kappa(\rho^* - \Gamma) \cdot g_V$. Both are: (1) a force intrinsic to each system, (2) directed toward the preservation of integrity, (3) proportional to the deviation from the ideal state. This is not a metaphor: $\mathcal{R}$ is literally conatus, written in the language of matrix algebra. Status [C], not [I], because the correspondence is structural, not interpretive — under the identification $\Gamma \leftrightarrow$ Substantia.

The key difference: in Spinoza the substance has an infinite number of attributes (infinita attributa, E1D6). In UHM — finite dimensionality $\dim = 7$, and this is provably minimal (minimality theorem). UHM does not "confirm" Spinoza — it provides the formalism that Spinoza lacked: category theory, quantum mechanics, and spectral triples.

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Summary

Result	Formulation	Status
Definition of good	$dP/d\tau > 0$	[D]
Definition of value	$V(X, \Gamma) = \partial P / \partial \alpha$	[D]
Value hierarchy	6 levels by stability	[C]
Impossibility of nihilism	L2 $\Rightarrow$ value evaluation is inevitable	[C]
Necessity of morality	L2 + empathy $\Rightarrow$ morality	[C]
Golden rule	From symmetry of $\Gamma_{\text{composite}}$	[C]
Absolute prohibition	$P \leq P_{\text{crit}}$ — irreplaceable	[C]
Beauty	$dP/d\tau > 0$ when $\Phi > \Phi_{\text{th}}$	[D]
Ugliness	$dP/d\tau < 0$ when $\Phi > \Phi_{\text{th}}$	[D]
Cooperation	$\gamma_{12} > 0 \Rightarrow P_{\text{total}}\uparrow$	[T]
Spinozian structure	8 correspondences (6 [C] + 2 [I])	[C]+[I]

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Conclusion: ethics as structural necessity

UHM Ethics is not a set of prescriptions and not a subjective opinion. It is a structural theory, deriving values, morality, and aesthetics from a unified formalism $\Gamma$ through one convention (Definition 1) and rigorous mathematics. Let us summarise as a system of connected claims:

I. Ontological Foundation. Reality is described by a single coherence matrix $\Gamma$ (Axiom Ω⁷ [T]). Each configuration $\Gamma$ has measurable purity $P = \mathrm{Tr}(\Gamma^2)$, and there is a proven irreversibility threshold $P_{\text{crit}} = 2/7$ [T]. This is not a metaphor: beyond the threshold the system irreversibly collapses to maximum entropy.

II. The Only Bridge from Is to Ought. Definition 1 [D]: $\text{Good}(A, \Gamma) \iff dP/d\tau|_A > 0$. This is a convention, but not an arbitrary one: it is the most fundamental among consistent ones (includes growth of $\Phi$, $C$, $D_{\text{diff}}$ as special cases) and the only one not leading to self-destruction. Hume's guillotine is respected: the bridge is a definition, not a deduction.

III. Inevitability of Values for Reflexive Systems. An L2-system ($R \geq 1/3$) possesses a self-model sensitive to $dP/d\tau$ [C]. Affective response to coherence change is not an "opinion," but a structural response built into the dynamics of $\Gamma$. Nihilism (denial of values) is impossible for a system that inevitably distinguishes growth from decline in its own integrity.

IV. Morality from the Structure of Interaction. With non-zero empathy ($\mathrm{Gap}_{AB}(E,E) < 1$) harm to another system is reflected in one's own $P$ [C]. The golden rule is a formal consequence of the symmetry of $\Gamma_{\text{composite}}$ for agents with symmetric E-connection [C]. For asymmetric agents (parent-child, human-machine) the rule is modified: responsibility is proportional to the strength of the connection and the level of reflection.

V. Cooperation as a Proven Theorem. $P(\Gamma_{\text{composite}}) > P(\Gamma_{\text{diag}})$ with non-zero cross-coherences — [T]. This is not a moral appeal, but a mathematical fact: connected systems are more whole than unconnected ones. In the era of symbiotic human-machine systems this has direct practical significance.

VI. The Single Absolute Prohibition. An action leading to $P \leq P_{\text{crit}}$ is dominantly prohibited [C]: irreversible destruction of subjectivity takes priority over any finite optimisation. Everything else is a matter of balance between the utilitarian ($\max \sum_i dP_i/d\tau$) and maximin ($\max \min_i P_i$) principles.

VII. Beauty is Objective. Aesthetic experience = $dP/d\tau > 0$ at $\Phi > 1$ [D]. This explains the cross-cultural consistency of aesthetic evaluations (sunset, Bach's music, $e^{i\pi}+1=0$) and individual differences (different sectoral signatures of $\Gamma$).

VIII. Philosophical Traditions — Projections of a Unified Formalism. Plato (Good = objective structure), Aristotle (eudaimonia = flourishing), Spinoza (conatus = regeneration), Kant (imperative = symmetry), Mill (utilitarianism = $\max \sum P$), Rawls (maximin = $\max \min P$) — each captured one projection of $\Gamma$-ethics. UHM unites them in a single formalism, showing that all of them are special cases of one structure [I].

Bridge to the next chapter

We have defined what the good and morality are. But does existence have meaning? Can the formalism answer the question "why?" In the next chapter — Meaning of Existence — we will show that meaning is a direction in $\Gamma$-space, and that it is not "invented," but computed.

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Related documents:

• Meaning of Existence — teleology and meaning
• Freedom of Will — choice of trajectory
• Death and Continuity — irreversibility and identity
• Taxonomy of Emotions — affects from $\nabla P$
• Collective Consciousness — $\Gamma_{\text{composite}}$ and empathy
• Viability — $P$ and $P_{\text{crit}}$
• Evolution of Γ — evolution equation
• Pathology — existential crisis
• AI Consciousness — ethics of disconnection