STANDALONE GEOMETRIC NOTE · V3 · 3 PAGES

One center,one axis.

An exact alignment test for off-center observers.

The result rests on explicit symmetry assumptions, not on accepting a cosmological research programme.

Assumptions

What must be symmetric

Consider a smooth Riemannian spatial slice (Σ, g), a center O and an observer M distinct from O, in a region where the radial direction is defined. The complete configuration includes geometry, fields, boundaries, propagation laws and the history relevant to constructing the observed sky.

This configuration is invariant under SO(3) rotations about O. The observer is at rest or moves radially, with no transverse velocity relative to OM. The observation map is constructed covariantly under rotations.

Result and proof

The stabilizer preserves one axis

The SO(2) rotations about OM leave the configuration and observer unchanged. Covariance therefore makes a scalar sky map invariant under these rotations.

A(Rφ n̂) = A(n̂)aℓm = 0   (m ≠ 0)dA × dB = 0

A covariantly extracted dipole vector must also remain invariant under SO(2). If nonzero, it is parallel or antiparallel to the axis. This conclusion requires neither a cosmological field equation nor a small-depth approximation.

Necessity, not proof of existence. Alignment fixes neither amplitude, sign nor the physical position of a center. A map affected by anisotropic selection no longer satisfies the same assumptions.

Source: Section I, Eqs. (1)–(3).

Local laws

The axis does not select the channel

On mobile: scroll the table horizontally to read all four columns.

Leading small-depth order, when its coefficient is nonzero
Channel	Quantity	Order	Channel assumption
Observer kinematics	Doppler contribution to Z	s⁰	Fixed observer velocity
Endpoint potential	Z⁻	s	Z = ln[N(M)/N(x_em)]
Integrated scalar	O_F⁻	s²	Smooth scalar, local geodesic path
Absolute counts	(dN/dΩ)⁻	s⁴	Smooth effective density ν
Relative count dipole	D_N	s	ν(M) > 0

OF− = (s2/2) ∇F(M) · n̂ + O(s4)(dN/dΩ)− = (s4/4) ∇ν(M) · n̂ + O(s6)DN = (3s/4) |∇ ln ν(M)| + O(s3)Z− = −s ∇ ln N(M) · n̂ + O(s3)

Odd parts are half the difference between opposite directions at equal depth s. Fields and geometry must be sufficiently smooth in a normal neighborhood; ν(M) > 0 is needed for the relative dipole. The law for Z assumes the stated positive endpoint scalar.

A zero gradient removes the leading term of its channel, not necessarily every higher order. These exponents apply to different quantities and are insufficient to identify a mechanism in a deep catalogue. The quantity cZ/s is a spatial proxy, not automatically an estimator of cosmological expansion.

Source: Section II, Eqs. (4)–(9).

A useful counterexample: temperature

Take a centered static spherical emitting surface with uniform temperature T_s. Under stationary endpoint transfer, for a stationary observer, with collisionless propagation preserving phase-space occupation and every viewing direction reaching this surface:

T_obs(n̂) = T_s N(R_s) / N(M)

The observed temperature is direction independent. An off-center observer in a radial potential therefore does not automatically obtain a temperature dipole. This example is not a model of the origin of the cosmic microwave background.

Source: Section III, Eq. (10).

What observations could exclude

Two non-kinematic residual dipoles that are significantly non-collinear and attributed to the same spherical structure exclude that common origin.

The analysis must first specify selection functions, correct kinematic and instrumental contributions, treat local structure and estimate uncertainty. Covariance, including between measurements, must accompany the comparison. Agreement of axes is a compatibility condition, not sufficient confirmation of a model.

Source: Section IV. No new empirical dataset is analyzed in the note.

Open One Center v3 · 3 pages