# Vacuum-energy accounting and stability audit

Treat regulated terms, renormalized observables, and cosmological inference as distinct entries in one auditable ledger.

## Instructor edition

## 1. Build the regulator-to-observable ledger

**Task type:** derivation

For a free relativistic field, derive the leading cutoff dependence of the zero-point energy density and show where the quartic scaling enters. Re-express the calculation as bare, counterterm, renormalized, and measured entries without claiming that a regulator-dependent term is directly observed.

### Deliverables

- Derivation with measure, degeneracy, and units shown
- Four-column renormalization ledger
- Comparison of two regulator choices

### Scoring criteria

- Leading scaling and dimensions: 8 points
- Renormalization bookkeeping: 7 points
- Regulator comparison without overclaiming: 5 points

### Solution outline

- Integrate the mode energy over momentum space to expose the cutoff scaling.
- Keep the gravitational counterterm and renormalization condition explicit.
- Note that different regulators rearrange intermediate terms while observables require a stated prescription.

## 2. Stress-test radiative stability

**Task type:** analysis

Create a sensitivity analysis in which field multiplicities, mass thresholds, subtraction scale, and an observational target vary. Report fine-tuning or mismatch diagnostics across the grid and identify which conclusions are prescription dependent.

### Deliverables

- A machine-readable grid of at least 24 rows
- A sensitivity ranking with uncertainty bands
- A claims table labeled calculation, inference, or observation

### Scoring criteria

- Complete and reproducible parameter grid: 7 points
- Sensitivity logic and uncertainty: 7 points
- Evidence labels remain accurate: 6 points

### Solution outline

- Track threshold contributions and signed multiplicities separately.
- Use logarithmic mismatch measures only after declaring the comparison target.
- Distinguish the technical naturalness problem from an experimental detection of vacuum energy.

## 3. Design a model-comparison program

**Task type:** design

Propose a cosmological model-comparison study that tests a restricted vacuum-energy mechanism against a baseline model. State the shared data, nuisance parameters, priors, forecast metric, and result that would count against the mechanism.

### Deliverables

- A bounded hypothesis and baseline
- Prior and nuisance-parameter table
- Mock-data validation, forecast, and falsification rule

### Scoring criteria

- Hypotheses are quantitatively distinguishable: 7 points
- Prior sensitivity and data leakage controls: 7 points
- Falsification and reporting rules: 6 points

### Solution outline

- Choose observables that the mechanism changes, not merely quantities it can fit after tuning.
- Validate the inference pipeline on simulated null and injected datasets.
- Report posterior predictive checks and prior dependence alongside any model score.

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