Gravity

Modelled[M/P] convention-conditional package — Doctrine v1.1. See /gravity/m13-package and /status.

Newton’s G as a Mach package

G is a named mixed-mass invert [M/P], not a free public theorem. Bricks α, √6, and H₀ stay explicit. One-convention Sciama+α does not recover H₀≃69.

Last updated: 17 August 2026

Reading this entry

This page assembles Newton G as a named mixed-mass Mach package: the double-copy ratio, compact S³ closure, and the placed invert that carries α, √6, and H₀ as explicit bricks. The same page records the older α² estimate as an order-of-magnitude companion to the CODATA value of G.

What this is used for

Use this page to see Gm_e²/(ke²) = α_G with α_G = α², the S³ closure 2GM_U/(R_Uc²) = 1, and the placed invert that relates G to α, H₀, ℏ, and the electron and proton masses.

Walk through

The double-copy block sets Gm_e²/(ke²) = α_G and, in the leading reading, α_G = α² with α ≈ 1/137. Mach closure on compact S³ is written 2GM_U/(R_Uc²) = 1. The panel starts from (Ṙ/R)² = 8πGρ/3 − c²/R², sets Ṙ = 0, inserts M_U = (4π/3)ρ R_U³, and reaches 0 = 2GM_U/R_U³ − c²/R_U². The published invert is G = (3α)/(8√6) × ℏ²H₀/(c m_e² m_p), with α placed as an input, √6 a Mach brick, and H₀ the clothes number, while a one-convention Sciama-plus-α pair sits near H₀ ∼ 11. The older table lists ke²/m_e² = 2.533×10⁻²⁸, α² = 5.325×10⁻⁵, G ≈ 6.67×10⁻¹¹, and CODATA G = 6.674×10⁻¹¹.

Integrity note (Doctrine v1.1):

is not a free public derivation. The published mixed-mass invert is convention-conditional. A one-convention Sciama+α evaluation duals to

km s⁻¹ Mpc⁻¹, not 69. See /status and M13 package.

The honest claim

In standard physics, Newton’s constant is a measured parameter. In this programme it is related to laboratory constants by a named Mach book (mixed-mass packing, double-copy [M], and a coefficient that is [E] only in the Sciama–Green slot at ). That is a package, not a closed theorem.

The Double-Copy Relation

The EM–gravity double copy (see EM–Gravity Double Copy) establishes that the gravitational coupling is related to the electromagnetic coupling by squaring the EM amplitude:

where is the gravitational fine-structure constant — the ratio of gravitational to EM coupling at the electron mass scale.

The double copy gives in the leading approximation (where is the EM fine-structure constant). This gives an immediate estimate of .

Mach Closure on S³

The universe has finite total mass (because is compact). The Mach closure condition — that the inertial frame is entirely determined by the universe’s matter content — gives:

Mach closure on S³ — the universe is exactly at its own Schwarzschild radius

Published invert (M13)

The live quantitative book is the mixed-mass invert

Bricks: is INPUT + placement [M]; is a Mach brick [M]; is the live prediction [P]. The 0.04% contact appears only if one recycles from this same invert — that is circular. External Planck / SH0ES values give percent-class residuals (see G prediction).

Lab census: a one-convention Green or raw book with does not hit the Hubble band. Putting the Green value into the raw slot is the splice that produces 69. That splice is not a derived coefficient.

Older double-copy estimates (order-of-magnitude only):

Quantity	Value	Source
	N·m²/kg²	Measured constants
		From fine-structure constant
	N·m²/kg²	Older α² estimate (order of magnitude)
	N·m²/kg²	Measured

The α² estimate is only an order-of-magnitude story. The quantitative public book is M13, and it is convention-conditional. See G prediction.

What is not claimed

A one-convention derivation of the published coefficient .
as an independent cosmology theorem.
Local minting of from this invert.

Integrity Status

[M]

Modelled / predictive package. Area packing and the Green potential on S³ are structure. The published G invert is not a one-convention theorem. Remainder k is closed as a derivation. See /status.

Source Documents

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