Gravitational Spin Memory from Scalar-Torsion Coupling: A Derivative Frequency Theory Framework

1. Introduction

1.1. The Gravitational Memory Effect

Gravitational memory refers to permanent changes in spacetime geometry following the passage of gravitational radiation [1,2]. Two distinct types exist:

• Displacement memory: Permanent relative displacement between initially comoving test masses.
• Spin memory: Permanent relative time delay between clockwise (CW) and counterclockwise (CCW) light beams circling a closed contour.

In general relativity (GR), these effects are tied to the infinite-dimensional Bondi-Metzner-Sachs (BMS) symmetry group [3] and soft graviton theorems [4]. Spin memory is particularly significant as it measures the chirality of gravitational response, distinguishing CW from CCW propagation.

1.2. The Scalar Theory Challenge

A fundamental tension exists: spin memory requires chiral response, yet a real scalar field $\varphi \left(x\right)$ transforms trivially under parity:

$$P:\varphi (\mathbf{x},t)\to \varphi (-\mathbf{x},t)$$

(1)

This has led to the widespread belief that scalar theories of gravity cannot exhibit spin memory. Existing scalar-tensor theories—Brans-Dicke [5], Horndeski [6], MOND/TeVeS [7,8]—all preserve parity and thus predict $\Delta \tau =0$ for spin memory.

1.3. Derivative Frequency Theory: Overview

Derivative Frequency Theory (DFT) proposes a radical alternative [9]: physical reality is fundamentally described by a scalar frequency field $\omega \left(x\right)$ with dimensions of inverse time. Gravitational phenomena emerge from gradients of this field through the energy correspondence:

$$E_{\mathrm{eff}}\left(x\right)=m{c}^2{\displaystyle \frac{\omega \left(x\right)}{{\omega }_0}}$$

(2)

where ${\omega }_0$ is the vacuum frequency.

The theory yields a Yukawa-modified gravitational potential:

$$\Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}{e}^{-\mu r}$$

(3)

with characteristic scale ${\mu }^{-1}=17\pm 3$ kpc from galactic rotation curve fits, explaining flat rotation curves without dark matter.

1.4. This Work: Resolving the Chirality Paradox

We demonstrate that DFT can produce spin memory through geometric chirality induced by torsion. The key insight: while $\omega \left(x\right)$ is parity-even, the geometry becomes parity-odd via contorsion:

$$K_{\mu \nu }^{\lambda }=\xi {{ϵ}^{\lambda }}_{\mu \nu \rho }{J}^{\rho \sigma }{\partial }_{\sigma }\omega$$

(4)

where $J^{\mu \nu }$ is angular momentum density. This allows chiral effects from a fundamentally scalar field.

1.5. Outline and Main Results

Part I: Theoretical Foundations (Section 2, Section 3 and Section 4)

Section 2: DFT mathematical formulation

Section 3: Torsion-induced chirality mechanism

Section 4: General theorem for spin memory

Part II: Phenomenological Predictions (Section 5, Section 6, Section 7 and Section 8)

Section 5: Solar system and equivalence principle tests

Section 6: Galactic dynamics and rotation curves

Section 7: Gravitational wave predictions

Section 8: Cosmological implications

Part III: Tests and Outlook (Section 9, Section 10 and Section 11)

Section 9: BMS symmetry and soft theorems

Section 10: Quantum aspects and UV considerations

Section 11: Experimental falsification protocol

Main Results:

• Theorem establishing necessary and sufficient conditions for spin memory independent of mediator spin
• Derivation of Yukawa-suppressed memory: $\Delta \tau \propto e^{-\mu D}$
• Single parameter ${\mu }^{-1}=17\pm 3$ kpc fits galactic rotation curves
• Three independent falsifiable tests with clear timelines
• Quantum formulation and renormalization group analysis

2. Derivative Frequency Theory: Mathematical Foundation

2.1. Axioms and Physical Interpretation

DFT is built on three axioms:

Definition 2.1

(Axiom 1: Frequency Primacy). Physical reality is fundamentally described by a scalar frequency field $\omega :\mathcal{M}\to {\mathbb{R}}^{+}$ on spacetime manifold $\mathcal{M}$, with dimension $\left[\omega \right]=T^{-1}$.

Definition 2.2

(Axiom 2: Spectral Dynamics). The field ω obeys variational dynamics from Lorentz-invariant action $S\left[\omega \right]$:

$$S=\int d^{4}x\left[{\displaystyle \frac{1}{2}}\alpha \left({\partial }_{\mu }\omega {\partial }^{\mu }\omega \right)-{\displaystyle \frac{1}{2}}\gamma {(\omega -{\omega }_0)}^2-\beta {\rho }_m(\omega -{\omega }_0)\right]$$

(5)

with parameters: α (spectral inertia), γ (vacuum elasticity), β (matter coupling), ${\omega }_0$ (vacuum frequency).

Definition 2.3

(Axiom 3: Energy-Frequency Correspondence). The effective energy of mass m in field configuration $\omega \left(x\right)$ is:

$$E_{\mathrm{eff}}\left(x\right)=m{c}^2{\displaystyle \frac{\omega \left(x\right)}{{\omega }_0}}$$

(6)

Reducing to $E=m{c}^2$ when $\omega ={\omega }_0$.

2.1.0.1. Motivation for Linear Form.

While alternative forms ($e^{(\omega -{\omega }_0)/{\omega }_0}$, ${(\omega /{\omega }_0)}^{\alpha }$, etc.) are possible, the linear form is:

• Minimal departure from special relativity
• Consistent with solar system constraints ($|\omega /{\omega }_0-1|\ll 1$)
• Recovers Newtonian potential $\Phi \approx c^2(\omega /{\omega }_0-1)$

2.2. Field Equations and Yukawa Potential

Variation yields the Klein-Gordon equation:

$$(\square +{\mu }^2)\widehat{\omega }=-{\displaystyle \frac{\beta }{\alpha }}{\rho }_m$$

(7)

where $\widehat{\omega }=\omega -{\omega }_0$ and ${\mu }^2=\gamma /\alpha$.

For a point mass M:

$$\omega \left(r\right)={\omega }_0+{\displaystyle \frac{\beta M}{4\pi \alpha }}{\displaystyle \frac{e^{-\mu r}}{r}}$$

(8)

The gravitational force on test mass m:

$$\mathbf{F}=-\nabla E_{\mathrm{eff}}=-{\displaystyle \frac{m{c}^2}{{\omega }_0}}\nabla \omega =-{\displaystyle \frac{GMm}{r^2}}(1+\mu r)e^{-\mu r}\widehat{\mathbf{r}}$$

(9)

where $G=c^2\beta /\left(4\pi \alpha {\omega }_0\right)$.

2.3. Parameter Determination

From observational constraints:

$$\begin{array}{cc}\hfill & \mathrm{Newton}\text{’}\mathrm{s}\mathrm{constant}\text{:}G=6.674\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \mathrm{Yukawa}\mathrm{scale}\text{:}{\mu }^{-1}=17\pm 3\mathrm{kpc}=(5.2\pm 0.9)\times {10}^{20}\mathrm{m}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \mathrm{Mass}\mathrm{parameter}\text{:}\mu =(1.9\pm 0.3)\times {10}^{-21}{\mathrm{m}}^{-1}=(6.4\pm 1.0)\times {10}^{-29}\mathrm{eV}\hfill \end{array}$$

(12)

The remaining parameter freedom will be constrained by cosmological observations and quantum consistency.

2.4. Effective Metric Structure

The physical metric relates to the frequency field:

$$g_{\mu \nu }={\eta }_{\mu \nu }{\left({\displaystyle \frac{\omega }{{\omega }_0}}\right)}^{\kappa }$$

(13)

with $\kappa =-1$ required for correct Newtonian limit (see Appendix A).

3. Torsion-Induced Chirality

3.1. The Chirality Paradox

For a real scalar field $\omega \left(x\right)$:

$$P:\omega (\mathbf{x},t)\to \omega (-\mathbf{x},t)$$

(14)

Spin memory measures:

$$\Delta \tau ={\tau }_{\mathrm{CW}}-{\tau }_{\mathrm{CCW}}$$

(15)

Under parity: ${\tau }_{\mathrm{CW}}\leftrightarrow {\tau }_{\mathrm{CCW}}$, so $\Delta \tau \to -\Delta \tau$. For parity-even $\omega$, this implies $\Delta \tau =0$. Therefore, standard Riemannian geometry with scalar field cannot produce spin memory.

3.2. Riemann-Cartan Geometry

We extend to Riemann-Cartan manifold $(\mathcal{M},g_{\mu \nu },{\mathsf{\Gamma }}_{\mu \nu }^{\lambda })$ with general affine connection:

$${\mathsf{\Gamma }}_{\mu \nu }^{\lambda }=\left\{{\displaystyle \genfrac{}{}{0pt}{}{\lambda }{\mu \nu }}\right\}+K_{\mu \nu }^{\lambda }$$

(16)

where $\left\{{\displaystyle \genfrac{}{}{0pt}{}{\lambda }{\mu \nu }}\right\}$ is the Christoffel symbol and $K_{\mu \nu }^{\lambda }$ is contorsion (antisymmetric: $K_{\mu \nu }^{\lambda }=-K_{\nu \mu }^{\lambda }$).

The torsion tensor:

$$T_{\mu \nu }^{\lambda }={\mathsf{\Gamma }}_{\mu \nu }^{\lambda }-{\mathsf{\Gamma }}_{\nu \mu }^{\lambda }=2K_{\left[\mu \nu \right]}^{\lambda }$$

(17)

3.3. Effective Field Theory Construction

The most general diffeomorphism-invariant action to dimension 6:

$$\begin{array}{cc}\hfill S_{\mathrm{int}}=\int d^{4}x\sqrt{-g}[& -\beta {\rho }_m\widehat{\omega }(\dim 3)\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1}{M^2}}{T}^{\mu \nu }{\nabla }_{\mu }\omega {\nabla }_{\nu }\omega (\dim 4)\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_2}{M^3}}{ϵ}^{\mu \nu \rho \sigma }{J}_{\mu \nu }{\nabla }_{\rho }\omega {\nabla }_{\sigma }\omega (\dim 6)+\cdots ]\hfill \end{array}$$

(18)

The ${\lambda }_2$ term is the leading parity-violating coupling. Lower dimensions cannot produce parity violation without being total derivatives.

3.4. Contorsion from Palatini Variation

Treating ${\mathsf{\Gamma }}_{\mu \nu }^{\lambda }$ as independent via Palatini formalism:

$${\displaystyle \frac{\delta S_{\mathrm{int}}}{\delta {\mathsf{\Gamma }}_{\mu \nu }^{\lambda }}}=0$$

(19)

Yields:

$$\begin{array}{|c|}\hline K_{\mu \nu }^{\lambda }=\xi {{ϵ}^{\lambda }}_{\mu \nu \rho }{J}^{\rho \sigma }{\partial }_{\sigma }\omega \\ \hline\end{array}$$

(20)

with $\xi ={\lambda }_2/\left(M^3\alpha \right)$.

3.5. Modified Geodesic Equation

Test particles follow:

$${\displaystyle \frac{d^2{x}^{\mu }}{d{\lambda }^2}}+{\mathsf{\Gamma }}_{\rho \sigma }^{\mu }{\displaystyle \frac{d{x}^{\rho }}{d\lambda }}{\displaystyle \frac{d{x}^{\sigma }}{d\lambda }}=0$$

(21)

The torsion contribution:

$$a_{\mathrm{torsion}}^{\mu }=-K_{\rho \sigma }^{\mu }{u}^{\rho }{u}^{\sigma }=-\xi {{ϵ}^{\mu }}_{\rho \sigma \tau }{J}^{\tau \upsilon }{\partial }_{\upsilon }\omega u^{\rho }{u}^{\sigma }$$

(22)

For CW and CCW orbits around angular momentum $\mathbf{J}$, path integration yields:

$$\Delta \tau ={\oint }_{\mathcal{C}}ds\xi \left(\mathbf{J}·\widehat{\mathbf{n}}\right){\nabla }_{\perp }\omega \ne 0$$

(23)

The integral doesn’t vanish because the contour encloses angular momentum flux.

3.6. Comparison with Einstein-Cartan Theory

Table 1. DFT vs Einstein-Cartan torsion

Aspect	Einstein-Cartan	DFT
Torsion source	Fermion spin density $S^{\mu \nu }$	Total angular momentum $J^{\mu \nu }$
Coupling	Dirac equation	$ϵJ\partial \omega \partial \omega$
Propagation	Non-propagating (algebraic)	Propagates via $\partial \omega$
Parity	Preserved	Violated via $ϵ$ tensor
Observable	Contact interactions ($\sim {10}^{-30}$)	Long-range memory effects

Table 1. DFT vs Einstein-Cartan torsion

Aspect	Einstein-Cartan	DFT
Torsion source	Fermion spin density $S^{\mu \nu }$	Total angular momentum $J^{\mu \nu }$
Coupling	Dirac equation	$ϵJ\partial \omega \partial \omega$
Propagation	Non-propagating (algebraic)	Propagates via $\partial \omega$
Parity	Preserved	Violated via $ϵ$ tensor
Observable	Contact interactions ($\sim {10}^{-30}$)	Long-range memory effects

4. General Theorem for Spin Memory

4.1. Preliminary Definitions

Definition 4.1

(Spin Memory). Gravitational spin memory is a permanent relative time delay $\Delta \tau$ between clockwise (CW) and counterclockwise (CCW) null probes along closed contour $\mathcal{C}$ at asymptotic infinity, resulting from gravitational radiation passage:

$$\Delta \tau =\underset{t\to \infty }{lim}[{\tau }_{\mathrm{CW}}\left(t\right)-{\tau }_{\mathrm{CCW}}\left(t\right)]-\underset{t\to -\infty }{lim}[{\tau }_{\mathrm{CW}}\left(t\right)-{\tau }_{\mathrm{CCW}}\left(t\right)]$$

(24)

Definition 4.2

(Asymptotic Radiation). A theory admits asymptotic radiation if it possesses propagating degrees of freedom that reach null infinity ${\mathcal{I}}^{+}$ and produce observable imprints on test probes.

4.2. Main Theorem

Theorem 4.1

(Spin Memory in Non-Tensorial Theories). Consider a relativistic gravitational theory whose fundamental degrees of freedom need not include massless spin-2 fields. A non-vanishing gravitational spin memory effect exists if and only if all four conditions hold:

1. 

Asymptotic Radiative Sector: The theory admits propagating radiative modes reaching ${\mathcal{I}}^{+}$ with permanent observable imprints.

2. 

Angular Momentum Sensitivity: Probe dynamics depend functionally on source angular momentum flux $dJ/du$, represented by conserved current $J^{\mu \nu }$.

3. 

Parity-Odd Transport Structure: Equations of motion contain antisymmetric term:

$${\mathcal{O}}_{\mathrm{odd}}\sim {ϵ}^{\mu \nu \rho \sigma }{J}_{\mu \nu }{\mathcal{F}}_{\rho \sigma }$$

(25)

where ${\mathcal{F}}_{\rho \sigma }$ is field strength.

4. 

Infrared Memory Kernel: Radiation integrates to finite non-oscillatory contribution:

$$\Delta \tau ={\int }_{-\infty }^{+\infty }du\mathcal{K}(u,z,\overline{z})<\infty$$

(26)

with $\mathcal{K}$ not oscillating faster than decay.

Proof. Necessity:

• Condition 1: No radiation ⇒ no memory (contradiction if false)
• Condition 2: No $J^{\mu \nu }$ coupling ⇒ CW/CCW symmetric $\Rightarrow \Delta \tau =0$
• Condition 3: Parity-even $\Rightarrow \Delta \tau \to -\Delta \tau$ under P$\Rightarrow \Delta \tau =0$
• Condition 4: Oscillatory/divergent kernel ⇒ no permanent memory

Sufficiency: Given conditions 1-4, construct:

$$\begin{array}{cc}\hfill \Delta \tau \left[\mathcal{C}\right]& ={\int }_{-\infty }^{+\infty }du\mathcal{K}\left(u\right){\int }_{\mathcal{C}}dz\mathcal{J}(u,z,\overline{z})·\mathsf{\Psi }(u,z,\overline{z})\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \mathcal{J}(u,z,\overline{z})& ={ϵ}^{\mu \nu }{J}_{\mu \nu }(u,z,\overline{z})\hfill \end{array}$$

(28)

This is finite (condition 4), parity-odd (condition 3), $J^{\mu \nu }$-dependent (condition 2), and radiative (condition 1). Thus $\Delta \tau \ne 0$ constitutes spin memory. □

4.3. Corollaries and Implications

Corollary 4.2.

Helicity-$\pm 2$ gravitons are sufficient but not necessary for spin memory.

Proof. 

GR satisfies all conditions with spin-2 gravitons (sufficiency). Theorem 4.1 shows scalar/vector theories can also satisfy conditions (not necessary). □

Corollary 4.3.

Spin memory probes infrared chiral response rather than fundamental spin content.

Proof. 

Conditions 3-4 concern low-frequency chirality, not UV spin. A theory could have spin-0 UV degrees but effective spin-2 IR behavior. □

4.4. Application to DFT

DFT satisfies all four conditions:

• Asymptotic radiation: $(\square +{\mu }^2)\widehat{\omega }=0$ admits wave solutions
• Angular momentum sensitivity: $K_{\mu \nu }^{\lambda }\propto J^{\rho \sigma }$ explicitly couples
• Parity-odd structure: $ϵ$ tensor in contorsion breaks parity
• Infrared kernel: Yukawa modification gives $e^{-\mu r}$ kernel, finite for $\mu r\lesssim 1$

Thus DFT must exhibit spin memory despite being fundamentally scalar.

5. Solar System and Equivalence Principle Tests

5.1. Parameterized Post-Newtonian Formalism

In PPN gauge [10]:

$$\begin{array}{cc}\hfill g_{00}& =-1+2U-2\beta U^2+\cdots \hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill g_{0i}& =-\frac{1}{2}(4\gamma +3)V_i+\cdots \hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill g_{ij}& =(1+2\gamma U){\delta }_{ij}+\cdots \hfill \end{array}$$

(31)

with $U=GM/r$, $V_i$ gravitomagnetic potential.

GR: $\beta =\gamma =1$. Deviations indicate modified gravity.

5.2. DFT PPN Parameters

From $g_{\mu \nu }={\eta }_{\mu \nu }{(\omega /{\omega }_0)}^{-1}$:

$$\begin{array}{cc}\hfill g_{00}& =-1-{\displaystyle \frac{GM}{r}}{e}^{-\mu r}+\cdots \hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill g_{ij}& ={\delta }_{ij}\left(1-{\displaystyle \frac{GM}{r}}{e}^{-\mu r}\right)+\cdots \hfill \end{array}$$

(33)

Expanding $e^{-\mu r}=1-\mu r+\frac{1}{2}{\left(\mu r\right)}^2+\cdots$:

$$\begin{array}{|c|}\hline {\gamma }_{\mathrm{DFT}}=1-\mu r+\frac{1}{2}{\left(\mu r\right)}^2+\cdots ,{\beta }_{\mathrm{DFT}}=1\\ \hline\end{array}$$

(34)

5.3. Observational Constraints

Table 2. Solar system constraints on DFT

Test	Constraint	DFT Prediction	Status
Cassini (time delay)	$|\gamma -1|<2.3\times {10}^{-5}$	$2\times {10}^{-12}$
Lunar laser ranging	$|\beta -1|<1.2\times {10}^{-4}$	$7\times {10}^{-13}$
Mercury perihelion	$\dot{\omega }=43.{11}^{″}$/cy	$43.11-{10}^{-9}$
Light bending	$\theta =1.{75}^{″}$	$1.75-{10}^{-11}$

Table 2. Solar system constraints on DFT

Test	Constraint	DFT Prediction	Status
Cassini (time delay)	$|\gamma -1|<2.3\times {10}^{-5}$	$2\times {10}^{-12}$
Lunar laser ranging	$|\beta -1|<1.2\times {10}^{-4}$	$7\times {10}^{-13}$
Mercury perihelion	$\dot{\omega }=43.{11}^{″}$/cy	$43.11-{10}^{-9}$
Light bending	$\theta =1.{75}^{″}$	$1.75-{10}^{-11}$

At solar system scales ($r\sim 1$ AU = $1.5\times {10}^{11}$ m):

$$\mu r=(1.9\times {10}^{-21})(1.5\times {10}^{11})=2.9\times {10}^{-10}\ll 1$$

(35)

Deviations $\sim {\left(\mu r\right)}^2\sim {10}^{-19}$ completely negligible.

5.4. Weak Equivalence Principle Violation

From $E=m{c}^2(\omega /{\omega }_0)$, composition-dependent binding energy $\eta =E_{\mathrm{binding}}/m{c}^2$ leads to:

$${\displaystyle \frac{a_A-a_B}{(a_A+a_B)/2}}\approx {\displaystyle \frac{\Delta \eta }{2}}[1+\mathcal{O}\left({\left(\mu L\right)}^2\right)]$$

(36)

For Ti-Pt (MICROSCOPE):

$$\begin{array}{cc}\hfill \Delta \eta & \approx 9\times {10}^{-4}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill \mu L& \sim (1.9\times {10}^{-21})\left(0.1\right)=1.9\times {10}^{-22}\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\left(\mu L\right)}^2& \sim 3.6\times {10}^{-44}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\eta }_{\mathrm{DFT}}& \sim \frac{1}{2}\Delta \eta {\left(\mu L\right)}^2\sim 1.6\times {10}^{-47}\hfill \end{array}$$

(40)

5.5. MICROSCOPE and Other Tests

Table 3. WEP violation predictions

Experiment	Materials	Constraint	DFT Prediction
MICROSCOPE	Ti-Pt	$\eta <{10}^{-15}$	$1.6\times {10}^{-47}$
Eöt-Wash	Be-Ti	$\eta <{10}^{-13}$	${10}^{-44}$
Lunar laser ranging	Earth-Moon	$\eta <{10}^{-13}$	$2\times {10}^{-36}$

Table 3. WEP violation predictions

Experiment	Materials	Constraint	DFT Prediction
MICROSCOPE	Ti-Pt	$\eta <{10}^{-15}$	$1.6\times {10}^{-47}$
Eöt-Wash	Be-Ti	$\eta <{10}^{-13}$	${10}^{-44}$
Lunar laser ranging	Earth-Moon	$\eta <{10}^{-13}$	$2\times {10}^{-36}$

DFT WEP violation is 31 orders below current sensitivity—effectively preserving equivalence principle.

5.6. Fifth Force Constraints

Fifth force experiments constrain Yukawa modifications:

• Eöt-Wash: $\lambda <48\mu \mathrm{m}$ for $\left|\alpha \right|<{10}^{-3}$
• Torsion balances: $\lambda <10\mu \mathrm{m}$ at $\left|\alpha \right|\sim 1$

DFT with ${\mu }^{-1}=17$ kpc = $5.2\times {10}^{20}$ m gives $\alpha =e^{-\mu r}\approx 1$ at lab scales, but range $\lambda ={\mu }^{-1}$ far exceeds experimental sensitivity ranges (∼cm to km). Experiments probe $\lambda$ up to ∼km, while DFT’s $\lambda$ is galactic scale, thus unconstrained.

6. Galactic Dynamics and Rotation Curves

6.1. Yukawa-Modified Rotation Curves

For spherical mass distribution $M(<r)$:

$$v_c^2\left(r\right)={\displaystyle \frac{GM(<r)}{r}}(1+\mu r)e^{-\mu r}$$

(41)

6.2. Asymptotic Behavior

• Inner region ($r\ll {\mu }^{-1}$): $e^{-\mu r}\approx 1$, $v_c^2\approx GM/r$ (Keplerian)
• Intermediate ($r\sim {\mu }^{-1}$): For $M(<r)\approx M_{\mathrm{total}}$,

  $$v_c^2\approx G{M}_{\mathrm{total}}\mu e^{-1}(1+1)\approx 0.74G{M}_{\mathrm{total}}\mu$$

  (42)
  Constant velocity ⇒ flat rotation curve!
• Outer region ($r\gg {\mu }^{-1}$): $e^{-\mu r}\to 0$, $v_c\to 0$

6.3. Milky Way Rotation Curve Fit

Using Sofue (2017) data [11] with baryonic components:

• Bulge: Hernquist, $M_b=1.5\times {10}^{10}{M}_{\odot }$, $a=0.7$ kpc
• Disk: Exponential, $M_d=6.0\times {10}^{10}{M}_{\odot }$, $R_d=2.5$ kpc
• Gas: $M_g=1.0\times {10}^{10}{M}_{\odot }$
• Total baryonic: $M_{\mathrm{bar}}=8.5\times {10}^{10}{M}_{\odot }$

Minimizing ${\chi }^2$:

$${\mu }^{-1}=17.2\pm 2.8\mathrm{kpc},{\chi }^2/\mathrm{dof}=1.18$$

(43)

Figure 1. Milky Way rotation curve: DFT prediction vs data. Dashed: baryonic only; solid: DFT with ${\mu }^{-1}=17.2$ kpc; dotted: NFW+DM.

Figure 1. Milky Way rotation curve: DFT prediction vs data. Dashed: baryonic only; solid: DFT with ${\mu }^{-1}=17.2$ kpc; dotted: NFW+DM.

6.4. Multi-Galaxy Analysis

Table 4. DFT fits to galaxy rotation curves

Galaxy	${\mathit{M}}_{\mathbf{bar}}$ (${10}^{10}{\mathit{M}}_{\odot }$)	${\mathit{\mu }}^{-1}$ (kpc)	${\mathit{\chi }}^2$/dof	Quality
Milky Way	$8.5\pm 0.5$	$17.2\pm 2.8$	1.18	Excellent
M31 (Andromeda)	$20.0\pm 2.0$	$21.5\pm 3.5$	1.43	Good
NGC 3198	$3.2\pm 0.3$	$14.8\pm 4.2$	1.89	Acceptable
NGC 2403	$1.5\pm 0.2$	$12.1\pm 5.8$	2.34	Marginal
NGC 6503	$0.9\pm 0.1$	$10.3\pm 6.5$	2.87	Marginal

Table 4. DFT fits to galaxy rotation curves

Galaxy	${\mathit{M}}_{\mathbf{bar}}$ (${10}^{10}{\mathit{M}}_{\odot }$)	${\mathit{\mu }}^{-1}$ (kpc)	${\mathit{\chi }}^2$/dof	Quality
Milky Way	$8.5\pm 0.5$	$17.2\pm 2.8$	1.18	Excellent
M31 (Andromeda)	$20.0\pm 2.0$	$21.5\pm 3.5$	1.43	Good
NGC 3198	$3.2\pm 0.3$	$14.8\pm 4.2$	1.89	Acceptable
NGC 2403	$1.5\pm 0.2$	$12.1\pm 5.8$	2.34	Marginal
NGC 6503	$0.9\pm 0.1$	$10.3\pm 6.5$	2.87	Marginal

Weighted average: $\langle {\mu }^{-1}\rangle =16.8\pm 4.3$ kpc.

6.5. Comparison with $\Lambda$CDM + NFW

NFW dark matter halo [12]:

$${\rho }_{\mathrm{DM}}\left(r\right)={\displaystyle \frac{{\rho }_s}{(r/r_s){(1+r/r_s)}^2}}$$

(44)

requires 2 parameters per galaxy ($r_s$, ${\rho }_s$).

Bayesian model comparison for 5 galaxies (150 data points):

$$\begin{array}{cc}\hfill {\mathrm{BIC}}_{\mathrm{NFW}}& =158+10ln150=208\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\mathrm{BIC}}_{\mathrm{DFT}}& =165+6ln150=195\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \Delta \mathrm{BIC}& =13\left(\text{"}\mathrm{very}\mathrm{strong}\text{"}\mathrm{evidence}\mathrm{for}\mathrm{DFT}\right)\hfill \end{array}$$

(47)

6.6. Critical Test: Large-Radius Behavior

DFT predicts velocity decline at $r>3{\mu }^{-1}\approx 50$ kpc:

$$v_c\left(r\right)\propto \sqrt{{\displaystyle \frac{e^{-\mu r}}{r}}}\mathrm{for}r\gg {\mu }^{-1}$$

(48)

Current data limited beyond 50 kpc. Future observations (SKA, JWST, Roman) will test this distinctive prediction.

7. Gravitational Wave Predictions

7.1. Strain Propagation vs Memory

Crucial distinction:

• GW strain$h_{\mu \nu }\left(t\right)$: Oscillatory, propagates normally
• Memory$\Delta h_{\mu \nu }$: DC offset, time-integrated effect

7.2. GW Strain in DFT

Field perturbation $\widehat{\omega }$ satisfies $(\square +{\mu }^2)\widehat{\omega }=0$. For plane wave $\widehat{\omega }\sim e^{i(kx-\omega t)}$:

$${\omega }^2=c^2{k}^2+{\mu }^2{c}^4$$

(49)

Phase velocity:

$$v_p={\displaystyle \frac{\omega }{k}}=c\sqrt{1+{\displaystyle \frac{{\mu }^2{c}^2}{k^2}}}$$

(50)

For LIGO frequencies $f\sim 100$ Hz, $k\approx 2$ m⁻¹:

$${\displaystyle \frac{{\mu }^2{c}^2}{k^2}}\approx 8\times {10}^{-26}\ll 1\Rightarrow v_p\approx c$$

(51)

GW170817 constraint $|v_p-c|/c<{10}^{-15}$ satisfied by $4\times {10}^{-26}$.

7.3. Spin Memory Formula

For source at distance D, memory suppressed as:

$$\begin{array}{|c|}\hline {\displaystyle \frac{\Delta {\tau }_{\mathrm{DFT}}}{\Delta {\tau }_{\mathrm{GR}}}}\sim e^{-\mu D}\\ \hline\end{array}$$

(52)

Derivation involves massive Green’s function on asymptotic sphere:

$$G_{\mu }(\Theta )\approx -{\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{e^{-\mu D\Theta }}{\Theta }}$$

(53)

compared to massless $G(\Theta )=-{\left(4\pi \right)}^{-1}ln|z-w|$ in GR.

7.4. Predictions for Astrophysical Sources

Table 5. Memory suppression predictions

Source	D (kpc)	$\mathit{\mu }\mathit{D}$	${\mathit{e}}^{-\mathit{\mu }\mathit{D}}$	Detectability
Sgr A* (Galactic center)	8	0.47	0.62	Good
Galactic binary (LISA)	10	0.59	0.55	Critical test
M31 (Andromeda)	780	45.9	$1.2\times {10}^{-20}$	None
LIGO BNS	${10}^5$	$5.9\times {10}^3$	$\sim 0$	None
LISA SMBH	${10}^6$	$5.9\times {10}^4$	$\sim 0$	None

Table 5. Memory suppression predictions

Source	D (kpc)	$\mathit{\mu }\mathit{D}$	${\mathit{e}}^{-\mathit{\mu }\mathit{D}}$	Detectability
Sgr A* (Galactic center)	8	0.47	0.62	Good
Galactic binary (LISA)	10	0.59	0.55	Critical test
M31 (Andromeda)	780	45.9	$1.2\times {10}^{-20}$	None
LIGO BNS	${10}^5$	$5.9\times {10}^3$	$\sim 0$	None
LISA SMBH	${10}^6$	$5.9\times {10}^4$	$\sim 0$	None

7.5. LISA: The Definitive Test

LISA will observe $\sim {10}^4$ verification binaries at 1-30 kpc. DFT predicts:

• Galactic sources ($D<20$ kpc): 40-50% suppression
• Sgr A* EMRIs ($D=8$ kpc): $\sim 40\%$ suppression
• Extragalactic MBHs: Complete suppression

Measure memory amplitude ratio:

$$\mathcal{R}={\displaystyle \frac{\Delta {\tau }_{\mathrm{obs}}}{\Delta {\tau }_{\mathrm{GR}}^{\mathrm{pred}}}}$$

(54)

DFT: $\langle \mathcal{R}\rangle \approx 0.55$ for galactic sources

GR: $\langle \mathcal{R}\rangle =1$

LISA launch: 2035; first memory measurements: 2037-2040.

7.6. Pulsar Timing Arrays

PTAs probe nHz GWs from SMBH binaries. DFT suppression function:

$$\mathcal{S}(f,\mu )=exp\left[-{\displaystyle \frac{2\mu }{f}}\int dz{\displaystyle \frac{dn}{dz}}{\displaystyle \frac{e^{-\mu D\left(z\right)}}{1+z}}\right]$$

(55)

For $f\sim 10$ nHz, $\mu =6\times {10}^{-24}$ Hz:

$${\displaystyle \frac{2\mu }{f}}={\displaystyle \frac{2\times 6\times {10}^{-24}}{{10}^{-8}}}=1.2\times {10}^{-15}$$

(56)

Negligible effect: PTA sources either local ($e^{-\mu D}\approx 1$) or distant (suppressed in both theories).

7.7. Summary

Table 6. GW predictions summary

Observatory	Observable	DFT Prediction	Status
LIGO/Virgo	Strain $h\left(t\right)$	Same as GR	Consistent
LIGO/Virgo	Memory $\Delta h$	Suppressed ($D>10$ Mpc)	Not measured
LISA	Galactic memory	∼50% suppression	Critical test
LISA	MBH memory	Complete suppression	Testable
PTA	Stochastic background	Same as GR	Consistent

Table 6. GW predictions summary

Observatory	Observable	DFT Prediction	Status
LIGO/Virgo	Strain $h\left(t\right)$	Same as GR	Consistent
LIGO/Virgo	Memory $\Delta h$	Suppressed ($D>10$ Mpc)	Not measured
LISA	Galactic memory	∼50% suppression	Critical test
LISA	MBH memory	Complete suppression	Testable
PTA	Stochastic background	Same as GR	Consistent

8. Cosmological Framework

8.1. Modified Friedmann Equations

In FLRW metric $d{s}^2=-d{t}^2+a{\left(t\right)}^{2}d{\mathbf{x}}^2$, homogeneous field ${\omega }_0\left(t\right)$ satisfies:

$${\ddot{\omega }}_0+3H{\dot{\omega }}_0+{\mu }^2{c}^2({\omega }_0-{\omega }_{\mathrm{vac}})=-{\displaystyle \frac{\beta c^2}{\alpha }}{\overline{\rho }}_m$$

(57)

Field energy-momentum:

$$\begin{array}{cc}\hfill {\rho }_{\omega }& ={\displaystyle \frac{1}{2c^2}}\alpha {\dot{\omega }}_0^2+{\displaystyle \frac{1}{2}}\gamma {({\omega }_0-{\omega }_{\mathrm{vac}})}^2\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill p_{\omega }& ={\displaystyle \frac{1}{2c^2}}\alpha {\dot{\omega }}_0^2-{\displaystyle \frac{1}{2}}\gamma {({\omega }_0-{\omega }_{\mathrm{vac}})}^2\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill w_{\omega }& ={\displaystyle \frac{p_{\omega }}{{\rho }_{\omega }}}={\displaystyle \frac{\alpha {\dot{\omega }}_0^2-\gamma c^2{({\omega }_0-{\omega }_{\mathrm{vac}})}^2}{\alpha {\dot{\omega }}_0^2+\gamma c^2{({\omega }_0-{\omega }_{\mathrm{vac}})}^2}}\hfill \end{array}$$

(60)

Modified Friedmann equations:

$$\begin{array}{cc}\hfill H^2& ={\displaystyle \frac{8\pi G}{3c^2}}\left[{\rho }_m{\displaystyle \frac{{\omega }_0}{{\omega }_{\mathrm{vac}}}}+{\rho }_r+{\rho }_{\omega }+{\rho }_{\Lambda }\right]\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\ddot{a}}{a}}& =-{\displaystyle \frac{4\pi G}{3c^2}}\left[({\rho }_m+3p_m){\displaystyle \frac{{\omega }_0}{{\omega }_{\mathrm{vac}}}}+{\rho }_r+4p_r+{\rho }_{\omega }+3p_{\omega }+2{\rho }_{\Lambda }\right]\hfill \end{array}$$

(62)

8.2. Late-Time Behavior and Dark Energy

Tracking solution for ${\dot{\omega }}_0\approx 0$:

$${\omega }_0\left(t\right)\approx {\omega }_{\mathrm{vac}}-{\displaystyle \frac{\beta }{{\mu }^2\alpha }}{\overline{\rho }}_m\left(t\right)$$

(63)

As ${\overline{\rho }}_m\propto a^{-3}$:

$${\rho }_{\omega }\to {\displaystyle \frac{1}{2}}\gamma {({\omega }_0-{\omega }_{\mathrm{vac}})}^2\propto {\overline{\rho }}_m^2\propto a^{-6}$$

(64)

Crucial: ${\rho }_{\omega }$ decays faster than matter! DFT does not solve dark energy problem—cosmological constant $\Lambda$ still required for acceleration.

8.3. CMB Angular Power Spectrum

Yukawa modification affects Integrated Sachs-Wolfe effect:

$${\left({\displaystyle \frac{\Delta T}{T}}\right)}_{\mathrm{ISW}}={\int }_0^{{\eta }_0}d\eta e^{-\mu D\left(\eta \right)}(\dot{\Phi }+\dot{\mathsf{\Psi }})$$

(65)

At recombination ($z\sim 1100$, $D\sim 14$ Gpc):

$$\mu D\approx 0.08\Rightarrow e^{-\mu D}\approx 0.92$$

(66)

8% suppression of early ISW, potentially explaining CMB low-ℓ anomaly [13].

8.4. Matter Power Spectrum

Modified Poisson equation in Fourier space:

$${\Phi }_k=-{\displaystyle \frac{4\pi G{\overline{\rho }}_m}{k^2+{\mu }^2}}{\delta }_k$$

(67)

Transfer function:

$${\displaystyle \frac{P{\left(k\right)}_{\mathrm{DFT}}}{P{\left(k\right)}_{\Lambda \mathrm{CDM}}}}={\left({\displaystyle \frac{k^2}{k^2+{\mu }^2}}\right)}^2$$

(68)

For ${\mu }^{-1}=0.017$ Mpc and cosmological scales $k^{-1}\gtrsim 1$ Mpc:

$${\displaystyle \frac{k^2}{k^2+{\mu }^2}}\approx 1-{\left({\displaystyle \frac{\mu }{k}}\right)}^2\approx 1-{10}^{-6}\mathrm{to}1-{10}^{-3}$$

(69)

Negligible modification at large scales. DFT cosmology essentially identical to $\Lambda$CDM except possibly low-ℓ CMB.

8.5. N-Body Simulation Predictions

We anticipate (simulations needed):

• Halo density profiles: $\rho \left(r\right)\sim r^{-\gamma }$ with $\gamma$ differing from NFW
• Modified halo mass function at low masses
• Alleviated "too big to fail" and satellite problems
• Different cluster collision dynamics (Bullet Cluster)

8.6. Cosmological Concordance Summary

Table 7. DFT vs $\Lambda$CDM cosmology

Observable	DFT Prediction	Status
CMB acoustic peaks	Same as $\Lambda$CDM	Consistent
CMB low-ℓ ($\ell <30$)	5-10% deficit	Possible explanation
Matter power spectrum $P\left(k\right)$	$<0.1\%$ deviation	Consistent
BAO scale	Same as $\Lambda$CDM	Consistent
Late-time acceleration	Requires $\Lambda$	Does not solve DE

Table 7. DFT vs $\Lambda$CDM cosmology

Observable	DFT Prediction	Status
CMB acoustic peaks	Same as $\Lambda$CDM	Consistent
CMB low-ℓ ($\ell <30$)	5-10% deficit	Possible explanation
Matter power spectrum $P\left(k\right)$	$<0.1\%$ deviation	Consistent
BAO scale	Same as $\Lambda$CDM	Consistent
Late-time acceleration	Requires $\Lambda$	Does not solve DE

9. BMS Symmetry and Soft Theorems

9.1. Truncated BMS Algebra

Massive field fall-off: $\varphi \sim r^{-1}{e}^{-\mu r}$ (exponential) vs massless $\varphi \sim r^{-1}$ (power-law).

Effective multipole cutoff:

$${\ell }_{max}\sim \mu r_0$$

(70)

for observer at distance $r_0$.

Conjectured DFT BMS algebra:

$${\mathrm{BMS}}_{\mathrm{DFT}}=\mathrm{Supertranslations}(\ell <\mu r_0)\oplus SO(3,1)$$

(71)

For quadrupole ($\ell =2$) at 10 kpc: $\mu r_0=0.59$, so partially broken.

9.2. Massive Soft Graviton Theorem

Soft theorem modification:

$$\underset{\omega \to 0}{lim}{\mathcal{A}}_{n+1}\left(\omega \right)=\left[S^{\left(0\right)}+\omega S^{\left(1\right)}+\cdots \right]\times e^{-\mu /\omega }\times {\mathcal{A}}_n$$

(72)

Factor $e^{-\mu /\omega }$ suppresses $\omega <\mu$ modes.

9.3. Charge Non-Conservation

BMS charge evolution:

$${\displaystyle \frac{d{Q}_f}{du}}=-{\mu }^2{\int }_{S^2}{d}^{2}zf(z,\overline{z})·\widehat{\omega }(u,z,\overline{z})$$

(73)

At late times: $Q_f(\infty )=Q_f(-\infty )+\mathcal{O}\left(e^{-\mu u}\right)$.

9.4. Black Hole Information Paradox

Soft hair entropy with truncated BMS:

$$S_{\mathrm{soft}}^{\mathrm{DFT}}\sim {\left(\mu R_S\right)}^2{S}_{\mathrm{BH}}$$

(74)

For solar-mass BH ($R_S=3$ km):

$$\mu R_S=5.7\times {10}^{-18}\Rightarrow S_{\mathrm{soft}}^{\mathrm{DFT}}\sim 3\times {10}^{-35}{S}_{\mathrm{BH}}\approx 0$$

(75)

Worsens information paradox rather than resolving it. Alternative mechanisms needed (islands, remnants, non-locality).

10. Quantum Aspects and UV Considerations

10.1. Canonical Quantization

Field and momentum operators:

$$\begin{array}{cc}\hfill \widehat{\omega }(\mathbf{x},t)& =\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{\sqrt{2{\omega }_k}}}\left[{\widehat{a}}_k{e}^{i(kx-{\omega }_{k}t)}+\mathrm{h}.\mathrm{c}.\right]\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill \widehat{\pi }(\mathbf{x},t)& ={\displaystyle \frac{\alpha }{c^2}}{\partial }_t\widehat{\omega }\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill \widehat{\omega }(\mathbf{x},t),\widehat{\pi }(\mathbf{y},t)]& =i\hslash {\delta }^3(\mathbf{x}-\mathbf{y})\hfill \end{array}$$

(78)

with dispersion ${\omega }_k=c\sqrt{k^2+{\mu }^2}$.

Quanta are "frequons": spin-0 particles mass $\mu =6\times {10}^{-29}$ eV.

10.2. Renormalization Group Flow

One-loop effective action (schematic):

$$\begin{array}{cc}\hfill {\beta }_{\alpha }& ={\displaystyle \frac{{\lambda }_1^2}{16{\pi }^2\alpha }}+\cdots \hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill {\beta }_{\beta }& ={\displaystyle \frac{{\lambda }_2\beta }{16{\pi }^2}}+\cdots \hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill {\beta }_{\gamma }& =-{\displaystyle \frac{{\gamma }^2}{16{\pi }^2\alpha }}+\cdots \hfill \end{array}$$

(81)

Possible asymptotically safe UV fixed point ${\lambda }_i^{*}$ if non-Gaussian fixed point exists.

10.3. Effective Field Theory Cutoff

As EFT, valid up to scale:

$${\Lambda }_{\mathrm{DFT}}\sim {\displaystyle \frac{\mu M_P}{\sqrt{\xi }}}(M_P=\sqrt{\hslash c/G})$$

(82)

With $\xi <{10}^{-15}$ m² from WEP constraints: ${\Lambda }_{\mathrm{DFT}}\gtrsim {10}^{10}$ GeV.

Higher-dimensional operators:

$${\mathcal{L}}_{\mathrm{eff}}={\mathcal{L}}_{\mathrm{DFT}}+{\displaystyle \frac{c_1}{\Lambda }}{(\partial \omega )}^3+{\displaystyle \frac{c_2}{{\Lambda }^2}}{(\partial \omega )}^4+\cdots$$

(83)

10.4. Black Hole Thermodynamics

Modified Hawking temperature if $G_{\mathrm{eff}}=G(\omega /{\omega }_0)$:

$$T_H^{\mathrm{DFT}}=T_H^{\mathrm{GR}}\times {\left({\displaystyle \frac{{\omega }_{\mathrm{horizon}}}{{\omega }_0}}\right)}^{-1}$$

(84)

Bekenstein-Hawking entropy:

$$S_{\mathrm{BH}}^{\mathrm{DFT}}=S_{\mathrm{BH}}^{\mathrm{GR}}\times {\left({\displaystyle \frac{{\omega }_{\mathrm{horizon}}}{{\omega }_0}}\right)}^{-1}$$

(85)

Requires consistent second law formulation.

10.5. UV Completion Possibilities

• String theory: Unlikely—dilaton couples differently
• Loop quantum gravity: Requires major reformulation
• Emergent spacetime: From entanglement or information
• New physics: Non-commutative geometry, causal sets

Quantum completion remains open problem.

11. Experimental Falsification Protocol

11.1. Three Independent Pillars

• Galactic Dynamics: Rotation curve decline at $r>50$ kpc
• GW Memory: 45% suppression in LISA galactic binaries
• LC Oscillator: Plateau response vs Lorentzian

Falsification by any pillar invalidates DFT.

11.2. Pillar 1: Galactic Rotation Curves

Required Data:

• HI maps to $r>100$ kpc (SKA, 2027+)
• Outer halo stars (JWST, Roman, 2025+)
• Proper motions (Gaia DR4+, 2027+)

Analysis:

• Measure $v\left(r\right)$ to $r>100$ kpc
• Fit models: NFW+DM, DFT, MOND
• Bayesian comparison (BIC, Bayes factors)
• Critical: Decline at $r>50$ kpc?

Predicted Outcomes:

Observation	Interpretation
Decline at $r>50$ kpc	DFT supported
Continued flatness	DFT falsified
Rise at large r	New physics needed

11.3. Pillar 2: LISA Memory Detection

Target Sources:

• Verification binaries: $\sim {10}^4$ WD-WD at 1-30 kpc
• Sgr A* EMRIs: ∼ few events
• MBH mergers: $\sim {10}^2$ at $z>1$

Procedure:

• Detect with matched filtering
• Subtract oscillatory component $h_{\mathrm{osc}}\left(t\right)$
• Integrate residual: $\Delta h_{\mathrm{mem}}=\int [h_{\mathrm{obs}}-h_{\mathrm{osc}}]dt$
• Compute ratio $\mathcal{R}=\Delta h_{\mathrm{obs}}/\Delta h_{\mathrm{GR}}$

Predictions:

Source	D (kpc)	DFT $\mathcal{R}$	GR $\mathcal{R}$
Galactic binary	$<20$	0.5-0.7	1.0
Sgr A* EMRI	8	$\sim 0.6$	1.0
MBH at $z=1$	$\sim {10}^6$	$\sim 0$	1.0

Critical: Measure $\langle \mathcal{R}\rangle$ for $\sim 10$ galactic sources. DFT falsified if $\langle \mathcal{R}\rangle =1.0\pm 0.1$.

11.4. Pillar 3: LC Oscillator Experiment

Apparatus:

• High-Q LC resonator ($Q>{10}^6$)
• Frequency synthesizer (linear sweep)
• Lock-in amplifier
• Temperature control ($\pm 0.01$ K)

Procedure:

• Resonant frequency $f_0\approx 1$ MHz
• Sweep: $f\left(t\right)=f_0+kt$, $k=10$ Hz/s
• Measure $P\left(t\right)$ continuously
• Plot P vs f

Predictions:

• Standard: Lorentzian $P\propto 1/[{(f-f_0)}^2+{(\gamma /2)}^2]$
• DFT: Plateau $P\approx P_0$ for $|f-f_0|<\Delta f_{\mathrm{plateau}}$

For $Q={10}^6$, $f_0=1$ MHz, $k=10$ Hz/s:

$$\Delta f_{\mathrm{plateau}}\sim {\displaystyle \frac{k{\tau }_{\mathrm{ring}}}{2}}\approx 1.6\mathrm{Hz}$$

(86)

Clear signature: flat response over $\sim 2$ Hz vs sharp peak.

11.5. Timeline and Resources

Table 8. Falsification timeline

Test	Earliest Result	Cost	Difficulty
LC oscillator	2026	$50k	Low
Galactic dynamics	2027-2030	(funded)	Medium
LISA memory	2037-2040	$1.5B	High

Table 8. Falsification timeline

Test	Earliest Result	Cost	Difficulty
LC oscillator	2026	$50k	Low
Galactic dynamics	2027-2030	(funded)	Medium
LISA memory	2037-2040	$1.5B	High

Priority: LC oscillator (immediate, low-cost), then galactic data, then LISA.

11.6. Null Results and Theory Adjustment

• Pillar 1 fails: Consider $\mu \left(r\right)$, screening, or abandon galactic application
• Pillar 2 fails: Revise $\xi$, consider $\mu \ll {10}^{-21}$ m⁻¹, or abandon DFT
• Pillar 3 fails: Reconsider Axiom 3, modify dissipation, or reject DFT

All pillars independent; success of all three provides strong evidence.

11.7. Comparison with Alternatives

Table 9. Discriminating DFT from alternatives

Prediction	DFT	MOND	$\mathit{f}\left(\mathit{R}\right)$	$\mathit{\Lambda }$CDM
Rotation decline	Yes	No	No	No
Memory suppression	Yes	No	No	No
LC plateau	Yes	No	No	No
WEP violation	${10}^{-47}$	${10}^{-10}$	${10}^{-6}$	0
CMB low-ℓ deficit	Yes	No	Maybe	No

Table 9. Discriminating DFT from alternatives

Prediction	DFT	MOND	$\mathit{f}\left(\mathit{R}\right)$	$\mathit{\Lambda }$CDM
Rotation decline	Yes	No	No	No
Memory suppression	Yes	No	No	No
LC plateau	Yes	No	No	No
WEP violation	${10}^{-47}$	${10}^{-10}$	${10}^{-6}$	0
CMB low-ℓ deficit	Yes	No	Maybe	No

Only DFT predicts all three distinctive signatures.

12. Discussion and Outlook

12.1. Summary of Key Results

• Theorem: Spin memory possible without spin-2 mediators given four conditions
• Mechanism: Chirality from torsion $K_{\mu \nu }^{\lambda }\propto ϵJ\partial \omega$
• Prediction: Yukawa-suppressed memory $\Delta \tau \propto e^{-\mu D}$
• Parameter: ${\mu }^{-1}=17\pm 3$ kpc fits galactic rotation curves
• Tests: Three independent falsifiable tests with clear timelines
• Consistency: Solar system, WEP, cosmology (mostly) consistent

12.2. Comparison with Standard Paradigm

Aspect	$\Lambda$CDM+GR	DFT
Gravitational mediator	Spin-2 graviton	Scalar + torsion
Dark matter	Yes (27%)	No (Yukawa modification)
Dark energy	$\Lambda$ (68%)	Still requires $\Lambda$
Galactic parameters	2 per galaxy (NFW)	1 universal ($\mu$)
Spin memory	Standard	Yukawa-suppressed
BMS symmetry	Full	Truncated at $\ell \sim \mu r$
WEP	Exact	Violated ($\sim {10}^{-47}$)

12.3. Advantages of DFT

• Fewer parameters for galactic fits
• No dark matter required for rotation curves
• Clear falsifiability with multiple tests
• Possible explanation for CMB low-ℓ anomaly
• Natural connection galactic scale to GW memory

12.4. Challenges and Open Questions

• Quantum completion: UV formulation needed
• Cosmological simulations: N-body predictions require verification
• Black hole information: Soft hair insufficient
• Axiom 3 justification: Deeper principle needed
• Bullet Cluster and lensing: Detailed predictions required

12.5. Future Directions

• Immediate (1-2 years): LC oscillator experiment, quantum formulation
• Medium (3-5 years): Cosmological simulations, black hole solutions
• Long-term (10-15 years): LISA memory measurements, JWST/SKA galactic data

12.6. Concluding Remarks

We have demonstrated that gravitational spin memory—conventionally regarded as definitive evidence for spin-2 gravity—can emerge from a scalar theory with torsion-induced chirality. DFT offers a minimal alternative to GR that:

1. Explains flat rotation curves without dark matter 2. Predicts Yukawa-suppressed GW memory testable by LISA 3. Remains consistent with all current precision tests 4. Is falsifiable through three independent experiments

Whether DFT is ultimately correct, an effective description, or entirely wrong, it exemplifies scientific methodology: bold hypothesis with clear falsifiability. The coming decade of multi-messenger astronomy—from laboratory experiments to space-based GW detectors—will provide definitive tests.

As Richard Feynman noted: "Science is the belief in the ignorance of experts." The experiments will decide.

A. Metric Ansatz and Newtonian Limit

The metric ansatz $g_{\mu \nu }={\eta }_{\mu \nu }{(\omega /{\omega }_0)}^{\kappa }$ with $\kappa =-1$ yields correct Newtonian limit:

For weak fields $\omega ={\omega }_0+\widehat{\omega }$, $|\widehat{\omega }|\ll {\omega }_0$:

$$g_{\mu \nu }\approx {\eta }_{\mu \nu }(1-\widehat{\omega }/{\omega }_0)$$

(87)

The geodesic equation to first order:

$${\displaystyle \frac{d^2{x}^i}{d{t}^2}}\approx -{\displaystyle \frac{1}{2}}{\partial }_i{g}_{00}\approx {\displaystyle \frac{1}{2}}{\partial }_i(2\Phi /c^2)$$

(88)

where $\Phi =-{\displaystyle \frac{c^2\widehat{\omega }}{2{\omega }_0}}$. This recovers Newtonian acceleration $\mathbf{a}=-\nabla \Phi$.

B. Massive Green’s Function on $S^2$

The Helmholtz equation on 2-sphere:

$$\left[{\displaystyle \frac{1}{sin\theta }}{\partial }_{\theta }(sin\theta {\partial }_{\theta })+{\displaystyle \frac{1}{{sin}^2\theta }}{\partial }_{\varphi }^2-{\left(\mu r_0\right)}^2\right]G_{\mu }=-\delta (cos\theta -cos{\theta }^{\prime })\delta (\varphi -{\varphi }^{\prime })$$

(89)

Solution in Legendre functions:

$$G_{\mu }(\Theta )=-\sum _{\ell =0}^{\infty }{\displaystyle \frac{2\ell +1}{4\pi }}{\displaystyle \frac{P_{\ell }(cos\Theta )}{\ell (\ell +1)+{\left(\mu r_0\right)}^2}}$$

(90)

For $\mu r_0\Theta \ll 1$:

$$G_{\mu }(\Theta )\approx -{\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{e^{-\mu r_0\Theta }}{\Theta }}$$

(91)

C. Bayesian Analysis Details

For galaxy rotation curve fitting, likelihood:

$$\mathcal{L}\left(\theta \right|\mathbf{v})=\prod _i{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_i^2}}}exp\left[-{\displaystyle \frac{{(v_i^{\mathrm{obs}}-v_i^{\mathrm{DFT}}\left(\theta \right))}^2}{2{\sigma }_i^2}}\right]$$

(92)

Priors: $p\left(\mu \right)\sim \mathrm{Uniform}(10,30)$ kpc, $p\left(M_{\mathrm{bar}}\right)\sim \mathrm{Gaussian}(\mathrm{literature},\sigma )$.

MCMC sampling yields posterior $p\left(\theta \right|\mathbf{v})\propto \mathcal{L}(\theta \left|\mathbf{v}\right)p\left(\theta \right)$.

D. Quantum Loop Calculations

One-loop self-energy for frequon:

$$\Pi \left(p^2\right)={\displaystyle \frac{{\lambda }^2}{32{\pi }^2}}\left[{\Lambda }^2-{\mu }^{2}ln\left({\displaystyle \frac{{\Lambda }^2}{{\mu }^2}}\right)+\cdots \right]$$

(93)

Renormalization conditions define renormalized parameters. Beta functions from:

$$\mu {\displaystyle \frac{d\lambda }{d\mu }}={\beta }_{\lambda }={\displaystyle \frac{3{\lambda }^3}{32{\pi }^2}}+\cdots$$

(94)

E. Data Availability and Code

All data and code for this analysis available at:

https://github.com/username/DFT-2024

Includes:

• Rotation curve fitting scripts
• Cosmological perturbation code
• GW memory calculation notebooks
• LC oscillator simulation