Derivation · Gravity as Natural Ratio

Derivation · Gravity as Natural Ratio

Stripping G reveals the law was always dimensionless

01

Newton's law in SI

F = G · mM / r²

G carries units m³ kg⁻¹ s⁻². Its entire job is to correct the dimensional mismatch between independently defined human unit scales.

↓

substitute   G = F_P · (l_P / m_P)²

02

G replaced by its Planck definition

F = F_P · (l_P/m_P)² · mM / r²

G is now visible as a product of Planck scaling factors — built entirely from SI unit definitions. No new physics. Only bookkeeping.

↓

divide both sides by F_P · expand scaling factors

03

In SI — unit scaling explicit

F/F_P  =  (m/m_P) · (M/m_P) · l_P² / r²

Every quantity expressed as a ratio to its own Planck scaling factor. The unit scaffolding is fully visible — and ready to cancel.

↓

F_P · m_P · m_P · l_P²  all cancel — leaving only the natural ratios

04

Cancellation — unit scaling vanishes

F/F_P  =  (m/m_P) · (M/m_P) · l_P² / r²

The Planck scales introduced in step 2 cancel exactly against the unit scaling. What remains is purely the physical relationship.

F  =  mM / r²

Dimensionless. No G. No Planck scales. No units of any kind.
m, M, r are pure natural ratios — the substrate relationship itself.

G is not the strength of gravity.
G is the cost of measuring gravity with mismatched rulers.

The law was always dimensionless underneath.

J. Rogers · S.E. Ohio