Mach Closure

Compact S³ lets a Mach sum converge. The Sciama–Green coefficient 3/8 is [E] only at Ω(HR/c)²=1. The universe is not thereby proved to sit at its Schwarzschild radius.

Reading this entry

Compact S³ gives the Mach principle a finite cosmic mass, so the inertia sum over distant bodies converges to one. The page keeps the Sciama–Green potential, with coefficient 3/8 at critical density and R = c/H, as the geometric book later Mach packages use.

What this is used for

Use this page when a later argument needs the closed Mach sum, the split between Φ = GM/(2πR) and the raw compactness GM/(Rc²) = 3π/4, or the closure condition 2GM_U/(R_U c²) = 1.

Walk through

The page separates Φ = GM/(2πR) from the raw compactness GM/R by 2π, and on compact S³ the Mach sum Σ Gm_i/(r_i c²) converges to 1. At critical density and R = c/H the Green value is |Φ|/c² = 3/8, while GM/(Rc²) = 3π/4, again split by 2π, and the closure statement is 2GM_U/(R_U c²) = 1. In flat space the integral of (Gρ/(rc²)) 4πr² dr from zero to infinity diverges. The estimate M_U ≈ 10⁵³ kg and R_U ≈ c/H₀ ≈ 1.3×10²⁶ m returns 2GM_U/(R_U c²) of order 0.5 to 2. The Brans–Dicke reading of that closure is ω_BD → ∞.

Integrity note (Doctrine v1.1): Distinguish the Green potential

from the raw compactness

. They differ by

. Do not park the Green 3/8 in the raw slot and call it a theorem.

Mach's principle

Inertia is not an intrinsic property — it arises from the interaction of each body with all other matter in the universe.

Why S³ satisfies Mach

S³ is compact → finite total mass → the Mach sum converges and equals 1 (the Mach closure condition).

Two books, not one number

On round S³ the exact Green potential is . At critical density and one finds . The raw compactness at the same point is . The factor between them is .

The slogan is an order-of-magnitude Mach statement, not the unique law. Writing the Green 3/8 into a raw slot is the splice behind the published 69 invert.

Why flat ℝ³ fails Mach

The Mach sum diverges for a uniform density infinite universe — the integral diverges.

Numerical estimate

kg (observed baryonic + dark matter), m → depending on mass estimate (order-of-magnitude consistent).

Connection to G

Mixed-mass packing plus the published coefficient book gives a named invert, not a determined G. See M13 package.

The Brans–Dicke connection

Mach closure in S³ gives (recovering GR-like behaviour) while satisfying Mach.

The Mach closure condition

Source Documents

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