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Exact massless scalar on a conformally flat external metric — Examples

1 · Prerequisites

2 · Summary

The constant conformal factor isolates the field and volume normalization. It verifies the $b^{3}$ test multiplier, $b^{4}$ volume density, and $b^{- 2}$ two-point kernel for $g = b^{2} η$ , without selecting a state for a general curved metric.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Constant conformal factor verifies field and volume normalization

Example

Take the theorem's external metric with a constant conformal factor $Ω = b > 0$ , so $g = b^{2} η$ . The scalar field is smeared by $Φ_{g} (f) = Φ_{0} (b^{3} f)$ relative to $d {v o l}_{g} = b^{4} d^{4} x$ . In particular, $R_{g} = 0$ and $P_{g} = b^{- 2} □_{η}$ . Relative to $d {v o l}_{g} (x) d {v o l}_{g} (y)$ , the two-point kernel is $W_{g} (x, y) = b^{- 2} W_{0} (x, y)$ , so its double smearing is $b^{6}$ times the flat covariance. This example checks the constant-factor normalization only; it asserts no preferred state for a general curved metric.

Facts & Assumptions

Given: ${A C}_{ω}$ , a positive constant $b$ , and the selected massless field construction of Exact massless scalar on a conformally flat external metric.

[A1]

${A C}_{ω}$ is the countable-choice assumption of The Axiom of Countable Choice ( ${A C}_{ω}$ ); it is inherited from the conditional QFT and EMG supplier chains.

[F1]

Exact massless scalar on a conformally flat external metric: $R_{g} = - 6 Ω^{- 3} □_{η} Ω$ and $P_{g} (Ω^{- 1} u) = Ω^{- 3} □_{η} u$ for smooth $u$ .

[F2]

Exact massless scalar on a conformally flat external metric: for every compact test $f$ , $Φ_{g} (f) = Φ_{0} (Ω^{3} f)$ .

[F3]

Exact massless scalar on a conformally flat external metric: relative to $d {v o l}_{g} (x) d {v o l}_{g} (y)$ , $W_{g} (x, y) = Ω (x)^{- 1} Ω (y)^{- 1} W_{0} (x, y)$ .

Verification

technique · direct

1.1givenA1F1algebra

Since $\partial_{a} b = 0$ , $□_{η} b = 0$ and [F1] gives $R_{g} = 0$ . Also $g^{a b} = b^{- 2} η^{a b}$ and $d {v o l}_{g} = b^{4} d^{4} x$ , so the density definition gives $□_{g} = b^{- 2} □_{η}$ and hence $P_{g} = b^{- 2} □_{η}$ .

2.1givenA1F2F3step 1.1algebra

Substituting $Ω = b$ into [F2] gives $Φ_{g} (f) = Φ_{0} (b^{3} f)$ ; [F3] gives the relative kernel $W_{g} = b^{- 2} W_{0}$ . In the double smearing the two volume factors contribute $b^{4} b^{4}$ , so the total coefficient is $b^{- 2} b^{8} = b^{6}$ , exactly the coefficient from the two test multipliers $b^{3} b^{3}$ .

3.1givenF2F3step 2.1∎

At $b = 1$ these formulas reduce to the flat metric, field, and covariance. Locality is inherited from the theorem and is not checked here; this example checks only the stated normalization and does not assert uniqueness of a vacuum on a nonconstant curved metric.