Unified Topological Mass Framework v3.0

Topological Gravity from Integer Winding: Axioms, Action, and Tests (with Numerical Supplement)

Authors: (Redacted for review)

Version: 3.0 (builds on but does not alter v2.3)

Keywords: topological gravity, scalar–tensor, universal mass coupling, Poisson–Yukawa, PPN, lensing, Shapiro delay, screening, cosmology, identifiability

Abstract

We extend the Unified Topological Mass Framework (UTMF) by promoting the integer winding sector to a universal gravitational scalar φ that multiplies rest scales and clock rates. A minimal generally-covariant action—Einstein–Hilbert plus a canonical kinetic term for φ and a simple analytic potential V(φ)—with matter coupling through φ yields:

  • a screened Poisson–Yukawa equation ∇²Φ - m_y²Φ = 4πGρ with range λ_y = 1/m_y, two-body potential V(r) = -GM/r(1 + αe^(-m_y r))
  • a scale-dependent PPN parameter γ(r) = (1-αe^(-m_y r))/(1+αe^(-m_y r))
  • a luminal GW theorem c_T = 1 for tensor waves
  • laboratory protocols for engineered gravitational redshift using controlled φ
  • a cosmological sector where slow residual unwinding reproduces Λ_eff with atomic-clock–level drift
  • a unified inference pipeline with sharp falsifiers

We also formalize the v2.3 kinetics link between the mean μ, variance, and cumulative drop I, clarifying that Var(N) = μ(1-μ/μ₀) while I = μ₀ - μ.

1. Introduction and motivation

UTMF v2.3 established a dominant topological mass law:

M_topo(N,w,T) = Λ_c e^(λ_c N) + α_c w + κ_c T²

2. Axioms (UTMF-G) and basic definitions

G1 (Universal coupling). All rest masses and clock intervals scale with φ:

m(x) = Λ_c y(x), dτ(x) = y(x) dt_proper, y(x) ≡ e^(λ_c N(x))

G2 (Potential). Define the Newtonian potential by:

Φ(x) = σ ln y(x)

G3 (Source principle). UTMF core energy (set by N) gravitates at leading order; pure holonomy phases do not.

G4 (Relativistic completion). Gravity follows an action (Sec. 3) with Einstein–Hilbert plus a kinetic term for φ and a simple potential V(φ); matter couples by replacing bare rest scales m₀ → m₀y.

G5 (Topological equivalence). Because the same φ sets both inertial mass and gravitational sourcing, free fall is composition-independent at O(α).

3. Relativistic action and matter coupling

We adopt the minimal, generally-covariant action:

S = ∫ d⁴x √(-g) [M*²/2 R - κ_y/2 ∇_μ(ln y) ∇^μ(ln y) - V(y)] + S_m[y,g,Ψ]
V(y) = V₀ y (V₀ ≥ 0)

4. Field equations, stability, and positivity

Variations yield:

M*² G_μν = T_μν^m + T_μν^(y)
κ_y □ ln y = V₀ - (1/√(-g)) δS_m/δy (1/y)
T_μν^(y) = κ_y ∂_μ ln y ∂_ν ln y - κ_y/2 g_μν (∂ ln y)² - g_μν V(y)
m_y² ≡ V₀/κ_y ≥ 0

5. Newtonian limit, Poisson–Yukawa, and two-body potential

In the weak, static, non-relativistic limit with φ = 1 + σΦ and rest mass density ρ:

(∇² - m_y²) Φ = 4π G_topo ρ
V(r) = -G_topo m₁m₂/r (1 + α e^(-m_y r))
ΔF/F_Newt = α(1 + m_y r) e^(-m_y r)
α = 2/(M*² κ_y)

G_N(r₀) ≡ G_topo(1 + α e^(-m_y r₀))

6. Post-Newtonian optics (PPN γ) and classical tests

In isotropic gauge for a static spherical source:

g₀₀ = -(1 + 2Φ), g_ij = (1 - 2γΦ) δ_ij
γ(r) = (1 - α e^(-m_y r))/(1 + α e^(-m_y r))

7. Gravitational waves: c_T = 1

Expanding the action to quadratic order in transverse–traceless modes on FLRW gives:

S_GW^(2) = M*²/8 ∫ a³[(ḣ_ij)² - (∇h_ij)²] dt d³x
c_T = 1

8. Laboratory predictions and mesoscopic tests

(L1) Engineered redshift via controlled φ.

Two identical clocks separated across a device imposing a topological step ΔN exhibit:

Δν/ν = σ λ_c ΔN

(L2) Mass tomography.

From force or phase maps infer φ(x), then N(x) and ρ_topo, directly imaging the integer field.

(L3) Sub-mm fifth-force.

At separation r ≪ λ_y the fractional deviation is ΔF/F ≈ α. At intermediate r ∼ λ_y, use ΔF/F = α(1 + m_y r)e^(-m_y r).

(L4) Atom interferometry.

Phase across a controlled φ-gradient: Δφ_quantum = (m_atom/ℏ) ∫ σ∇Φ·dr.

9. Astrophysics and cosmology

(A1) Lensing–dynamics consistency.

Given M_lens and one G_N calibration, dynamical masses and lensing converge; γ(r) controls small departures from GR in deflection.

(A2) Homogeneous evolution and dark-energy mimic.

Linking to v2.3 kinetics (Sec. 11), the slow drift of y obeys:

ẏ/y = -λ_c Γ₀ e^(-β A y), A = Λ_c(1 - e^(-λ_c))
w_y = -1 + λ_c Γ₀/(3H) e^(-β A y)

(A3) Structure growth.

Linear growth sees γ(r) in the Poisson kernel; δ̈ + 2Hδ̇ = 4πGγρ̄δ.

10. Screening regimes and fifth-force compliance

UTMF-G admits several paths to existing bounds:

Weak coupling: α ≪ 1 suppresses deviations universally.

Short range: λ_y ≪ solar-system scales suppresses γ(r) and Yukawa tails.

Environmental (topological chameleon): with V(φ) ∝ φ^n, m_eff grows in dense regions (screened) while remaining small cosmologically.

11. Link to v2.3 kinetics in N: mean, variance, and drift

Let μ = ⟨N⟩, σ² = Var(N). In the cold-tail linear-death closure:

a_N ≈ 0, b_N = r(t)N, r(t) = Γ₀ e^(-β A y(t))
μ̇ = -r μ, μ(t) = μ₀ e^(-∫₀ᵗ r(s) ds)
Var(N) = μ(t)(1 - μ(t)/μ₀)
I(t) ≡ ∫₀ᵗ r(s)μ(s) ds = μ₀ - μ(t)
ẏ/y = -λ_c Γ₀ e^(-β A y) ⇒ Ei(β A y(t)) = Ei(β A y₀) - λ_c Γ₀ β A t

12. Falsifiers and inference pipeline

Falsifiers (any one is fatal):

  1. Composition-dependent free fall at O(α).
  2. Post-fit mismatch in γ(r) or Shapiro delay after a single G_N calibration.
  3. Engineered redshift fails Δν/ν = σλ_c ΔN.
  4. Inconsistent α across lab/solar/galactic scales.
  5. GW speed c_T ≠ 1 at leading order.
  6. Violation of geometric transition ratios from v2.3 spectral ladders.

Inference pipeline:

Parameters: θ = 100

Data blocks: lab forces, interferometry, light bending/Shapiro, lensing+dynamics, clocks

Likelihood: block-diagonal Gaussians built from the Yukawa potential, γ(r), and φ maps

Identifiability: v2.3 spectral usage requires σ² ≫ 1 for unique nearest-integer N̂

Priors: α ∈ [10⁻⁶, 1], m_y ∈ [10⁻¹⁵, 10⁶] m⁻¹

Numerical Supplement — Worked Examples

S1. Solar-System Safe (Cassini-class deflection compliance)

α = 0.10, m_y⁻¹ = 10⁸ m (range = 10⁵ km)

m_y r_SS ≈ 1.496 × 10³ ⇒ e^(-m_y r_SS) ≈ e^(-1496) ≈ 0

γ(r_SS) = (1-αe^(-m_y r_SS))/(1+αe^(-m_y r_SS)) ≈ 1

Δγ ≡ γ-1 ≈ -2αe^(-m_y r_SS) ≈ 0

S2. Lab-Testable (sub-mm torsion balance / AFM)

α = 5.0 × 10⁻³, m_y⁻¹ = 0.20 mm ⇒ m_y = 5.0 × 10³ m⁻¹

ΔF/F_Newt = α(1+m_y r)e^(-m_y r) = (5×10⁻³) × (1.5) × e^(-0.5) ≈ 4.56×10⁻³

Conclusion: a clean, order-10⁻³ fractional deviation at sub-mm separations while remaining totally invisible at solar-system scales.

S3. Cosmology-Consistent slow drift and w_y

ẏ/y = -λ_c Γ₀ e^(-β A y) ≡ -ε_drift

w_y + 1 = ε_drift/(3H₀) ≲ 10⁻¹⁷/(3 × 6.94×10⁻¹¹) ≈ 4.8×10⁻⁸

This condition constrains the combination λ_c Γ₀ e^(-β A y₀).

A0. Nomenclature

N: integer winding

φ: universal scalar

Λ_c, λ_c: UTMF mass scale and slope

κ_y: coupling in S[φ]

M*: emergent Planck mass

V₀: stiffness of φ

V(φ): scalar potential

m_y: screening mass

α: Yukawa strength

γ: topological PPN

G_N: measured Newton constant

Ei: exponential integral