Subjective Time

Bridge from the previous chapter

In the Emotion taxonomy we showed that emotions are the "interior projection" of the dynamics of viability $dP/d\tau$. But this very dynamics unfolds in time. How exactly does the subject experience time? Why do some minutes "fly by" while others "drag"? The answer lies in the coherence $\gamma_{OE}$ between the Ground dimension (O, the internal clock) and Interiority (E, experience). If $\gamma_{OE}$ is high — each "tick" of the clock is filled with experience and time "slows down". If it is low — ticks pass by consciousness and time "flies".

On notation

• $\gamma_{OE}$ — coherence between Ground (O) and Interiority (E)
• $\gamma_{OO}$, $\gamma_{EE}$ — populations of dimensions O and E
• $\tau$ — emergent time, derived from the structure of the category $\mathcal{C}$
• $P = \mathrm{Tr}(\Gamma^2)$ — purity
• $\Gamma$ — coherence matrix
• Full notation table — in Notation

Chapter roadmap

1. Philosophical history — from Augustine to Husserl
2. Subjective tempo $\mathcal{T}$ — definition and derivation from first principles
3. Temporal dilation — formula for stretching/compression of time
4. Flow state (flow) — why time "slows down and speeds up" simultaneously
5. Boredom — the antipode of flow
6. Meditation — systematic management of temporal coherences
7. Danger and time slowing — why time "stops" during a fall
8. Temporal memory window — the "depth of the present"
9. Connection to physical time — four equivalent constructions

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Philosophical History: What Is Time?

Augustine (354–430): the paradox of time

Saint Augustine in the "Confessions" (Book XI) formulated one of the most famous paradoxes:

"What then is time? If no one asks me, I know; if I wish to explain it to one that asks, I know not."

Augustine noted a fundamental difficulty: the past no longer exists, the future does not yet exist, and the present is merely a fleeting point without extension. Where, then, does time exist? His answer: time exists in the soul — as memory (the past), perception (the present), and expectation (the future). Time is not an objective river in which we swim, but a structure of our consciousness.

Bergson (1889): duration vs spatial time

Henri Bergson in "Time and Free Will" (1889) drew a radical distinction:

• Spatial time (temps) — the time of physics, measured by clocks. It is homogeneous: every second is equal to every other. It can be decomposed into points, like space.
• Duration (durée) — the time of consciousness, the time of experience. It is non-homogeneous: a minute of waiting is not equal to a minute of joy. It cannot be decomposed into points — it is a continuous flow where the past penetrates the present.

Bergson insisted: genuine reality is durée, not temps. Physical time is a spatial metaphor imposed on duration. When we say "five minutes have passed", we are already spatialising time, slicing it into pieces.

Husserl (1905): retention, protention, primal impression

Edmund Husserl in the lectures "On the Phenomenology of the Internal Time-Consciousness" (1905) gave the most subtle analysis. Every moment of consciousness contains three layers:

• Primal impression (Urimpression) — the experience of "now", the fleeting point of the present
• Retention — the "just-passed", still held in consciousness (not a recollection, but the "tail" of the present)
• Protention — the "about-to-come", anticipation of the immediate future

Retention is not recollection. When you hear a melody, the previous note is not "remembered" — it still sounds in consciousness, gradually fading. It is precisely thanks to retention that you hear a melody, not separate sounds.

UHM position: time from O-E coherence

UHM formalises the intuitions of all three thinkers:

• Augustine: time exists "in the soul" — in UHM subjective time is defined by the coherence $\gamma_{OE}$, connecting the "clock" ($O$) and "experience" ($E$)
• Bergson: duration is non-homogeneous — in UHM the subjective tempo $\mathcal{T}$ changes depending on the state $\Gamma$
• Husserl: retention and protention — in UHM the "temporal window" $T_{\text{mem}}$ defines the depth of retention; the autocorrelation $\rho_E(\tau) \cdot \rho_E(\tau - \Delta\tau)$ formalises the "tail" of the present

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Motivation: Two Times

In UHM, physical time $\tau$ is not postulated but derived from the structure of dimension O (Ground) via the Page–Wootters mechanism. However, the subjective experience of time — "how fast/slow time flows" — depends not on $\tau$ as such, but on the coherence between O and E: on how closely the "internal clock" is linked to "experiential content".

An everyday analogy. Imagine a station clock with a second hand. Physical time is the uniform ticking of this clock. Subjective time is how you perceive these ticks. If you are absorbed in an interesting book ($\gamma_{OE}$ is high), each tick is filled with content — an hour passes in "five minutes". If you are waiting for a delayed train ($\gamma_{OE}$ is low, but $\gamma_{LE}$ is high — you are aware of the waiting), each tick is empty — five minutes drag like an hour.

Definition of Subjective Tempo (D.1)

Derivation of the formula from first principles

Let us begin with the question: what should subjective tempo measure? It should answer: "how much experiential content corresponds to one tick of the internal clock?"

Step 1. In UHM, the "internal clock" is dimension $O$ (Ground). The population $\gamma_{OO}$ characterises the "resource" invested in timekeeping. The higher $\gamma_{OO}$, the more "ticks" the system produces per unit of physical time.

Step 2. "Experiential content per tick" is the coherence $\gamma_{OE}$ between the clock ($O$) and experience ($E$). If $\gamma_{OE} = 0$, the clock ticks but experience is in no way linked to it — the subject "does not notice" the passage of time. If $|\gamma_{OE}|$ is high, each tick is filled with content.

Step 3. The natural measure is the ratio of content to number of ticks:

$$\mathcal{T} = \frac{|\gamma_{OE}|}{\gamma_{OO}}$$

This ratio is dimensionless and shows what fraction of the "clock resource" is linked to experience.

Definition D.1 (Subjective tempo) [D]

Subjective tempo is a dimensionless quantity characterising the relative speed of subjective time:

$$\mathcal{T}(\tau) := \frac{|\gamma_{OE}(\tau)|}{\gamma_{OO}(\tau)}$$

where:

• $|\gamma_{OE}|$ — modulus of the coherence between Ground and Interiority
• $\gamma_{OO}$ — population of the Ground dimension

Range: $\mathcal{T} \in [0, 1]$ (from the Cauchy–Schwarz inequality: $|\gamma_{OE}|^2 \leq \gamma_{OO} \gamma_{EE}$, given $\gamma_{EE} \leq 1$).

Breakdown of each symbol

For complete clarity let us unpack the formula $\mathcal{T} = |\gamma_{OE}|/\gamma_{OO}$ symbol by symbol:

• $\mathcal{T}$ — subjective tempo (calligraphic T from "tempo")
• $\gamma_{OE}$ — element of the coherence matrix $\Gamma$ at the intersection of row $O$ (Ground) and column $E$ (Interiority). It is a complex number: $\gamma_{OE} = |\gamma_{OE}|e^{i\theta_{OE}}$
• $|\gamma_{OE}|$ — the modulus of this complex number: the "strength" of the connection between clock and experience, without regard to the "angle" (perspective)
• $\gamma_{OO}$ — diagonal element of $\Gamma$: the population of dimension $O$. A real number showing how much "resource" is invested in timekeeping

Interpretation

$\mathcal{T}$	Subjective effect	Description	Example
$\mathcal{T} \to 1$	Time "slows down"	Rich O-E coherence: each "clock tick" is filled with experience	The moment of an accident, the first parachute jump
$\mathcal{T} \to 0$	Time "flies"	Weak O-E coherence: "ticks" pass by consciousness	Deep sleep, anaesthesia
$\mathcal{T} \approx \text{const}$	Normal pace	Stationary O-E connection	Calm wakefulness

Numerical example. Three states of one person over the course of a day:

State	$\gamma_{OO}$	$\lvert\gamma_{OE}\rvert$	$\mathcal{T}$	Experience
Morning coffee	$0.12$	$0.06$	$0.50$	Normal tempo — a familiar morning
Car accident	$0.14$	$0.12$	$0.86$	"Time slowed down" — every moment is detailed
Falling asleep	$0.10$	$0.01$	$0.10$	"Time disappears" — an instantaneous void

Note: during the accident $\gamma_{OO}$ increases slightly (adrenaline intensifies timekeeping), while $|\gamma_{OE}|$ rises sharply (each tick is linked to intense experience). As a result $\mathcal{T}$ almost doubles — the subject experiences a "slowing" of time.

Temporal Dilation (C.1)

Statement C.1 (Subjective time dilation) [Т] — upgraded from [C]

The ratio of subjective to physical time increments is proportional to the subjective tempo:

$$\frac{\delta\tau_{\text{subj}}}{\delta\tau_{\text{phys}}} \propto \mathcal{T}(\tau) = \frac{|\gamma_{OE}|}{\gamma_{OO}}$$

Derivation [Т]. The connection of $\gamma_{OE}$ to the experience of time is not a semantic postulate but a theorem following from three [Т]-results:

1. T-87 [Т]: O is the clock dimension — time $\tau$ emerges from correlations between O and the rest via the Page–Wootters mechanism.
2. T-186(a) [Т]: E is the experience dimension — the phenomenal functor $F \cong \&|_{\mathcal{D}}$ extracts the E-sector as the carrier of subjective content.
3. T-88 [Т]: The coupling $\gamma_{OE}$ appears in the regeneration rate $\kappa_0 = \omega_0 \cdot |\gamma_{OE}| \cdot |\gamma_{OU}| / \gamma_{OO}$, which governs the speed at which the system updates its experiential state.

The ratio $|\gamma_{OE}|/\gamma_{OO}$ is therefore the rate of experiential content production per clock tick — a derived quantity, not a convention. $\square$

At high $\mathcal{T}$ subjective time "stretches" (more experience per unit of physical time). At low $\mathcal{T}$ subjective time "compresses".

Mechanism

Dimension O, via the Page–Wootters mechanism, acts as the internal clock:

$$\mathcal{H}_{\text{total}} = \mathcal{H}_O \otimes \mathcal{H}_{6D}$$

Conditional state at a fixed "tick" $|\tau_n\rangle_O$:

$$\Gamma(\tau_n) = \frac{\mathrm{Tr}_O\!\left[(|\tau_n\rangle\langle\tau_n|_O \otimes \mathbb{1}_{6D}) \cdot \Gamma_{\text{total}}\right]}{p(\tau_n)}$$

The coherence $\gamma_{OE}$ determines how non-trivial the E-component of the correlation with the O clock is. If $\gamma_{OE} = 0$, the Interiority dimension is "disconnected" from the clock — subjective time is not registered.

Analogy. Imagine a metronome (O) and a dancer (E). If the dancer is listening to the metronome ($\gamma_{OE}$ is high), each beat is filled with movement — "time is marked out". If the dancer is wearing headphones ($\gamma_{OE} = 0$), the metronome ticks, but the dance is not linked to it — for the dancer "there is no time", even though the metronome keeps running.

Danger and Time Slowing

One of the most vivid and widely known phenomena of subjective time is its "slowing" in moments of danger. People who have survived car accidents, falls, and attacks often report: "time stopped", "I saw everything in slow motion".

Mechanism in UHM terms

At the moment of sudden danger, a sharp reorganisation of the $\Gamma$-profile occurs:

Parameter	Before danger	During danger	What happens
$\gamma_{OO}$	$0.12$	$0.15$	Adrenaline intensifies timekeeping
$\lvert\gamma_{OE}\rvert$	$0.06$	$0.13$	Each "tick" is linked to experience
$\gamma_{DD}$	$0.14$	$0.24$	Dynamics mobilised
$\gamma_{AE}$	$0.10$	$0.28$	Apperception is maximal — "I see every detail"
$\gamma_{LL}$	$0.15$	$0.06$	Logic suppressed — "no time for thinking"
$\mathcal{T}$	$0.50$	$0.87$	Subjective time almost doubled

This explains why:

• A second of falling is experienced as "a whole minute" ($\mathcal{T}$ sharply increased)
• Details are remembered with photographic accuracy ($\gamma_{AE}$ is maximal)
• Considered decisions are impossible ($\gamma_{LL}$ is suppressed — reflex acts, not reason)

Numerical example. A climber falls. The physical fall lasts 3 seconds. Subjectively he experiences:

$$\delta\tau_{\text{subj}} = \delta\tau_{\text{phys}} \times \frac{\mathcal{T}_{\text{danger}}}{\mathcal{T}_{\text{normal}}} = 3 \times \frac{0.87}{0.50} \approx 5.2 \text{ subj. seconds}$$

He "manages" to see the ledge, grab it, become aware of what is happening — in "3 physical seconds" he lived through 5 subjective ones. This is not mysticism — it is the mathematics of $\gamma_{OE}$.

Flow States (Flow)

The flow state (flow by Csikszentmihalyi, 1990) is one of the most studied altered states of consciousness. Mihaly Csikszentmihalyi described it as a state of complete immersion in an activity, when time "flows differently".

$\Gamma$-profile of flow

$$\text{Flow:} \quad \gamma_{DE} \gg \overline{\gamma}, \quad \mathcal{T} \text{ elevated}, \quad \mathrm{Gap}(D,E) \approx 0$$

Parameter	Value in Flow	Typical estimate	Interpretation
$\gamma_{DE}$	High	$\sim 0.30$	Strong connection of dynamics and experience — "immersion"
$\mathcal{T} = \lvert\gamma_{OE}\rvert/\gamma_{OO}$	Elevated	$\sim 0.7$	Subjective time expanded — "much experience"
$\mathrm{Gap}(D,E)$	$\approx 0$	$< 0.05$	Minimal gap — "transparency" between action and experience
$\gamma_{AE}$	High	$\sim 0.25$	Concentration of attention
$\gamma_{DU}$	High	$\sim 0.20$	Teleology — the sense of a goal
$\gamma_{LL}$	Low	$\sim 0.06$	Logical tracking weakened

Resolving the flow paradox

The flow state contains a famous paradox: time simultaneously "slows down" and "speeds up". During flow each moment seems infinitely rich (time slowed), but after the activity ends it seems that "an instant flew by" (time accelerated).

Resolution in UHM: separation into two mechanisms:

1. During flow: $\mathcal{T}$ is elevated (each tick is filled with experience) — subjectively each moment "lasts a long time"
2. Retrospectively: low $\gamma_{LL}$ (logical control weakened) means that "time markers" were not being placed. When recalling, the brain estimates duration by the number of markers — there are few, so "it passed quickly"

Analogy. In the flow state you are a jazz musician improvising. Each note (each moment) is filled with meaning ($\mathcal{T}$ is high). But you are not counting bars ($\gamma_{LL}$ is low). Therefore, after a two-hour concert it seems that 20 minutes have passed, even though during the playing each second was infinitely rich. This is not a contradiction — it is two different aspects of the same $\Gamma$-profile.

Numerical example. A programmer in the flow state (3 hours of physical time):

Moment	$\mathcal{T}$	$\gamma_{LL}$	Experience
During flow (each minute)	$0.70$	$0.06$	Each minute is saturated, $\delta\tau_{\text{subj}} \approx 1.4 \times \delta\tau_{\text{phys}}$
Retrospectively (after exiting)	—	—	"What? Already 3 hours? It felt like half an hour!"

Boredom

Boredom is the state that is the antipode of flow:

$$\text{Boredom:} \quad \gamma_{DE} \approx 0, \quad \gamma_{DL} \text{ low}, \quad \mathcal{T} \text{ reduced}$$

Parameter	Value during boredom	Typical estimate	Interpretation
$\gamma_{DE}$	$\approx 0$	$< 0.03$	Dynamics disconnected from experience — "nothing is happening"
$\gamma_{AE}$	Low	$< 0.05$	Attention defocused
$\mathcal{T}$	Reduced	$\sim 0.2$	Little experience per "tick" — time "drags"

Paradox of boredom [I]

Subjectively during boredom time "drags", even though $\mathcal{T}$ is low (the prediction: time should "fly"). Resolution: during boredom $\gamma_{LE}$ is elevated — reflexive monitoring of the passage of time. The awareness "I am bored" amplifies the subjective assessment of duration through a metacognitive loop. This is consistent with the L2 condition $R \geq 1/3$ — boredom is impossible below L2.

Numerical example. During boredom: $\mathcal{T} \approx 0.2$ (little content), but $\gamma_{LE} \approx 0.25$ (reflection "I am bored"). The system is in a paradoxical regime: low $\mathcal{T}$ means little experience per tick, but high $\gamma_{LE}$ means that the absence of experience is itself experienced as content. This is precisely why boredom is the privilege of conscious beings (L2+): an amoeba does not get bored, because it has no metacognitive loop.

Comparison of flow and boredom:

Parameter	Flow	Boredom
$\gamma_{DE}$	$0.30$	$0.02$
$\gamma_{AE}$	$0.25$	$0.04$
$\gamma_{LL}$	$0.06$	$0.05$
$\gamma_{LE}$	$0.08$	$0.25$
$\mathcal{T}$	$0.70$	$0.20$
Time (during)	"The moment lasts forever"	"Minutes drag"
Time (after)	"An instant flew by"	"It dragged on endlessly"

Meditation and Temporal Perception

Meditative practices systematically alter temporal coherences. For more on altered states see ASC.

Concentration (shamatha)

Shamatha (Skt. "calm abiding") — the practice of one-pointed attention: focus on an object (the breath, a mantra, a point) while letting thoughts go.

$$\text{Shamatha:} \quad \gamma_{AE} \uparrow, \quad \gamma_{DE} \downarrow, \quad \gamma_{EO} \uparrow$$

Focusing attention ($\gamma_{AE} \uparrow$) with a decrease in dynamic content ($\gamma_{DE} \downarrow$) and a deepening of the connection with the ground ($\gamma_{EO} \uparrow$). Subjectively: time "disappears" — a transition to a stationary $\Gamma$.

Numerical example. Before meditation: $\gamma_{AE} = 0.10$, $\gamma_{DE} = 0.15$, $\gamma_{EO} = 0.05$, $\mathcal{T} = 0.50$. After 30 minutes of shamatha: $\gamma_{AE} = 0.25$, $\gamma_{DE} = 0.05$, $\gamma_{EO} = 0.15$, $\mathcal{T} = 0.35$. Attention strengthened 2.5-fold, dynamic content decreased 3-fold — "thoughts quieted, but awareness sharpened". $\mathcal{T}$ decreased (less content per tick), but subjectively time does not "drag" (unlike boredom), because $\gamma_{LE}$ is not elevated — there is no reflexive monitoring of "I am bored".

Insight (vipassanā)

Vipassanā (Skt. "clear seeing") — the practice of observing the stream of consciousness without attachment to an object.

$$\text{Vipassanā:} \quad \gamma_{LE} \uparrow, \quad R \uparrow, \quad \gamma_{EO} \uparrow$$

An increase in understanding ($\gamma_{LE}$) and reflection ($R$) with a deepening of the connection with the ground. Subjectively: time is simultaneously "saturated" and "transparent".

Numerical example. An experienced vipassanā practitioner: $\gamma_{LE} = 0.28$, $R = 0.65$, $\gamma_{EO} = 0.20$, $\mathcal{T} = 0.60$. Subjective tempo is moderately elevated, but the key difference from flow is a high $\gamma_{LE}$ (awareness is present) and a high $R$ (reflection is deep). The meditator is simultaneously "in flow" and "observing themselves" — a state impossible without $R \geq 1/3$.

Temporal Memory Window

Definition D.2 (Temporal window) [D]

Temporal window $T_{\text{mem}}$ is the duration of the interval over which the autocorrelation of experiential content is significant:

$$T_{\text{mem}} := \inf\left\{\Delta\tau > 0 : \mathrm{Tr}\!\left(\rho_E(\tau) \cdot \rho_E(\tau - \Delta\tau)\right) < \epsilon\right\}$$

where $\epsilon$ is the correlation threshold, $\rho_E(\tau) = \mathrm{Tr}_{-E}(\Gamma(\tau))$ is the reduced experience matrix.

The temporal window defines the "depth of the present" — how many "ticks" of the past are simultaneously present in experience. This is the mathematical formalisation of Husserlian retention: what "tail" of the past still "sounds" in the present.

This corresponds to the History component in the quadruple of experiential content $\mathrm{Exp}(\Gamma, \tau) = (\mathrm{Intensity}, \mathrm{Quality}, \mathrm{Context}, \mathrm{History})$ from interiority theory. The connection to types of memory is discussed in Attention and memory.

Factors influencing $T_{\text{mem}}$

Factor	Influence on $T_{\text{mem}}$	Mechanism	Example
High $\gamma_{SL}$	Increase	Stable logical structure preserves correlations	A logical chain of reasoning is remembered longer
Strong decoherence $\mathcal{D}_\Omega$	Decrease	Rapid destruction of correlations	Under stress, the previous moment is quickly "erased"
High $\gamma_{EO}$	Increase	Deep connection stabilises memory	Meditative states — "expanded present"
$P \to P_{\text{crit}}$	Decrease	Low coherence — short memory	In dementia, the "present" shrinks to seconds

Specious present and O-dynamics [H]

The phenomenological "present" (~300 ms according to Varela, Pöppel) may be derived from the O-sector. The subjective time formula $dt_{\text{subj}}/dt_{\text{phys}} = |\gamma_{OE}|/\gamma_{OO}$ [C] defines the integration time window. At typical values $\gamma_{OO} \sim 1$ and $|\gamma_{OE}| \sim 0.3$ (awareness threshold), the characteristic time: $T_{\text{present}} \sim 1/(\omega_0 \cdot \gamma_{OO}) \sim 300$ ms at $\omega_0 \sim 3$ Hz (theta rhythm). Status: [H]. Calibration of $\omega_0$ is required.

Connection to Physical Time

Emergent time $\tau$ in UHM is defined via four equivalent constructions (see Emergent time):

1. Page–Wootters: correlation with the O-dimension
2. Information-geometric: Bures metric on $\mathcal{D}(\mathcal{H})$
3. Categorical: chains of morphisms in $\mathrm{Exp}_\infty$
4. Stratificational: collapse of strata to the terminal object $T$

Subjective time is not an alternative to physical time, but its interior projection: the same dynamics $\Gamma(\tau)$, perceived "from within" through the E-sector. This is the direct realisation of Augustine's idea: time exists both "in the world" ($\tau$) and "in the soul" ($\mathcal{T} \cdot \tau$) — but it is the same time seen from different sides.

The emotional experience of time (anxious waiting, joyful anticipation) is determined by the combination of $\mathcal{T}$ and $dP/d\tau$ — for details see Emotion taxonomy. Applied consequences for cognitive architecture are in the CC theorems.

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What we learned

1. The problem of time — from Augustine through Bergson to Husserl — receives in UHM a formal solution through the coherence $\gamma_{OE}$
2. Subjective tempo $\mathcal{T} = |\gamma_{OE}|/\gamma_{OO}$ — a dimensionless measure of the "speed" of subjective time, derived from first principles
3. High $\mathcal{T}$ — time "slows down" (each tick is filled with experience); low $\mathcal{T}$ — time "flies"
4. Danger sharply raises $\mathcal{T}$ through an increase in $|\gamma_{OE}|$ — the formal explanation for "time slowing during a fall"
5. Flow state: $\gamma_{DE} \gg \overline{\gamma}$, $\mathrm{Gap}(D,E) \approx 0$, $\mathcal{T}$ elevated — the "stretching-compression" paradox is resolved through the separation of $\gamma_{LL}$ and $\mathcal{T}$
6. Boredom: $\gamma_{DE} \approx 0$, $\mathcal{T}$ low, but $\gamma_{LE}$ high — metacognitive monitoring of "emptiness" creates the sensation of stretched time
7. Temporal window $T_{\text{mem}}$ defines the "depth of the present" — depends on $\gamma_{SL}$, $\gamma_{EO}$, and the rate of decoherence

Bridge to the next chapter

We have considered what is experienced (qualia), how it is experienced (emotions), when it is experienced (subjective time). It remains to answer the question: about what is the experience? The directedness of consciousness toward an object — intentionality — is examined in the next chapter: Intentionality. There we will show that intentionality is a morphism in the category $\mathbf{Hol}$ satisfying a condition on the E-sector.

Related Documents

• Ground (O) — the clock dimension, source of $\gamma_{OO}$
• Emergent time — four constructions and the Page–Wootters mechanism
• Coherence matrix — definition of $\gamma_{OE}$ and coherences
• Interiority theory — the History component in $\mathrm{Exp}(\Gamma, \tau)$
• Emotion taxonomy — dynamics $dP/d\tau$ and sectoral signature
• Gap semantics — $\mathrm{Gap}(D,E)$ in the flow state
• Attention and memory — temporal window and types of memory
• Theorems of Coherence Cybernetics — applied consequences of temporal dynamics