Cosmology

Modelled[M] geodesic packaging. October 2026 distances, lunar-laser ratio, and vacuole distinction are algebra or cited comparisons. The join stays open.

S³ Cosmology

Cosmological redshift is the conformal ratio 1+z=1/a(t)=R₀/R, with the coasting profile H(z)=H₀(1+z). Tag [M]. The one-component equivalent is w=−1/3. The join stays open.

Last updated: 19 October 2026

Reading this entry

The S³ cosmology page gives the framework its coasting redshift, the conformal ratio 1+z = 1/a(t) = R₀/R with H(z) = H₀(1+z), as the modelled geodesic package [M]. At the clothes baseline H₀ = 68.973 the same profile fixes the luminosity distance and the flat one-component equivalent w = −1/3.

What this is used for

Later distance tables, the Hubble splice, and the microwave comparisons all start from this coasting profile. The clothes radius R₀ = √6 c/H₀ = 10646.75 Mpc and c/H₀ = 4346.52 Mpc are the scales those pages share.

Walk through

The page opens on 1+z = 1/a(t) = R₀/R and H(z) = H₀(1+z) [M], states the luminosity distance d_L = (c/H₀)(1+z) ln(1+z), and gives the flat one-component equivalent w = −1/3. At H₀ = 68.973 km s⁻¹ Mpc⁻¹ it prints c/H₀ = 4346.52 Mpc and R₀ = √6 c/H₀ = 10646.75 Mpc, with the Heegaard neck at η = π/4 and the hyperspherical equator at χ = π/2. A separate Weber package then writes ρ_W(z) = ρ₀(1+z)³ [1 − 3H(z)²/(H₀²(1+z)²)] and H(z)/H₀ = √[(Ω_m(1+z)³ + (1−Ω_m)(1+z)²)/(1 + 3Ω_m(1+z))] / E(0), whose deceleration is q₀ ≈ −0.55. The October table lists coasting d_L = 6025.55 Mpc at z = 1.00 and 9132.58 Mpc at z = 1.40, and the modulus difference at z = 1.40 is about −0.233 mag.

Integrity note (Doctrine v1.1.7): The redshift definition on this page is modelled [M]. It is not a residual-chip flip. Hold, the hard join, and fl-heavy stay as they are. Foot δ is [E INPUT]. Foot K/L is [E], the algebra of that input. A frozen product with constant radius pays spectroscopic redshift 0. That score is not a reading of 1+z=R₀/R. The join stays open. The geometry spoke is linked from measey.com.

1. The framework definition of cosmological redshift

Cosmological redshift is the conformal scale-factor ratio of the everting vacuum torus-knot lattice,

[M] geodesic packaging. Coasting profile. Not tired light.

Because the stretch is metric, event durations and a blackbody both scale with that same factor, and the blackbody stays a blackbody. The luminosity distance of this profile, , is algebra of the profile. A flat one-component history with has . The figure is not that inversion. The pair is not that output.

The Heegaard neck is the Clifford torus at . The hyperspherical equator is the great two-sphere. Matter in one solid torus and antimatter in the other is a modelled package: dual-vortex initial data, mutual Ampère sorting, a ballistic split, and rest-mass locking. The baryon-to-photon ratio is not a derived output. Pair creation on the neck is net-zero.

New clothes-radius numbers use km s⁻¹ Mpc⁻¹ [P]. At that baseline Mpc and Mpc. The lock stays [M]. A frozen product still pays spectroscopic redshift 0. Identifying the coasting profile with that stationary radius stays open. If and that radius expands, is 928 times the quoted lunar-laser half-width . A stationary container does not write .

2. A separate Weber package, not selected

A non-linear Weber density correction on compact is a further modelled package. It is not selected against the coasting profile above, and it is not a match to the DESI dark-energy pair.

The Mach anchor on this page's older diagram used a round . That diagram is not recomputed here. New clothes numbers stay on 68.973.

3. The Non-Linear Weber Friedmann Expansion Rate

Combining the Weber density correction with the curvature term yields the normalized expansion rate:

Weber S³ Friedmann Expansion Formula

4. Older comparison table, not a DESI certificate

Observable	Standard Lambda-CDM	Weber package, unselected	Status
Dark Energy $\Omega_\Lambda$	0.685 (Ad-hoc)	not selected	unselected package
Deceleration $q_0$	-0.53	-0.55	Modelled [M]
DESI 2024 BAO ($z=0.51$)	12.38	13.12	5.96% Error
SNIa $d_L(z=1.0)$	6680.4 Mpc	6725.1 Mpc	unselected; not the coasting d_L

5. October 2026 numbers

At km s⁻¹ Mpc⁻¹ the coasting distance and a flat ΛCDM column with , , and the same , are:

z	Coasting d_L	Flat ΛCDM d_L
1.00	6025.55 Mpc	6706 Mpc
1.40	9132.58 Mpc	10167.74 Mpc

The modulus difference at is about mag. That is model against model. A Pantheon+ χ² is not evaluated. The same gives ΛCDM ages 472.8 Myr at , 368.1 Myr at , and 297.0 Myr at . Extending the coasting profile from redshift to infinity gives Myr at . That integral is not an age of the stationary container. No formation clock is exhibited.

The boxed radius is the Einstein–de Sitter mass match: 121.83 pc for the Sun. The balance is an outward acceleration , and it gives 96.70 pc for the Sun and 1.395 pc for the Earth. On the coasting profile , so that acceleration is not the scale-factor tide. The depth of an orbit inside either radius does not write a leak. Inside a time-independent index the orbital scale rate is already zero.

The microwave pages state the scored peak and quadrupole. An inserted 147 Mpc cell on this subtends , which is 3.2% from the quoted peak 220.5 and about 14 times the quoted half-width ±0.5. The native acoustic comparison against ~220 stays that comparison. The placed quadrupole pair is 225.895 against 1016.730 μK², a 77.78% deficit. The 5.56% S³ factor covers 7.14% of it.

The quoted lithium centrals 4.68×10⁻¹⁰ and 1.58×10⁻¹⁰ stand in the ratio 2.962. No equilibrium abundance is computed. The neutrino candidate meV is one scale under [M]. It lies under the quoted KATRIN ceiling 0.45 eV. The three absolute masses stay open.

The index is the constitutive sentence in use. For the weak-field bend has magnitude . Isotropic elasticity does not set that exponent. The Novello–Zel'dovich 4D element may be set aside. The Fermat principle on Riemannian 3-space still uses that 3-metric. Ω_CDM=0 is not written.

STATUS: [M] MODELLED PACKAGE

The coasting redshift definition is [M]. The Weber expansion rate on this page is a separate modelled package and is not selected. The coasting luminosity distance is the table in §5, not the 6725 Mpc row above. Neither package is an [E] match to DESI. The join stays open.

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