Abstract

TTG v1.1 extends the minimal anchor core (v1.0) by introducing nonlinear, even-parity coupling functionsA(), such as:

Series at -0: () = tanh() = ' (/3) + O(); A() = cosh() = 1 + ('/2) ' + (/24) + O().

(A.1)A() = cosh(")orA() = exp("'2M')

The scalar-matter coupling is governed by the derivative:

(A.2)() d(lnA)/d

In the small-field limit, () admits an even-parity expansion:

(A.3)() = " + " + ...where = ', = 3(forA() = cosh("))

This structure enables environmental screening: in high-density regions (e.g., Solar System), 0 implies:

(A.4)() - 0suppressed fifth forces

The core formalism remains unchanged; the extension lies solely in the functional form ofA(), ensuring full backward compatibility:

(A.5)() = constTTG v1.1 TTG v1.0

To characterize the spatial range of scalar-mediated interactions, we define the Yukawa interaction range:

(A.6)_ 1m_(Yukawa interaction range)

We also provide explicit parameter ranges, compare TTG with other screening mechanisms, and discuss quantum stability.

Keywords

Time; time density; time flow; time pressure; time vortex; temporal theory; TTG; TTE; electrostatics; gravity; quantum nonlocality; CP violation; muonic hydrogen; alpha decay; regularization; substance; causality; ontology; quantum gravity; experimental verification

Units & Conventions

We work in natural units = c = 1 throughout the theoretical formulation (Sections 2--3, 5, 7--9). In phenomenological, postNewtonian, and gravitationalwave expressions (Sections 4, 6, 10), constants c and are reinstated explicitly to maintain SI units. The Newtonian potential has dimensions [] = m'"s', and its dimensionless postNewtonian counterpart is defined as U /c'. The scalar mass m_ sets a Yukawa interaction range _ /(m_ c) in SI units, which reduces to _ = 1/m_ in natural units.

Table of Contents

1. Introduction: From Anchor to Adaptation
2. Screening Mechanism: Coupling Functions and ()
   2.1 Comparison with Other Screening Mechanisms
3. Field Equations and Dynamics
4. Newtonian and Post-Newtonian Limits
5. Cosmology and Large-Scale Dynamics
6. Gravitational Waves and Dipole Radiation
   6.1 Screening Behavior
7. Falsifiable Predictions and Signal Table
8. Parameter Space and Priors
   8.1 Core Parameters
   8.2 Scalar Coupling Behavior
   8.3 Priors for Screening Consistency
   8.4 Explicit Parameter Bounds
   8.5 Environmental Regimes
   8.6 Scalar-Mediated Force
   8.7 Constraints and Estimation
9. Quantum Corrections and Theoretical Stability
   9.1 Symmetry Protection
   9.2 Radiative Stability
   9.3 UV Completion
   9.4 Summary of Stability Features
10. Comparison with Dark Matter and MOND
    10.1 Galactic Rotation Curves
    10.2 Effective Mass Interpretation
    10.3 Comparison with MOND
    10.4 Applicability and Limitations
    10.5 Hybrid Interpretation Table
11. Conclusion: TTG as a Living Framework

References

Appendix A. Comparison with TTG v1.0

1. Introduction: From Anchor to Adaptation

TTG v1.0 established a minimal, falsifiable framework based on universal linear coupling between a scalar field and matter, governed by a constant parameter . This formulation yielded clean predictions across cosmological and gravitational domains, but faced tension with precision Solar System tests (e.g., MICROSCOPE, Cassini), where fifth-force constraints demand near-zero scalar coupling.

TTG v1.1 addresses this by introducing nonlinear, even-parity coupling functions A(), such as:

(1.1)A() = cosh(")orA() = exp("'2M')

The scalar-matter interaction is controlled by the derivative:

(1.2)() d(lnA)/d

which admits a small-field expansion:

(1.3)() = " + " + ...with = ', = 3(for A() = cosh("))

This structure enables density-dependent screening: in high-density environments (e.g., Solar System), 0 implies:

(1.4)() - 0suppressed fifth forces

(*The Yukawa interaction range _ 1/m_ is defined in the Units & Conventions block above.*)

The extension preserves the effective field theory (EFT) framework and opens new phenomenological windows: Galactic rotation curves (unscreened regime) Asymmetric gravitational wave binaries (dipole radiation) Hierarchical falsifiability across scales

TTG v1.1 retains the core formalism of v1.0 and ensures full backward compatibility:

(1.6)() = constTTG v1.1 TTG v1.0

2. Screening Mechanism: Coupling Functions and ()

To enable density-dependent screening, TTG v1.1 introduces nonlinear, even-parity coupling functionsA(), such as:

(2.1)A() = cosh(")orA() = exp("'2M')

The scalar-matter interaction strength is governed by the derivative:

(2.2)() d(lnA)/d

For even-parity couplings, () admits a small-field expansion:

(2.3)() = " + " + ...with = ', = 3(for A() = cosh("))

This structure implements the screening principle:

• High-density environments (e.g., Solar System): 

(2.4) - 0() - 0suppressed fifth force Consistent with MICROSCOPE, Cassini constraints

• Low-density environments (e.g., galactic, cosmological scales): 

(2.5) 0() 0observable deviations from GR

To formalize the screening condition:

(2.6)lim0() = 0(screening condition)

This mechanism allows TTG v1.1 to remain active on large scales while automatically suppressing deviations in regions of high matter density.

2.1 Comparison with Other Screening Mechanisms

Unlike chameleon or Vainshtein mechanisms, which rely on fine-tuned potentials or higher-derivative interactions, TTG screening arises naturally from the even-parity couplingA(). This avoids need for environment-dependent mass or strong coupling scales, simplifying the theoretical structure.

3. Field Equations and Dynamics

The dynamical structure of TTG v1.1 preserves the Einstein equations from v1.0. The modification enters solely through the scalar coupling function(), replacing the constant.

• Einstein equations: Unchanged from TTG v1.0; the metric sector remains governed by:

(3.1)G_ = 8G"(T_ + T_^)

• Scalar field equation: The scalar fieldobeys a sourced Klein--Gordon-type equation:

(3.2) V() = ()"T

whereT T^_is the trace of the matter energy-momentum tensor.

• Energy-momentum exchange: Non-conservation ofT_due to scalar coupling:

(3.3)_T^ = ()"T"^

This reflects the exchange of energy-momentum between matter and the scalar field.

• Scalar field stress-energy tensor: The contribution of to the total energy-momentum is given by:

(3.4)T_^ = " g_"["()' + V()]

• Compatibility note: All TTG v1.0 equations remain valid under the substitution:

(3.5) ()

This ensures full backward compatibility and preserves the original structure of the theory.

4. Newtonian and Post-Newtonian Limits

In the weak-field regime, TTG v1.1 modifies the Newtonian potential via a Yukawa-type correction sourced by the scalar field:

(4.1)(r) = ()' " (G"Mr) " exp(m_"r)

To characterize the range of this correction, we define the Yukawa interaction range:

(4.2)_ 1m_

This correction is suppressed in high-density environments due to screening:

• Solar System screening:

(4.3) - 0() - 0Yukawa term vanishes at required precision

• PPN consistency: In screened regions, post-Newtonian parameters recover their GR values:

(4.4)(_) - 0_PPN - 1,*PPN - 1 + O((*)')

• Fifth force behavior:

(4.5)High-density:() - 0force suppressed Low-density:() 0force active

• Total gravitational potential: The full potential includes both Newtonian and scalar contributions:

(4.6)_total(r) = G"Mr + (r)

This ensures TTG v1.1 remains consistent with precision tests while allowing deviations from GR on larger scales.

5. Cosmology and Large-Scale Dynamics

The scalar fieldevolves under a nonlinear couplingA(), enabling late-time cosmological activation while remaining screened during early epochs.

• Nonlinear coupling: As introduced earlier:

(5.1)A() = cosh(")orA() = exp("'2M')

• Scalar evolution: At late times,departs from zero, activating scalar dynamics and modifying the effective equation of state:

(5.2)w(z) 1

This behavior mimics dynamical dark energy and allows deviations from CDM.

• Example potential: A quintessence-like form:

(5.3)V() = V"exp(")

This drives slow-roll evolution ofand can be tuned to match late-time acceleration.

• Scalar coupling behavior: For exponential coupling, the derivative takes the form:

(5.4)() = "M'(for A() = exp("'2M'))

• Early-universe compatibility: In high-density regimes (BBN, CMB), screening ensures:

(5.5)() 0 effectively frozen, preserving standard early-universe physics

• Modified Friedmann equation: Scalar energy density contributes to cosmic expansion:

(5.6)H'(z) = (8G3) " [m(z) + (z)]

with scalar energy density:

(5.7)_(z) = "(/t)' + V()

• Effective equation of state of : Determines acceleration behavior:

(5.8)w_ = ["(/t)' V()]["(/t)' + V()]

This dual behavior --- screened at early times, active at late times --- allows TTG v1.1 to remain consistent with precision cosmology while introducing testable deviations at large scales.

6. Gravitational Waves and Dipole Radiation

In TTG v1.1, the tensor sector remains unmodified:

• Gravitational wave speed:

(6.1)c_gw = c

The scalar fieldintroduces a dipole radiation channel in asymmetric binaries, modifying the gravitational wave phase:

(6.2)(f) = _GR(f) + _dip " (_A _B)' " ("M"f)^73

where:

• _GR(f) --- standard GR waveform
• _dip --- 1PN coefficient, proportional to scalar charges and binary asymmetry
• _A, _B --- scalar couplings of the two bodies
• Dipole phase correction: Expressed in terms of chirp mass: --- here the chirp mass is (m m)^{3/5}/(m+m)^{1/5}, and _GR(f) denotes the GR baseline phase.

(6.3)(f) (_A _B)' " f^73

6.1 Screening behavior

• Compact objects (NS, BH): High compactness leads to strong screening:

(6.4)_A, _B 0dipole term suppressed in symmetric systems (e.g., NS--NS, BH--BH)

• Asymmetric binaries (e.g., NS--BH, He-WD--NS): Scalar charges differ:

(6.5)_A _Bobservable dipole radiation

This makes TTG v1.1 testable via waveform deviations in systems with unequal screening profiles.

7. Falsifiable Predictions and Signal Table

TTG v1.1 yields distinct, falsifiable signals across gravitational and cosmological domains. The scalar coupling()varies with environment, producing a hierarchy of regimes:

• Screened: High-density regions (e.g., Solar System)
• Unscreened: Low-density regions (e.g., galaxies, cosmology)
• Asymmetric: Systems with unequal scalar charges (e.g., NS--BH binaries)

Signal Table

Scalar Coupling Behavior

• (7.1)(_) - 0fifth force suppressed
• (7.2)_A _Bdipole GW radiation observable
• (7.3)() 0modified dynamics on large scales

Fifth Force Amplitude

To quantify deviations from Newtonian gravity, we define the scalar-mediated fifth force:

(7.4)F_5th(r) = ()' " (G"Mr') " exp(r_)

This expression shows how the force is suppressed at short distances (via screening) and activated at large scales (via unscreened ()).

This structure enables hierarchical falsifiability, distinguishing TTG v1.1 from v1.0 and offering clear experimental windows across regimes.

8. Parameter Space and Priors

TTG v1.1 introduces a compact set of parameters governing scalar dynamics and coupling behavior.

8.1 Core Parameters

The scalar--gravity interaction is governed by three primary quantities:

Here, M denotes a mass scale, typically taken to be the Planck mass M_Pl for naturalness, unless stated otherwise.

(8.1){, , M}--- control the shape and scale of the coupling functionA()

The scalar potential determines the evolution of :

(8.2)V() = V " exp(")

These parameters define the dynamical landscape of TTG v1.1.

8.2 Scalar Coupling Behavior

The interaction strength is encoded in the derivative of the coupling function:

(8.3)() d(lnA)/d

For cosh-type coupling:

(8.4)() = " + " + ...with = ', = 3

For exponential coupling:

(8.5)() = "M'

This flexibility allows TTG v1.1 to interpolate between screened and unscreened regimes.

8.3 Priors for Screening Consistency

To ensure compatibility with Solar System tests and allow large-scale deviations:

(8.6)(_) 10(screened regime)
(8.7)(_gal) 10'(unscreened regime)

These priors separate environments by scalar activity and observational constraints.

8.4 Explicit Parameter Bounds

These bounds preserve naturalness and observational viability.

8.5 Environmental Regimes

These regimes define where TTG v1.1 becomes observationally active.

8.6 Scalar-Mediated Force

To quantify deviations from Newtonian gravity, we define the scalar fifth force:

(8.8)F_5th(r) = ()' " (G"Mr') " exp(r_)

This expression links coupling strength, range, and environmental suppression.

8.7 Constraints and Estimation

Parameter estimation is performed via MCMC fits to combined datasets:

• Planck
• BAO
• SN Ia
• Galactic rotation curves (SPARC)

This structure ensures TTG v1.1 remains predictive and testable across scales, with clear priors separating screened and unscreened regimes.

9. Quantum Corrections and Theoretical Stability

TTG v1.1 is formulated as a low-energy effective field theory (EFT), valid below the Planck scale:

(9.1)E M_Pl

Its even-parity structure suppresses dangerous quantum corrections and ensures theoretical stability.

9.1 Symmetry Protection

• Radial symmetry: The coupling function depends only on'at leading order:

(9.2)A() = A(') + higher-order terms

This protects against linear tadpole terms and ensures that quantum corrections respect parity.

• Naturalness: Small values ofandare technically natural in the sense of 't Hooft: the symmetry is restored when, 0.

(9.3), 0 symmetry restored

This ensures that loop corrections do not destabilize the scalar sector.

9.2 Radiative Stability

To estimate quantum corrections to the scalar potential, we consider the 1-loop Coleman--Weinberg term:

(9.4)V_1loop() " ln[A()]

This correction remains controlled as long as A() grows slowly and M_Pl.

9.3 UV Completion

TTG is expected to be UV-completed within a fundamental theory of quantum gravity, such as:

• String theory
• Loop quantum gravity
• Higher-dimensional moduli frameworks

In such settings, the scalar fieldmay arise as a modulus or geometric degree of freedom, and the couplingA()may reflect compactification effects.

9.4 Summary of Stability Features

These features ensure that TTG v1.1 remains quantum-mechanically stable under radiative corrections, unlike many other modified gravity models.

10. Comparison with Dark Matter and MOND

TTG v1.1 can mimic dark matter effects on galactic scales through the unscreened scalar fifth force.

10.1 Galactic Rotation Curves

The Yukawa-like correction to the gravitational potential modifies orbital velocities:

(10.1)(r) = ()' " (G"Mr) " exp(m_"r)

The corresponding circular velocity becomes:

(10.2)v_rot(r) = -[r " d(total)/dr] = -[G"Mr + ()' " G"Mr " exp(m_"r)]

At galactic scales (r _), the exponential term flattens the velocity profile, reproducing observed rotation curves without invoking particle dark matter.

10.2 Effective Mass Interpretation

Scalar effects can be interpreted as an emergent mass profile:

(10.3)M_eff(r) r' " d()/drG

This allows TTG v1.1 to reproduce galactic dynamics via gravitational modification rather than matter content.

10.3 Comparison with MOND

The functional form of()can be tuned to reproduce MOND-like behavior:

(10.4)() 1r(at intermediate scales)

This mimics MOND phenomenology while avoiding strong constraints from:

• Galaxy clusters
• Cosmic microwave background (CMB)
• Large-scale structure

Unlike MOND, TTG retains a relativistic EFT structure and remains compatible with gravitational wave propagation and cosmological expansion.

10.4 Applicability and Limitations

TTG v1.1 does not eliminate the need for dark matter entirely. It offers an alternative gravitational explanation for galactic anomalies, while retaining dark matter for:

• Cluster-scale lensing
• Structure formation
• CMB acoustic peaks

Unless further extended, TTG v1.1 operates as a partial substitute for dark matter on galactic scales.

10.5 Hybrid Interpretation Table

11. Conclusion: TTG as a Living Framework

TTG v1.0 defined a fixed, minimal, and falsifiable anchor --- a scalar--gravity framework with universal linear coupling and clean predictions across scales.

TTG v1.1 builds on this foundation by introducing nonlinear, density-dependent coupling functionsA(), enabling environmental screening while preserving the core formalism. The scalar derivative()governs interaction strength and allows TTG to remain silent in high-density regimes and active on galactic and cosmological scales.

Backward Compatibility

The linear limit of TTG v1.1 recovers TTG v1.0:

(11.1)() = constTTG v1.1 TTG v1.0

This ensures that all predictions of v1.0 remain embedded within v1.1 as a limiting case.

Archival Integrity and Version Discipline

TTG v1.1 preserves the editorial clarity of v1.0 by maintaining strict version boundaries:

(11.2)TTG v1.1 is not a hybrid extension of v1.0no retroactive modification

Instead, TTG v1.1 is a layered generalization, with falsifiability preserved at each level. This protects the archive from drift, romanticization, and hybrid collapse.

Structured Evolution

TTG evolves from a static proposal into a living framework, capable of absorbing new constraints and generating new predictions. Its falsifiability is preserved through:

• Hierarchical screening
• Domain-specific signals
• Parameter priors separating regimes

Future Directions (TTG v1.2+)

• Quantum corrections: Renormalization of(), loop stability, and EFT control
• Thermodynamic sourcing: Entropy gradients as scalar triggers
• Matter sector coupling: Extension to neutrinos, dark sector, or baryon asymmetry

These directions remain grounded in falsifiability and version discipline.

TTG as a Process, Not a Breakthrough

TTG is no longer a candidate for final theory. It is a documented process --- a trace of scientific attempt, marked by:

• Structural honesty
• Editorial closure
• Falsifiability across versions

Its value lies not in coherence, but in the courage to record its limits. TTG v1.1 thus becomes a living archive, where each version is a fold, not a fix.

Version Structure Summary

TTG remains anchored in falsifiability, but open to structured generalization --- a theory not of fixed form, but of principled evolution.

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Appendix A. Comparison with TTG v1.0

This appendix serves as an archival marker distinguishing the minimal anchor (v1.0) from its adaptive extension (v1.1). It preserves version discipline and clarifies the structural evolution of TTG.

Version Comparison Table

Limiting Behavior

TTG v1.1 reduces to TTG v1.0 in the linear limit:

(A.1)() = constTTG v1.1 TTG v1.0

This ensures backward compatibility while preserving falsifiability and structural clarity.