Definition:
The directional energy potential (SEP) is a fundamental spatial vector field that defines the interaction capability of any physical configuration. It is determined by the directional energy distribution and the mass configuration of the system.

\[ \vec{\varphi}(\vec{r}) := \frac{{\vec{E}}_{\mathrm{TRR}}(\vec{r})}{m} \cdot \hat{n} \]

Where:

• \({\vec{E}}_{\mathrm{TRR}}(\vec{r})\) is the directional energy vector at position \(\vec{r}\),
• \(m\) is the local mass associated with the configuration,
• \(\hat{n}\) is the direction of projection or interaction selectivity.

Physical Meaning and Units:
The SEP vector expresses how much directionally projectable energy is available per unit of mass, aligned with the projection axis.

\[ [\vec{\varphi}] = \left[\frac{\mathrm{kg \cdot m^2}}{\mathrm{kg}}\right] = [\mathrm{m^2}] \]

Interaction Condition (SEP Projection Law):
An interaction is possible at a given location \(\vec{r}\) only if the scalar projection of SEP onto an external field (e.g. photon, probe, structure) exceeds a critical threshold:

\[ {\vec{\varphi}}_{\text{internal}}(\vec{r}) \cdot {\vec{\varphi}}_{\text{external}} \ge \varphi_{\text{crit}} \]

Where:

• \({\vec{\varphi}}_{\text{internal}}(\vec{r})\): SEP vector of the internal structure at location \(\vec{r}\),
• \({\vec{\varphi}}_{\text{external}}\): SEP vector of the incoming field (e.g. photon, probe, environment),
• \(\varphi_{\text{crit}} \in \mathbb{R}^+\): scalar threshold required for interaction.

This condition defines a realized interaction node and governs all physical phenomena in the TRR–NOTIME framework. No temporal variable is involved.

Interpretation:

• SEP is not derived from time or force.
• It does not generate motion, but determines the structure of possible interaction.
• Only directional compatibility of SEP vectors creates observable phenomena.
• Where no projection exists, nothing happens – regardless of energy or configuration presence.

Every physical phenomenon is fully determined by the configuration of the directional energy potential (SEP) and its projection selectivity. Time is not part of physical reality; it does not exist as an independent quantity, causal parameter, or integration variable.

All interactions, changes, and observable manifestations are outcomes of the spatial structure of SEP, not of any change in time.

Formally:

Implications:

• Time cannot be used to explain physical phenomena; it has no interactive role.
• Frequency, motion, action, or decay are secondary, non-temporal manifestations of SEP structure.
• Equations involving time derivatives must be converted into structural relations in the space of projection layers (NTL).
• Anything that does not change in SEP does not change at all — regardless of the flow of observer time.

There exists an initial state of reality, denoted as \(\varphi_0\), which is:
– directionally perfectly balanced,
– non-reactive,
– without localization.

This state represents maximal energetic balance without any manifestation — a state of zero directional momentum:

The φ₀–state is a potential of zero projection distinguishability — not a result, not a point, not a geometry. It is an unmanifested possibility of reality, from which something may become projected — only if and where interaction allows.

Fundamental Statement on Balance:

Balance is not a relation.
It is not a relationship between objects.
It is a state in which no relation can be realized — because no distinguishability is present.

Terminological Distinction:
To avoid confusion, the term "balance" must be strictly distinguished from "equilibrium":

The \(\varphi_0\)-state is thus a state of balance — not equilibrium.
It is an unmanifested potential in which nothing can arise, because there is nothing to overcome, change, balance, or detect.

Note on the Status of the Axiom:

• Axiom 0 has a metatheoretical nature:
• – It cannot be derived from any existing structure,
• – But its negation leads to logical contradiction.
• It is a necessary foundation — without it, projection, differentiation, or directionality, i.e., existence itself, could not emerge.

The vacuum in TRR–NOTIME is not the absence of energy, but a structurally balanced field of directional energy potentials (SEP), where each component may be active but the total projection remains zero.

Formally:

\( \text{Vacuum} \;≔\; \left\{ \vec{\varphi}_k \in \mathbb{R}^3 \;\middle|\; \|\vec{\varphi}_k\| > 0,\ \sum_k \vec{\varphi}_k = \vec{0} \right\} \)

• Each \( \vec{\varphi}_k \) is a nonzero directional SEP vector,
• The sum of all vectors in the configuration equals zero: \( \sum_{k} \vec{\varphi}_k = \vec{0} \),
• Therefore, the configuration has no net projectional asymmetry and cannot interact or manifest any effect.

TRR–NOTIME does not interpret vacuum as a lack or negative pressure, but as the default structure of space itself. As such, vacuum cannot lie below a reference level — it is not below normal, it defines the normal.

SEP components within vacuum may be structurally active, but their projectional balance renders them interactionally silent.

The vacuum is not absence.
It is a balanced SEP configuration that appears silent — but contains latent directional structure.

Relation to Axiom 0:

• φ₀-state: has \( \vec{\varphi}_k = 0 \;\forall k \) → no SEP.
• TRR vacuum: has \( \vec{\varphi}_k \neq 0 \), but \( \sum_k \vec{\varphi}_k = \vec{0} \) → balanced latent SEP field.

Thus, the vacuum is a projectionally latent state — it structurally exists, but does not produce any detectable asymmetry.

Whenever a local closure of directional energy potential vectors (SEP) occurs — i.e., a subset of SEP vectors becomes internally projectively consistent and externally asymmetric — a stable configuration emerges, called a φ–center.

A φ–center is characterized by:

• Localization in space,
• A defined configurational mass \( m \),
• The potential to serve as a carrier of interaction (e.g., particle, black hole core, orbital system).

This structure is the first detectable configuration in TRR–NOTIME, emerging naturally from differentiation within the latent vacuum state.

Context within the Axiom Set:

A φ–center thus represents the realization of interaction — a locally bound projectional structure, made possible by internal SEP compatibility and projectional closure.

The TRR–NOTIME framework defines all measurable physical quantities without reference to time.

All quantities are derived from three primary spatial constructs:

Interpretation:

None of these quantities require time for their definition or measurement. They are all derived from structural characteristics of space and mass, and represent physical observables through projectional compatibility.

In TRR–NOTIME, time is neither a base unit nor a derived dimension.
All physical reality is expressed through spatial, directional, and interactional properties.

TRR–NOTIME defines two fundamental derived quantities that describe gravitational effects without using time or force:

Interpretation and Significance

In the TRR–NOTIME framework, gravitational phenomena are not interpreted as forces or spacetime curvature, but as projectional differentiation in the directional energy potential (SEP).

The quantity \( g_{\mathrm{SEP}} \) describes the rate of change in SEP with respect to height — in other words, how many units of directional potential (in m²) change per 1 meter of altitude.
It is an internal, purely structural quantity, independent of time, motion, or force.

\[ [g_{\mathrm{SEP}}] = \frac{[m^2]}{[m]} = [m] \]

The quantity \( g_{\mathrm{TRR}} \) represents the observable gravitational effect at a given point, computed from the known SEP field and present mass. It is used for direct comparison with empirically determined values — for example, in modeling the gravitational effect of a planet.

Both quantities have the unit [m], since they do not express acceleration but spatial imbalance in the projectional SEP field.

In TRR–NOTIME, gravity is not a force, nor an effect of curvature.
It is a structural asymmetry in SEP, projected across spatial layers.

Definition:

An interaction occurs only if the directional energy potential (SEP) field satisfies the condition of directional compatibility between internal and external configurations.

Let the SEP vector field be defined as:

\[ \vec{\varphi}(\vec{r}) = \frac{{\vec{E}}_{\mathrm{TRR}}(\vec{r})}{m(\vec{r})} \]

• \( \vec{\varphi}(\vec{r}) \) — directional energy at position \( \vec{r} \)
• \( m(\vec{r}) \) — interacting mass at the same position
• Unit: \( [\vec{\varphi}] = \mathrm{[m^2]} \)

Interaction Condition:

A point \( \vec{r} \) becomes a candidate for interaction if:

\[ \vec{\varphi}_{\mathrm{local}}(\vec{r}) \cdot \vec{\varphi}_{\mathrm{external}} \geq \varphi_{\mathrm{critical}} \]

• \( \varphi_{\mathrm{critical}} \in \mathbb{R}^+ \) — scalar interaction threshold
• \( \vec{\varphi}_{\mathrm{local}}(\vec{r}) \) — SEP vector of the structure at point \( \vec{r} \)
• \( \vec{\varphi}_{\mathrm{external}} \) — incoming directional SEP (e.g., photon, probe)

Definition of a Realized SEP Node:

An SEP node is a spatial point where this condition is satisfied and interaction is physically realized:

\[ \mathrm{SEP\ node} \Longleftrightarrow \vec{\varphi}_{\mathrm{local}}(\vec{r}) \cdot \vec{\varphi}_{\mathrm{external}} \geq \varphi_{\mathrm{critical}} \]

This defines the moment of interaction, not merely its potentiality.

Any physical quantity that cannot be directly and simply derived from the configuration of directional energy potential (SEP) is, within TRR–NOTIME, considered a derived projectional quantity.

Such quantities – including time, force, or geometry – are not treated as causes of physical phenomena, but as projectional manifestations arising from the interaction between the SEP structure and the observer's projectional boundary.

The observer defines the limit at which a SEP projection becomes an observable quantity, but has no influence over the cause itself.

A physical cause is independent of observation. It must be directly contained in SEP, or else it is an interpretation of a consequence.

TRR–NOTIME does not deny the existence or utility of derived quantities, but requires that they be reconstructible as projectional consequences of SEP, rather than introduced as independent inputs.

Interpretation Note:

This axiom defines a strict distinction between what exists as a cause (SEP), and what emerges as a consequence of observable projection (such as time, force, or spatial form).

Purpose:

This axiom is required for any observable interaction to emerge from the SEP field. It defines the fundamental selectivity that governs realization within the TRR–NOTIME framework.

Statement:
Energy is not a substance but a directional structural projection of SEP. Conservation of energy is the result of projectional balance in the SEP field.

\[ \sum_{\text{in}}{\vec{\varphi}}_i = \sum_{\text{out}}{\vec{\varphi}}_j \]

This equality holds for every realized interaction (emergence or dissolution of a φ–center), but is trivially satisfied in the vacuum or \( \varphi₀ \)-state.
Energy is neither created nor destroyed; it is projected differently.

Statement:
A configuration cannot spontaneously evolve into interactional asymmetry without a directional cause. Irreversibility is not temporal, but structural.

A system is interactionally complete if:
Entropy is not randomness. It is the inability of SEP components to cancel.

\[ \sum_{k}{\vec{\varphi}_k} = \vec{0} \quad \Rightarrow \quad \text{No directional preference, no asymmetry, no entropy increase} \]

Statement:
If SEP is projectionally balanced and shows no divergence:

\[ \sum_{k}{\vec{\varphi}_k} = \vec{0} \quad \wedge \quad \nabla \cdot \vec{\varphi} = 0 \]

Then the system is fully stable, has zero projectional drive, and is structurally saturated.
There is no need to formulate a separate third law.
Absolute equilibrium is not unattainable – it is structurally defined.

Definition D₁ — TRR–NOTIME Temperature

\[ T_{\mathrm{TRR}} \coloneqq \nabla \cdot \vec{\varphi} \]

Where:
\( \vec{\varphi} \) is the directional energy potential (SEP) [m²],
\( \nabla \cdot \vec{\varphi} \) is the spatial divergence of SEP, i.e., the net scalar outflow per unit volume at a given point in space.

Physical Interpretation:
\( T_{\mathrm{TRR}} \) represents the interactional openness of a configuration.
It is not related to molecular motion, entropy, or time.
It is defined as a purely structural scalar resulting from the spatial configuration of SEP.

Interpretation Table:

Statement:
Any directional SEP configuration that is physically present but creates no directional selectivity with respect to a given observer (i.e., its projection onto the observer’s interaction frame is zero) is latent to that observer.
This latent presence manifests structurally as dark energy.

\[ \vec{\varphi}_k \cdot \hat{n}_{\text{observer}} = 0 \;\;\Rightarrow\;\; \text{SEP}_i \text{ is latent} \;\;\Rightarrow\;\; \text{dark energy} \]

\[ \sum_{\text{latent}}{\vec{E}_{\mathrm{SEP}}} \neq 0 \;\;\Rightarrow\;\; \text{Observer perceives missing energy} \;\;\Rightarrow\;\; \text{dark energy} \]

Derivation:
Follows from:

• Axioms 0–2: existence of SEP structures,
• CT–1: rejection of time as causal,
• Axiom 7: interaction requires directional projection.

Conclusion:
Dark energy is not a separate substance, but a relative structural condition defined by the absence of projectional compatibility with the observer's frame of interaction.

Statement:
Dark matter is the observable manifestation of SEP configurations originating from the latent component of dark energy, which exhibit nonzero but incomplete projection selectivity. They are partially projected, yet not fully interacting.

\[ 0 < \vec{\varphi}_i \cdot \hat{n}_{\text{observer}} \ll \max\left(\vec{\varphi}_j\right) \;\;\Rightarrow\;\; \text{dark matter} \]

Where:

• \(\vec{\varphi}_i\) is the SEP vector of the configuration (e.g., φ–structure),
• \(\hat{n}_{\text{observer}}\) is the observer's direction of selectivity.

Derivation:
Follows from the same base as DE–1, with additional reference to Axiom A3 (projection threshold).

Conclusion:
Dark matter is a projection-shifted component of latent energy — not a separate class of matter, but a structurally intermediate state between full projection and latent silence.

Statement:
In the TRR–NOTIME framework, every apparent temporal evolution corresponds uniquely to a projectional differentiation in the SEP configuration space. Therefore, time can be formally redefined as a non-causal, non-dynamic ordering over discrete SEP layers. It is not a physical quantity, but a projectionally invariant structural relation.

Formal Definition:

• \({\vec{\varphi}}_k \coloneqq \frac{{\vec{E}}_k}{m}\) — SEP configuration in projection layer \(\Lambda_k\),
• Where \({\vec{E}}_k\) is directional energy, and \(m\) is structural mass reference.

\[ {\vec{\varphi}}_{k+1} \ne {\vec{\varphi}}_k \;\Longleftrightarrow\; \text{real physical change (interaction)} \\ {\vec{\varphi}}_{k+1} = {\vec{\varphi}}_k \;\Longleftrightarrow\; \text{no physical change} \]

Therefore:

\[ \frac{d\vec{\varphi}}{dt} \ne 0 \quad \text{(classical time evolution)} \quad \Longrightarrow \quad {\vec{\varphi}}_{k+1} \ne {\vec{\varphi}}_k \]

Implication:
The use of time as a variable \(t \in \mathbb{R}\) is structurally redundant and can be replaced by an ordering function over SEP projection layers:

\[ \text{Time} \coloneqq \text{order relation } \prec \text{ such that } \varphi_A \prec \varphi_B \iff \exists k: \varphi_A = \varphi_k, \; \varphi_B = \varphi_{k+1} \]

This ordering is:

• Transitive
• Reflexive
• Antisymmetric
• But non-causal, non-dynamic, and non-interactive

Conclusion:
This theorem provides the missing formal bridge in Axiom CT–1:
It does not negate time, but shows that time is structurally equivalent to layer indexation within SEP, and therefore not necessary for any physical explanation.

Clarification – Scientific Status of Time in TRR–NOTIME:
Unlike speculative claims that time does not exist, TRR–NOTIME provides a formal definition of what time is. Time is not dismissed but reclassified.

Time is a projectionally invariant relation between SEP layers.
It has a well-defined structure, but no generative power. It indexes differentiation — it does not cause it.
In this sense, TRR–NOTIME distinguishes between descriptive relations and causal physical entities. Time belongs to the former. SEP belongs to the latter.