PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Exact positive constant solution of the CMC constraint equation

Physical scope

Exact algebraic solution of the zero-cosmological-constant CMC vacuum constraint equation in the stated constant-coefficient case; it is not a universal solvability theorem.

Statement

Let the vacuum conformal data be as in the definition, so that $\tilde{σ}$ is TT with respect to $\tilde{h}$ . Suppose the conformal metric has $\tilde{R} = 0$ and $∣ \tilde{σ} ∣_{\tilde{h}}^{2} = A$ is a positive constant, with constant $τ \neq 0$ . Then the Lichnerowicz equation has exactly one positive constant solution, $ϕ_{0} = {(\frac{3 A}{2 τ^{2}})}^{1 / 12} .$ With a fixed length unit, $[A] = [τ^{2}] = m^{- 2}$ , so $ϕ_{0}$ is dimensionless. Uniqueness is only among positive constant solutions; no uniqueness claim is made for arbitrary positive functions.

Statement references

Vacuum conformal constraint data

Conformal reduction of the vacuum constraint equations

Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$

Rational powers $a^{r}$ of a positive base

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: Vacuum conformal data as in the definition, including a smooth TT covariant tensor $\tilde{σ}$ on a nonempty compact smooth three-manifold, with $\tilde{R} = 0$ , $∣ \tilde{σ} ∣_{\tilde{h}}^{2} = A > 0$ , and constant $τ \neq 0$ . Assume ${A C}_{ω}$ for the inherited embedded Lorentzian constraint interpretation.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ), Vacuum conformal constraint data: ${A C}_{ω}$ is the axiom that every countable family of nonempty sets has a choice function; the local definition requires it only for an embedded Lorentzian interpretation, not for this algebraic solution.

[F1]

Vacuum conformal constraint data: $ϕ$ is positive and dimensionless, $τ$ is the constant trace parameter, and the zero- $Λ$ vacuum scalar residual is defined using $\tilde{R}$ and $∣ \tilde{σ} ∣_{\tilde{h}}^{2}$ ; the input $\tilde{σ}$ is symmetric, trace-free and divergence-free with respect to $\tilde{h}$ .

[F2]

Conformal reduction of the vacuum constraint equations: the scalar residual vanishes exactly when $- 8 \tilde{Δ} ϕ + \tilde{R} ϕ - ∣ \tilde{σ} ∣_{\tilde{h}}^{2} ϕ^{- 7} + (2 / 3) τ^{2} ϕ^{5} = 0$ .

[F3]

Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ : every positive real number has a unique positive twelfth root.

[F4]

Rational powers $a^{r}$ of a positive base: for a positive base, $x^{1 / 12}$ denotes its unique positive twelfth root.

Proof

1.1A1F1F2algebra

For a constant $ϕ > 0$ , $\tilde{Δ} ϕ = 0$ . Substituting $\tilde{R} = 0$ and $∣ \tilde{σ} ∣_{\tilde{h}}^{2} = A$ in [F2], its equation becomes $- A ϕ^{- 7} + (2 / 3) τ^{2} ϕ^{5} = 0$ . Multiplication by $ϕ^{7} > 0$ is reversible, so this is equivalent to $ϕ^{12} = 3 A / (2 τ^{2})$ .

2.1F3F4step 1.1algebra∎

The right side $3 A / (2 τ^{2})$ is positive because $A > 0$ and $τ \neq 0$ . By [F3], it has exactly one positive twelfth root, and by [F4] that root is $(3 A / (2 τ^{2}))^{1 / 12}$ . Thus this is the unique positive constant solution.

Remarks

This is an exact algebraic solution for the stated constant seed data. It does not establish solvability for other conformal metrics or TT tensors.

Exact positive constant solution of the CMC constraint equation

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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