# Temporal Theory of the Universe: Mathematical Foundations

Андрей Викторович Лемешко

This work presents the Temporal Theory of the Universe (TTU), a novel paradigm that unifies fundamental interactions through the internal structure of time. By redefining forces as acts of temporal regime alignment, TTU replaces the concept of field-based transmission with instantaneous realization at points of phase coincidence. The theory is operationalized through three parameters (Δτ, δτ, κ) and a field-theoretic Lagrangian, reproducing predictions from general relativity to nuclear binding energies. Crucially, TTU is falsifiable, providing clear experimental pathways for validation, from phase-locked energy transfer to precision tests of gravitational effects.

Temporal Theory of the Universe: Mathematical Foundations

Lemeshko Andriy

Doctor of Philosophy, Associate Professor

Taras Shevchenko National University of Kyiv, Ukraine

ORCID: 0000-0001-8003-3168

Abstract This work presents the Temporal Theory of the Universe (TTU), a novel paradigm that unifies fundamental interactions through the internal structure of time. By redefining forces as acts of temporal regime alignment, TTU replaces the concept of field-based transmission with instantaneous realization at points of phase coincidence. The theory is operationalized through three parameters (, , ) and a field-theoretic Lagrangian, reproducing predictions from general relativity to nuclear binding energies. Crucially, TTU is falsifiable, providing clear experimental pathways for validation from phase-locked energy transfer to precision tests of gravitational effects.

Keywords

TTU, Temporal Theory of Unification, TTG, TTN, TTEM, TTT, , , , temporal ontology, phase conjugation, wave-free transmission, energy teleportation, quantum foundations, unified interactions.

Сontent

• Introduction

• Core Ideas

• 2.1 Fields of Temporal Structure

• .2 Origin of Parameters and

• 2.3 TTU Index

• 2.4 Gravity as Effective Synchrony (TTG)

• 2.5 Nuclear Autoconjugation (TTN)

• 2.6 Electromagnetism as Phase Coupling (TTEM)

• 2.7 Wave-Free Teleportation (TTT)

• 2.8 Quantization Roadmap

• 2.9 Toward the Standard Model

• 2.10 Experimental Handles and Falsifiability

• 2.11 Notation and Intuition at a Glance

• Mathematical Foundation of TTU: Field Formalism and Variational Principle

• 3.1 Action and Lagrangian

• 3.2 EulerLagrange Equations

• 3.3 Effective Metric and Gravitational Limit

• 3.4 Canonical Momenta and Hamiltonian

• 3.5 Quantization Roadmap

• Phenomenology and Experimental Predictions

• 4.1 Parameters Recap

• 4.2 TTG: Observable Effects

• 4.3 TTN: Binding and Stability

• 4.4 TTEM: Phase Coupling

• 4.5 TTT: Teleportation Correlations

• 4.6 TTU Index

• Summary Table: Experimental Predictions

• Discussion and Future Work

• 6.1 Conceptual Implications

• 6.2 Relation to Existing Theories

• 6.3 Open Questions

• 6.4 Experimental Roadmap

• 6.5 Future Work

• Open Theoretical Extensions and Calibration Pathways

• 7.1 Calibration of Potentials

• 7.2 UV Behavior and Planckian Regime

• 7.3 Topological Sectors of Time

• 7.4 TTU and Consciousness (TTO)

• 7.5 Quantization Beyond Canonical

• Conclusion and Outlook

• 8.1 Summary of Contributions

• 8.2 Achievements of v.3

• 8.3 Future Directions

• 8.4 Outlook

• Reference

• Appendixs

1. Introduction

What if time is not merely a parameter that clocks advance through, but the medium of realization the substance within which events come into being? In the Temporal Theory of the Universe (TTU), reality is not passed along by carriers traveling through fields; it flashes into realization wherever temporal regimes align. Space, fields, and even familiar forces are then not primary objects but secondary patterns carved by the internal structure of time.

This shift of perspective reframes several pillars of physics:

• Gravity (TTG) becomes ultimate synchrony the tendency of temporal layers to compress and align near mass-energy.

• Nuclear forces (TTN) appear as auto-conjugation self-sustained, ultra-dense realizations in narrow temporal windows.

• Electromagnetism (TTEM) is read as phase coupling energy transfer as an act of phase alignment rather than wave transport.

• Teleportation (TTT) stops being fiction: information and, in principle, energy re-emerge wave-free at distant nodes when temporal alignment is established.

Three operational parameters organize this ontology:

• Transition tension how hard it is to switch temporal regimes.

• Realization window the thickness of the temporal layer where an event is realized.

• Synchrony an effective permeability of the temporal medium.

Intuitively, narrow windows ( ), high synchrony ( ), and well-tuned phase () produce intense realization the signature of strong interactions and coherent transfer.

.

2. Core Ideas 2.1) Fields of temporal structure

We introduce two macroscopic fields that describe the state of time rather than of space:

• (x) a temporal density (or pressure) field controlling how thick or thin the realization window tends to be.

• (x) a phase field tracking the position within a local temporal cycle; its rate encodes phase evolution.

A minimal, Lorentz covariant effective density (to be calibrated later) reads:

(1)L = "A()"" + "B()"" U() W(, )

Here U() shapes windows of realization; W(, ) governs phase locking and synchrony.

Canonical potentials (phenomenology mechanism) A concrete, falsifiable starting point is:

(2)U() = "m²"( )² + "( ) + W(, ) = f()"[1 cos]

with:

(3)f() = g + g"( ) + 2"g"( )² +

• Minima of U() at or near * identify narrow realization windows (TTN wells).

• The phase term W(, ) prefers = 2"n and penalizes detuning, quantifying phase synchronization (TTEM).

Interpretation: Equation (2) sets where the temporal medium likes to realize events; Equation (3) sets how strongly it locks phases there.

2.2) Where do and come from?

In v.3 we tie the phenomenological parameters to the curvatures of the potentials at the operative point *. Define reference scales _U, _W > 0 (set by calibration to TTG/TTN data):

(4) _U² " U() W " W(, 0) |{}

• measures how stiff the temporal density well is (narrower, steeper U() higher synchrony/throughput).

• measures how effectively phase locking converts temporal pressure into observable transfer (force/energy).

These dimensionless coefficients now have a microstructural origin (via U() and W(, )) while remaining experimentally calibratable.

2.3) The TTU index: when realization becomes robust

To summarize how realizable a regime is, we use a bounded phase alignment factor:

(5)() cos²(2)() [0, 1]

and define a dimensionless TTU index:

(6)I_TTU ( / ) " " " ()

where is a reference thickness.

Regimes with I_TTU 1 self-sustain coherent realization (TTN, TTG, conscious TTO), while I_TTU 1 correspond to dilute, fragile phases (photons, neutrinos).

2.4) Gravity as effective synchrony (TTG)

Instead of postulating geometric curvature from the outset, TTU derives an effective metric from gradients of the temporal fields:

(7)g^eff_{} = {} + "" + ""_ +

with , fixed by calibration to classical tests (perihelion, redshift). In regions where shrinks and peaks, geodesics of g^eff_{} reproduce gravitational phenomena as flows toward ultimate temporal synchrony.

A useful operational relation is the temporal gradient:

(8)_ _r

which captures how realization thickness changes spatially. Empirically, free fall acceleration tracks the pressure drop of time:

(9)g "c²"_

with a scale factor fixed so that Eq.(9) matches GM / r² in weak fields. In v.3, Eq.(9) is not a postulate but a limit of geodesic motion in g^eff_{}.

2.5) Nuclear auto conjugation (TTN)

In TTU, nuclear binding corresponds to realization in ultra-narrow windows near a minimum of U(). A compact estimator of auto conjugation strength is:

(10)_auto _min " " ()

which explains why dense, phase-aligned configurations are extraordinarily stable.

The energetic scale follows the temporal uncertainty:

(11)E_bind / _min

Together, Eqs.(10)(11) connect binding systematics to how thinly time can be realized in the nuclear regime.

2.6) Electromagnetism as phase coupling (TTEM)

If electromagnetism is a phase alignment process, then the effective current should be proportional to phase flow weighted by temporal permeability:

(12)J^ " " f() " _

with f() from Eq.(3). Energy transfer is then not a wave marching through space but a coincident realization at phase-synchronized nodes.

In practice, the observable burst scales like:

(13)R(t) (, ) "

peaking when windows are thin and phases match.

2.7) Wave-free teleportation (TTT)

For two nodes A and B that share a prepared temporal alignment, the correlation of outcomes obeys:

(14)P_corr(, ) = sync " exp[()² / (2"²)] " cos²(2")

where = _B _A and = _B _A.

Equation(14) predicts how robustly information reappears at B as a function of frequency detuning and phase mismatch a direct experimental handle.

2.8) Quantization roadmap

The field formalism in Eq.(1) is designed for canonical and path integral quantization. Two physically relevant regimes:

• Infrared (IR): small fluctuations around * and = 0 quasi-free quanta of temporal density and phase, testable as corrections to TTEM and TTG.

• Ultraviolet (UV): near-Planckian limit 0 TTT domain with strong, non-Gaussian fluctuations; here, operator ordering and non-perturbative effects can be probed by phase teleportation experiments (cf. Eq.(14)).

We will specify canonical momenta and constraints in the Mathematical Foundation section.

2.9) Toward the Standard Model

Coupling TTU to matter suggests minimal temporal coupling:

(15)L_SM+TTU = L_SM + y_"" + y_""^ + ()"F{}"F^{} +

where denotes Standard Model fermions and F_{} the electromagnetic field strength. The terms encode how temporal density and phase flow modulate masses, currents, and gauge propagation.

All couplings y_, y_, and () are testable via precision spectroscopy and astroparticle bounds.

2.10) Experimental handles and falsifiability

TTU makes concrete, checkable statements:

• Gravitational redshift & perihelion: reproduced via g^eff_{} (Eq.(7)) with a transparent calibration path.

• Binding energies & nuclear systematics: scale with _min (Eq.(11)) and alignment factor _auto (Eq.(10)).

• Wave-free correlations: frequency/phase-dependent P_corr (Eq.(14)) offers a laboratory protocol to bound _sync, _.

• EM transfer without carriers: bursts R(t) (Eq.(13)) under phase locking provide a direct TTEM signature.

• Anomalous thrust / pressure of time: gradients _ (Eq.(8)) tie mechanical responses to temporal cavitation.

2.11) Notation and intuition at a glance

• how thick is the layer where reality happens right now?

• how permeable is time to forming links across layers?

• , where are we in the temporal cycle, and how fast do we move along it?

• how pressurized is time, biasing realization to be thin or thick?

If these quantities are aligned (high , thin , locked , stable ), the world does not need to send anything: it realizes the same pattern there as here.

## 3.Mathematical Foundation of TTU: Field Formalism and Variational Principle (with Explanations)

3.1) Action and Lagrangian density

The Lagrangian density is given by:

(16)L = "A()""^ + "B()""^ U() W(, )

Interpretation:

• The first two terms represent the kinetic energy of two fields:

• temporal density (thickness of the realization window)

• temporal phase state

• The potentials U() and W(, ) define where and how events are realized an energy landscape for time.

The corresponding action is:

(17)S[, ] = L " -(g)dx

Interpretation: The action S integrates the dynamics over all spacetime. Minimizing S yields the equations of motion.

3.2) EulerLagrange equations

The general principle of variational dynamics yields:

(18)(L / (X)) L / X = 0

Interpretation: Variation of the action equals zero gives the field equations for each variable.

For :

(19)[A()"^] + "A()""^ + "B()""^ + U() + W(, ) = 0

Interpretation:

• First term: wave-like propagation of temporal density

• Remaining terms: how potentials and phase effects influence the thickness of the realization window

For :

(20)[B()"^] + W(, ) = 0

Interpretation: The phase tends to synchronize (minimize W) but can flow through gradients.

3.3) Effective metric and gravitational limit

The effective metric is defined as:

(21)g^eff_{} = {} + "" + ""_

Interpretation: The metric depends on gradients of temporal fields gravity emerges as an effect of temporal pressure.

The gravitational potential is approximated by:

(23)_grav - 2"()² + 2"()²

Interpretation: Gravitational potential is energy stored in temporal gradients, not an a priori curvature of space.

3.4) Canonical momenta and Hamiltonian

The canonical momenta are defined as:

(24) = A() " = B() "

Interpretation: Momenta show how fast temporal density and phase evolve.

The Hamiltonian is given by:

(25)H = " + " L

Interpretation: The Hamiltonian represents the energy of time in terms of its density and phase.

3.5) Quantization roadmap

In the quantum version, temporal density and phase become operators. The canonical commutation relations are:

(27)[(x),(y)] = i""(x y) [(x),(y)] = i""(x y)

Interpretation: Temporal density and phase become operator-valued fields possible quanta of time.

4.1) Phenomenological Parameters Recap

• thickness of the realization window (how thick time is locally)

• synchrony coefficient (temporal permeability)

• phase-locking efficiency

• () = cos²( / 2) phase alignment factor

These parameters control the intensity and stability of realization across TTU domains.

4.2) Gravity (TTG): Observable Effects

Gravitational acceleration is proportional to the spatial gradient of temporal thickness:

(29)g "c²" _r

Interpretation: Stronger pressure drop of time stronger gravity.

The effective metric is defined as:

(30)g^eff_{} = {} + "" + ""_

Interpretation: Gravity emerges as a deformation of the effective metric caused by temporal gradients, not as a fundamental curvature of space.

4.3) Nuclear Auto-Conjugation (TTN)

Auto-conjugation strength and nuclear binding energy are governed by:

(31)_auto = " () / _minE_bind / _min

Interpretation:

• Auto-conjugation strength _auto grows as the realization window narrows (_min ) and phase alignment improves (() ).

• Binding energy E_bind scales inversely with minimal temporal thickness explains nuclear stability in ultra-aligned regimes.

4.4) Electromagnetism (TTEM): Phase Coupling

Electromagnetic transfer is governed by phase flow and temporal permeability:

(32)J^ " " f() " _R(t) (, ) "

Interpretation: Electromagnetic transfer is a phase-driven process: energy appears at synchronized nodes without wave propagation.

4.5) Wave-Free Teleportation (TTT)

The correlation between two temporally aligned nodes is governed by:

(33)P_corr(, ) = sync " exp[()² / (2"²)] " cos²(2")

Interpretation: Correlation strength depends on phase mismatch () and frequency detuning (). Equation(33) predicts measurable thresholds for teleportation-like effects without wave propagation via synchronized temporal windows.

4.6) TTU Index for Realization Robustness

A dimensionless index quantifies the stability of a realization regime:

(34)I_TTU = ( " " () " ) /

Interpretation:

• I_TTU 1 indicates strong, self-sustained realization typical of nuclear and gravitational domains.

• The index combines synchrony (), phase-locking (), alignment (()), and local thickness () relative to a reference scale .

5. Summary Table: Experimental Predictions


Domain


Observable


TTU Prediction


Test Method


Gravity (TTG)


Redshift, perihelion shift


Matches GR viag^eff_{}calibration


Astrophysical observations


Nuclear (TTN)


Binding energy vs.


E_bind / _min


Nuclear spectroscopy


EM (TTEM)


Phase-locked energy bursts


R(t) "


High-Q resonator experiments


Teleportation (TTT)


Correlation vs.,


Eq.(33): Gaussian in, cosine in


Quantum optics / phase-locking setups


Pressure of Time


Anomalous thrust vs._


g _


Precision accelerometry


## 6.Discussion and Future Work

## 6.1) Conceptual Implications

TTU reframes the ontology of physics:

• Time as primary: Instead of being a passive parameter, time becomes the active substrate of realization.

• Fields of temporal structure: Gravity, nuclear forces, and electromagnetism emerge as different regimes of temporal alignment rather than separate fundamental interactions.

• No carriers, no waves: Information and energy transfer occur through phase synchronization, not through propagation in space.

Interpretation:

If TTU is correct, the unification problem is not about adding spatial dimensions but about understanding the internal degrees of freedom of time.

6.2) Relation to Existing Theories

• General Relativity: TTU reproduces GR predictions in the weak-field limit via the effective metric g^eff_{} (see Eq.(30)).

• Quantum Mechanics: Phase alignment in TTU parallels quantum coherence, but without requiring a probabilistic ontology.

• Quantum Gravity: TTU suggests that quantization of temporal fields (, ) may resolve singularities by replacing geometric collapse with temporal thinning.

6.3) Open Questions

• Calibration: How can the functions A(), B(), U(), and W(, ) be fixed from empirical data? What observables constrain their form?

• UV Behavior: Does TTU remain predictive near Planck scales, or must additional constraints (e.g. non-perturbative regularization or topological protection) be introduced?

• Topology of Time: Are there nontrivial phase sectors analogous to instantons that could explain dark energy, inflation, or vacuum selection?

• Consciousness Link: Can TTU provide a physical basis for temporal coherence in neural systems (TTO domain)? Is there a measurable signature of phase alignment in cognitive processes?

6.4) Experimental Roadmap

• Short-term:

• High-Q resonator tests for phase-locked EM bursts (TTEM)

• Quantum optics experiments to verify the correlation law (see Eq.(33))

• Mid-term:

• Precision accelerometry to detect pressure of time anomalies (see Eq.(29))

• Long-term:

• Nuclear spectroscopy to probe -scaling of binding energy (see Eq.(31))

• Astrophysical tests for deviations in the effective metric (see Eq.(30))

6.5) Future Work

• Develop numerical simulations of TTU field equations (see Eqs.(19)(20)) for realistic physical and cosmological scenarios.

• Explore quantization schemes beyond canonical formalism including path integrals and lattice TTU to probe non-perturbative regimes.

• Investigate coupling to the Standard Model (see Eq.(15)) for observable deviations in particle physics, especially in mass generation and gauge propagation.

• Publish benchmark datasets for TTU parameter estimation, enabling reproducible calibration and cross-domain comparison.

## 7.Open Theoretical Extensions and Calibration Pathways

7.1) Empirical Calibration of TTU Potentials

While the canonical forms of the potentials U() and W(, ) are specified, their coefficients m_, _, g, g, g remain to be calibrated. A proposed roadmap:

• TTG calibration: Fit , in the effective metric g^eff_{} using perihelion shift and gravitational redshift data.

• TTN calibration: Use nuclear binding energies and isotope systematics to constrain _min, , and the curvature U()*.

• TTEM calibration: Extract f() from high-Q resonator experiments measuring phase-locked energy bursts.

This empirical anchoring would allow TTU to move from phenomenological to predictive enabling falsifiable modeling across gravitational, nuclear, and electromagnetic domains.

7.2) Ultraviolet Behavior and Planckian Regime

TTU predicts strong non-Gaussian fluctuations in the limit 0, relevant for TTT and early-universe physics. This regime probes the deepest structure of time and its coupling to matter.

Open questions:

• Does the TTU field theory remain renormalizable near Planck scales, or does it require new regularization principles?

• Are there natural cutoffs or self-regularizing mechanisms in the temporal domain e.g., saturation of (, ) or bounded curvature in U()?

• Can TTU resolve classical singularities (e.g., black holes, Big Bang) via temporal thinning rather than geometric collapse?

Numerical simulations of Eqs.(19)(20) in high-gradient regimes are essential to explore these questions. Such simulations may reveal emergent stability, chaotic transitions, or novel attractors in the 0 limit.

7.3) Topological Sectors of Time

The phase field (x) admits nontrivial configurations analogous to instantons or solitons. These topological structures may encode deep cosmological and physical phenomena.

Hypotheses:

• Temporal instantons: Localized phase jumps in (x) that could seed inflation or contribute to dark energy via abrupt shifts in realization density.

• Phase vortices: Persistent temporal misalignments that behave like dark matter storing energy in rotational phase sectors without direct coupling to standard fields.

• Global phase sectors: Large-scale configurations of (x) that may encode cosmological boundary conditions or multiverse-like domains, potentially influencing the topology of spacetime itself.

These structures could be probed via the topology of the potential W(, ) and its minima especially in regimes where _ and _ interact nonlinearly.

7.4) TTU and Temporal Coherence in Conscious Systems (TTO)

TTU suggests that robust realization characterized by high I_TTU may underlie coherent cognitive states. This opens a speculative but testable bridge between physics and consciousness.

Proposed model:

• Neural ensembles function as localized TTU domains, exhibiting synchronized (x) and minimized , forming temporally coherent substrates.

• Conscious episodes correspond to transient peaks in I_TTU, sustained by recursive feedback loops across neural layers and phase-locked cortical regions.

• Experimental test: Correlate EEG phase coherence with TTU-index-like metrics derived from neural timing, such as local variability and phase alignment ().

This framework invites a new class of neurophysical experiments, where temporal topology and realization dynamics become measurable correlates of awareness.

## 7.5. Quantization Beyond Canonical

While canonical quantization is outlined (Eq. 27), TTU may benefit from:

• Path integral formulation: to capture nonperturbative effects in TTT.

• Lattice TTU: discretized simulations of temporal fields for cosmology and condensed matter analogs.

• Operator algebra: explore commutation relations in curved temporal backgrounds.

These approaches could reveal new quantum regimes of time itself.

## 8.Conclusion and Outlook

8.1) Summary of Core Contributions

The Temporal Theory of Unification (TTU) introduces a paradigm shift in foundational physics and ontology:

• Time as the primary substrate: Physical reality emerges through the structured realization of time , , rather than through static spatial geometry. Space is not fundamental; it is a derivative of temporal coherence.

• Unified interpretation of forces: Gravity (TTG), nuclear forces (TTN), and electromagnetism (TTEM) are interpreted as distinct regimes of temporal alignment governed by gradients, phase-locking, and realization thickness.

• Predictive framework: TTU provides testable equations (Eqs.2934) linking temporal parameters (, , , ) to observable phenomena including gravitational redshift, nuclear binding energy, phase-driven energy transfer, and teleportation-like correlations.

## 8.2) Key Achievements of v.3

• Established field formalism with Lagrangian (Eq.16) and effective metric (Eq.30).

• Defined phenomenological parameters and TTU index (Eq.34) for experimental calibration.

• Proposed falsifiable predictions for gravity, nuclear binding, EM phase coupling, and teleportation-like correlations.

8.3) Future Directions

TTU opens a multidimensional roadmap for theoretical, experimental, and interdisciplinary expansion:

## Mathematical Development:

• Full quantization of TTU fields (, ) and rigorous renormalization analysis to assess UV completeness.

• Exploration of topological sectors (instantons, vortices, global phase domains) and their implications for cosmology, dark energy, and vacuum structure.

## Experimental Validation:

• High-precision tests of phase-locked EM bursts (TTEM) in resonator systems.

• Quantum optics experiments to verify the correlation law (see Eq.(33)) across varying and .

• Astrophysical constraints on effective metric deviations (see Eq.(30)) via gravitational lensing, redshift, and perihelion data.

## Interdisciplinary Links:

• Potential connection to consciousness studies via temporal coherence in neural ensembles (TTO domain).

• Applications in quantum information and teleportation protocols, leveraging TTUs non-wave phase transfer mechanisms.

## 8.4) Outlook

If TTU withstands empirical scrutiny, it could redefine the foundations of physics by replacing the geometry-centric worldview with a temporal ontology. This shift would not only unify gravity and quantum theory but also open pathways to technologies based on phase synchronization of timefrom energy transfer without carriers to robust quantum communication.

## 9.References

## Foundational Works on Temporal Physics (Kozyrev and Successors)

• Kozyrev, N. A. (1971). Time as a Physical Factor. Astronomical Herald, 7(3), 2327. Retrieved from http://elib.gnpbu.ru/text/kozyrev_vremya-kak-faktor_1971/go,0/ (Accessed: August 10, 2025).

• Kozyrev, N. A. (1977). Astronomical Observations by Means of the Physical Properties of Time. In Flaring Stars: Proceedings of the Symposium, Byurakan, October 58, 1976 (pp. 209227). Yerevan: Academy of Sciences of the Armenian SSR. Retrieved from https://djvu.online/file/NGh6VKtGHHN3j (Accessed: August 10, 2025).

• Kozyrev, N. A. (1971). Time as a Physical Phenomenon. Leningrad: GPNTB. Retrieved from https://nkozyrev.ru/bd/130.pdf (Accessed: August 10, 2025).

• Miroshnikov, A. N. (1985). Temperature Anomalies of Mass. Journal of Experimental and Theoretical Physics, (4), 112118. Retrieved from http://elib.gnpbu.ru/text/miroshnikov_temperaturnye-anomalii-massy_1985/go,0/ (Accessed: August 10, 2025).

• Anonymous. (1987). Experimental Detection of Entropy Gradient. Reports of the Academy of Sciences of the USSR, 297(4), 865868.

• Anonymous. (1991). Entropic Aspects of Symmetry in Non-Equilibrium Processes. In Problems of Universe Exploration (Issue 15, pp. 4559). Leningrad: Nauka.

• Anonymous. (1991). Change in Gyroscope Weight under Vibrations. Technology for the Youth, (89), 1214.

• Batsiev, V. A. (Ed.). (2008). Time and Stars: To the 100th Anniversary of N. A. Kozyrev. St. Petersburg: Asterion. Retrieved from https://archive.org/details/kozyrev (Accessed: August 10, 2025).

## Contemporary Development: Temporal Theory of Unification (TTU)

• Lemeshko, A. (2025). TTU: Temporal Unification Theory. Zenodo. https://doi.org/10.5281/zenodo.16732254

• Lemeshko, A. (2025). TTE: Temporal Theory of Everything. ResearchGate. https://doi.org/10.13140/RG.2.2.35468.83847

• Lemeshko, A. (2025). TTG: Temporal Theory of Gravitation. Zenodo. https://doi.org/10.5281/zenodo.16044168

• Lemeshko, A. (2025). TTU and the Enigmas of Black Holes. ResearchGate. https://doi.org/10.13140/RG.2.2.25445.10726

• TTU-Group Repository. (2025). All materials, preprints, and data. Retrieved from https://zenodo.org/communities/ttg-series (Accessed: August 10, 2025).

## Additional Literature (Fundamental and Related Works)

• Rovelli, C. (2018). The Order of Time. Riverhead Books.

• Wheeler, J. A., & Feynman, R. P. (1945). Interaction with the Absorber as the Mechanism of Radiation. Reviews of Modern Physics, 17(23), 157181. https://doi.org/10.1103/RevModPhys.17.157

• Aharonov, Y., & Bohm, D. (1959). Significance of Electromagnetic Potentials in the Quantum Theory. Physical Review, 115(3), 485491. https://doi.org/10.1103/PhysRev.115.485

• Dirac, P. A. M. (1958). The Principles of Quantum Mechanics (4th ed.). Oxford University Press.

• Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman.

• t Hooft, G. (2000). The Holographic Principle. arXiv:hep-th/0003004

• Verlinde, E. (2011). On the Origin of Gravity and the Laws of Newton. Journal of High Energy Physics, 2011(4), 29. https://doi.org/10.1007/JHEP04(2011)029

10. Appendix

Appendix A. TTU Equations and Interpretations


Formula


Interpretation / Context


(1)


L = "A()"" + "B()"" U() W(, )


Effective Lagrangian of TTU


(2)


U() = "m_²"( )² + _"( ) +


Temporal density potential


W(, ) = f()"[1 cos ]


Phase coupling potential


(3)


f() = g + g"( ) + 2"g"( )² +


Tunable coupling function


(4)


_U² " U(), W " W(, 0)


Synchronization parameters


(5)


() cos²(2)


Phase alignment factor


(6)


I_TTU ( / ) " " " ()


TTU realization index


(7)


g^eff_{} = _{} + "" + ""


Effective metric


(8)


_ _r


Gradient of realization thickness


(9)


g "c²"_


Gravitational acceleration


(10)


_auto _min " " ()


Auto-coupling strength


(11)


E_bind / _min


Binding energy in nuclear regime


(12)


J^ " " f() " _


Electromagnetic current via phase


(13)


R(t) (, ) "


Energy release from phase coupling


(14)


P_corr(, ) = sync " exp[()² / (2"²)] " cos²(2")


Teleportation correlation function


(15)


L_SM+TTU = L_SM + y_"" + y_""^ + ()"F_{}"F^{} +


Minimal coupling to the Standard Model


(16)


L = "A()""^ + "B()""^ U() W(, )


Lagrangian for variational principle


(17)


S[, ] = L " -(g) dx


TTU action


(18)


(L / (X)) L / X = 0


EulerLagrange equation


(19)


[A()"^] + "A()""^ + "B()""^ + U() + _ W = 0


Equation of motion for


(20)


[B()"^] + _ W(, ) = 0


Equation of motion for


(21)


g^eff_{} = _{} + "" + ""


Effective metric (repeated)


(23)


_grav - 2"()² + 2"()²


Gravitational potential


(24)


= A() " , = B() "


Canonical momenta


(25)


H = " + " L


Temporal Hamiltonian


(27)


[(x), (y)] = i""(x y),[(x), (y)] = i""(x y)


Quantum commutation relations


(29)


g "c²"_


Gravitational acceleration (repeated)


(30)


g^eff_{} = _{} + "" + ""


Effective metric (repeated)


(31)


_auto = " () / _min,E_bind / _min


Auto-coupling and binding energy (repeated)


(32)


J^ " " f() " _,R(t) (, ) "


Phase current and energy release (repeated)


(33)


P_corr(, ) = sync " exp[()² / (2"²)] " cos²(2")


Teleportation correlation (repeated)


(34)


I_TTU = ( " " () " ) /


TTU realization index (repeated)


Appendix B: Glossary of TTU Terms and Symbols


Symbol / Term


Meaning / Interpretation


(Theta)


Temporal density field; encodes the local thickness of realization


(phi)


Phase field; governs alignment and coherence across temporal domains


Realization thickness; minimal temporal interval for physical manifestation


(kappa)


Curvature parameter derived from second derivative of U(); governs temporal stiffness


(xi)


Phase coupling strength; derived from gradient of W(, ) with respect to


()


Phase alignment factor; typically defined as cos²(2)


I_TTU


TTU realization index; composite measure of coherence, curvature, and thickness


A(), B()


Kinetic coefficients for and fields; modulate propagation and interaction strength


U()


Temporal potential; defines preferred values and stability of


W(, )


Phase coupling potential; governs interaction between and


f()


Modulation function within W(, ); tunable via empirical calibration


g, g, g


Coefficients in f(); control linear and nonlinear phase coupling


,


Metric coupling constants; determine how and affect spacetime geometry


g^eff_{}


Effective metric; modified spacetime geometry induced by TTU fields


_


Gradient of ; source of gravitational acceleration in TTG


_auto


Auto-coupling strength; internal force from phase alignment and realization thickness


E_bind


Binding energy; inversely proportional to _min in nuclear regime


J^


Electromagnetic current derived from phase gradients


R(t)


Energy release rate from phase-locked transitions


P_corr(, )


Correlation probability for teleportation-like effects


L_SM+TTU


TTU-extended Lagrangian including coupling to Standard Model fields


,


Canonical momenta for and fields


S[, ]


TTU action; integral over spacetime of the TTU Lagrangian


TTG


Temporal Theory of Gravitation


TTN


Temporal Theory of Nuclear interactions


TTEM


Temporal Theory of Electromagnetism


TTT


Temporal Theory of Topology and early-universe physics


TTO


Temporal Theory of Ontology and consciousness


## Appendix С: TTU Parameters Units and Dimensions


Symbol


Name / Description


SI Unit


Physical Dimension


Temporal density field


[unitless or s]


[T] or [1]


Phase field


[radian]


[1] (dimensionless angle)


Realization thickness


[second]


[T]


Reference realization thickness


[second]


[T]


Temporal curvature parameter


[second²]


[T²]


Phase coupling strength


[second]


[T]


()


Phase alignment factor


[unitless]


[1]


I_TTU


TTU realization index


[unitless]


[1]


A()


Kinetic coefficient for


[second²]


[T²]


B()


Kinetic coefficient for


[second²]


[T²]


U()


Temporal potential


[joule]


[ML²T²]


W(, )


Phase coupling potential


[joule]


[ML²T²]


f()


Modulation function in W


[joule]


[ML²T²]


g, g, g


Coupling coefficients in f()


[joule], [J"s], [J"s²]


[ML²T²], [ML²T], [ML²T]


,


Metric coupling constants


[meter²"second²]


[L²T²]


g^eff_{}


Effective metric tensor


[unitless]


[1]


_


Gradient of realization thickness


[second"meter]


[TL]


Mass density


[kg"m]


[ML]


g


Gravitational acceleration


[m"s²]


[LT²]


_auto


Auto-coupling strength


[unitless or N]


[1] or [MLT²]


E_bind


Binding energy


[joule]


[ML²T²]


J^


Electromagnetic current


[ampere]


[I]


R(t)


Energy release rate


[watt]


[ML²T]


Phase difference


[radian]


[1]


Frequency difference


[radian"s]


[T]


_


Frequency spread


[radian"s]


[T]


P_corr


Correlation probability


[unitless]


[1]


,


Canonical momenta


[joule"second]


[ML²T]


H


Hamiltonian


[joule]


[ML²T²]


S[, ]


Action


[joule"second]


[ML²T]