# Instructor guide: Vacuum cutoff and curvature-mismatch ledger

Course: Vacuum energy and the cosmological constant

Suggested time: 65–85 minutes

## Learning target

Learners quantify regulator sensitivity and distinguish formal vacuum contributions, renormalized cosmological parameters, observed curvature, and extractable work.

## Prepare

- Review energy-density dimensions and quartic scaling.
- Declare the regulator and field content.
- Provide the observational comparison scale with provenance.

## Facilitation moves

- Require a four-orders-per-decade prediction.
- Label bare, counterterm, and renormalized quantities.
- Ask what operational protocol would define work.

## Misconception checks

- **The cutoff estimate is a measured reservoir density.** It is regulator- and scheme-dependent until embedded in a renormalized observable calculation.
- **Renormalization makes the cosmological-constant problem disappear.** It defines finite parameters; radiative stability and the observed small value remain deep explanatory problems.

## Accessibility and participation

- Translate logarithmic ratios into orders of magnitude.
- Use labeled layers for bare, counterterm, and observed values.
- Offer dimensional-analysis scaffolding.

## Evidence of learning

- A verified quartic scaling result
- A bare-versus-renormalized ledger
- An observable-versus-extractable distinction

## Extension

Compare a hard cutoff with dimensional regularization and identify which physical sensitivity survives scheme changes.

## Evidence boundary

Assess the learner's reasoning only within the declared model and recorded observations. Do not upgrade a simulation result into a claim about an unmodeled physical system.