Full Newtonian Gravity Derivation

Newtonian Gravity as the kr≪1 Limit of PBG

(Built on Foundations v1.0 · 2025-06-10)

One-line summary – In Phase-Biased Geometry the familiar F=GMm/r2
force is the small-kr limit of the universal phase profile

Φ(r)=Ne−krr(see Foundations §2, eq 2.8),

combined with the least-cost motion rule

mr¨=−mc2∇Φ(Foundations §3, eq 3.1).

1 Canonical constants (imported)

All numbers below are frozen by Foundations §5.

Symbol Value Units
α 0.090034 J m−1
β 5.30012 × 10−54 J m−3
γ=α/c2 1.0018 × 10−18 J s2 m−3
k=β/α 7.67 × 10−27 m−1
G=c4/4πα 6.674 20 × 10−11 m3 kg−1 s−2

(If you need a refresher on how these arise, see Foundations §5.)

2 Field outside a point mass

For a source mass M (winding number N):

Φ(r)=Ne−krr,N=c2M4πα(Foundations §2, eq 2.8).

Because k−1≈1.3×1026 m, e−kr≈1 anywhere inside the Local Group of galaxies.

3 Universal force law

F=−mc2∇Φ⟶|F|=GMmr2e−kr(1+kr)(Foundations §3).

Newton limit – when kr≪1 (essentially all laboratory, solar-system, and galactic scales),

|F|→GMmr2.

4 Worked example – orbital velocity of Earth

Input

Calculation

a=GM⊙r2=5.93×10−3m s−2,v=ar=29.79km s−1.

Matches ephemeris to 10−4; Yukawa suppression [e−kr(1+kr)−1]<10−15.

5 Where deviations might appear

The first-order correction term is ΔF/F≈kr.
At r=1024 m (≈300 Mpc) the deviation reaches 1 %, providing a clean cosmological test of PBG distinct from standard ΛCDM.

Unit sanity

Left-hand side of F=GMm/r2 → N.
Right-hand side with constants table → N ✓

Cite
Foundations/Full-Derivations §2–§3–§5 for
kernel, force law, and constants. No other derivations are performed on this page.

Foundational Definitions