1 Canonical Invariant

Let

μ > 0, γ > μ,

and let r denote the evaluation variable. Define

σ(r) = ln( μ²(1 + γr) / (μ + γ) ) − μr.

The admissible domain is

1 + γr > 0, μ + γ > 0.

2 Exponential Manifestation

Define

ρ(r) = eσ(r).

Hence

ρ(r) = [μ²(1 + γr) / (μ + γ)] e−μr.

At the origin,

ρ(0) = μ² / (μ + γ),

therefore

0 < ρ(0) < ∞.

The invariant is finite and positive at the origin.

3 Differential Structure

Define

σ1 = ∂σ / ∂r, σ2 = ∂²σ / ∂r².

Direct differentiation gives

σ1 = γ / (1 + γr) − μ, σ2 = −γ² / (1 + γr)².

Since

σ2 < 0,

the invariant is strictly concave throughout its admissible domain.

4 Closure

Closure is defined by the stationary condition

σ1 = 0.

Equivalently,

γ / (1 + γr) = μ.

Therefore,

r* = 1/μ − 1/γ.

Since γ > μ > 0, it follows that r* > 0.

Evaluating the second derivative at closure,

1 + γr* = γ / μ, σ2(r*) = −μ².

The closure is therefore unique.

5 Recovery

Since

−σ2 = γ² / (1 + γr)²,

it follows that

√(−σ2) = γ / (1 + γr).

The defining constants are recovered directly from the derivatives,

γ = √(−σ2) / [1 − r√(−σ2)], μ = −σ1 + √(−σ2).

Substitution reproduces the original defining constants identically.

6 Dimensional Resolution

Define

f = √ρ.

The radial differential operator in D spatial dimensions is

ℒD = f″/f + [(D − 1)/x] f′/f,

where x is the radial coordinate of the spatial realization.

Using r = x², together with the previously established derivatives, the operator reduces identically to

ℒD = rσ1² + 2rσ2 + Dσ1 + (D − 1)(D − 3) / 4x².

The first three terms are determined entirely by the invariant. The remaining term depends only on the dimensionality of the spatial realization.

It vanishes precisely when

(D − 1)(D − 3) = 0,

giving

D ∈ {1, 3}.

The one-dimensional branch is degenerate.

The unique nondegenerate spatial solution is therefore D = 3.

7 Spatial Realization

Only after the dimensional result has been established is the evaluation variable realized geometrically. For the three-dimensional solution,

r = x² + y² + z².

At closure, r = r*, therefore

x² + y² + z² = r* = 1/μ − 1/γ.

This is the spatial realization of the previously established invariant closure.

No additional assumptions are introduced.

8 Multiplicativity

Define

Mσ(a, b) = σ(n).

Then

Mσ(a, b) = σ(ab),

or equivalently,

Mσ(a, b) ⇔ ab = n.

9 Divisibility

For a ∣ n, there exists k ∈ ℕ such that

n = ak.

Therefore,

a ∣ n ⇔ ∃ k ∈ ℕ : Mσ(a, k) = σ(n).

10 Nested Evaluation

Suppose

a ∣ b, b ∣ n.

Then

b = ac, n = bd,

and therefore

n = acd.

Consequently,

Mσ(a, cd) = Mσ(ac, d) = Mσ(b, d) = σ(n).

Associativity gives

(ab)c = a(bc) = n,

and therefore

Mσ(ab, c) = Mσ(a, bc) = σ(n).

The evaluation is independent of the parenthesization of the factorization.