# Mean-field gravity validity tribunal

Determine when a renormalized mean source is informative and when fluctuations or curvature invalidate the semiclassical description.

## Instructor edition

## 1. Reconstruct the semiclassical source equation

**Task type:** derivation

Vary a gravity-plus-matter effective action to obtain the semiclassical field equation, identifying the renormalized stress tensor and local geometric counterterms. Reduce the equation to a declared one-scale estimate.

### Deliverables

- Action, variation, and sign conventions
- Renormalized equation with each term classified
- One-scale reduction with dimensional check

### Scoring criteria

- Variation and tensor structure: 8 points
- Renormalization terms correctly located: 7 points
- Scale estimate and caveats: 5 points

### Solution outline

- Separate classical gravity, higher-curvature terms, and the expectation value of the matter source.
- State the quantum state and renormalization prescription needed to define the expectation value.
- Use the reduced estimate only as a validity diagnostic, not a substitute for the field equation.

## 2. Chart the mean-versus-fluctuation regime

**Task type:** analysis

Sweep curvature radius, mean stress, and stress-fluctuation ratio. Define thresholds for negligible backreaction and acceptable source fluctuations, then construct a regime map with borderline cases and threshold sensitivity.

### Deliverables

- At least 18 evaluated parameter points
- A two-gate regime map
- Threshold-sensitivity appendix

### Scoring criteria

- Diagnostics are dimensionless and reproducible: 7 points
- Borderline and uncertainty cases handled: 7 points
- Interpretation respects approximation limits: 6 points

### Solution outline

- Test the mean-source backreaction ratio and a normalized fluctuation measure separately.
- Mark valid, backreacting, fluctuation-dominated, and indeterminate regions.
- Show how classifications move when each chosen threshold changes.

## 3. Challenge an induced-gravity signature

**Task type:** design

Design an observational or analogue-gravity study that could constrain one induced-gravity prediction while separating it from classical environmental effects and detector noise.

### Deliverables

- Forward models for target and strongest conventional alternative
- Calibration and environmental-control matrix
- Decision rule plus independent-analysis plan

### Scoring criteria

- Target prediction is bounded: 7 points
- Alternative model receives a fair test: 7 points
- Blinding, stop, and replication rules: 6 points

### Solution outline

- Select a measurable contrast with different parameter dependence under the two models.
- Use negative controls and environmental monitors before interpreting residuals.
- Require an independent analysis implementation and publish null results under the same rule.

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Evidence rule: distinguish calculation, model-dependent inference, experimental observation, and unresolved claim in every response.