# Observer, state, and backreaction defense

Separate state choice, detector response, and geometric consistency before interpreting particles in curved spacetime.

## Instructor edition

## 1. Reconstruct the observer-dependent particle model

**Task type:** derivation

Starting from two complete mode bases related by a Bogoliubov transformation, derive the normalization constraint and the expected occupation number seen by the second basis. Then connect the field result to a finite-time detector response without treating either quantity as observer independent.

### Deliverables

- A line-by-line derivation with conventions and inner product stated
- A table separating mode occupation, detector response, and stress-energy
- Two limiting-case checks

### Scoring criteria

- Correct transformation and normalization: 8 points
- Explicit assumptions and observer labels: 6 points
- Detector comparison and limiting checks: 6 points

### Solution outline

- Use the conserved field inner product to obtain the alpha-beta normalization relation.
- Compute the transformed number expectation from the beta coefficients.
- Show that a switched detector samples a response function rather than a global particle count.

## 2. Map the hierarchy before trusting the approximation

**Task type:** analysis

Construct a dimensionless hierarchy using curvature radius, detector gap, switching time, and renormalized stress scale. Evaluate at least six parameter points and identify where adiabatic reasoning, detector resolution, or negligible backreaction fails first.

### Deliverables

- A reproducible parameter table or dataset export
- Three dimensionless diagnostics with units canceled explicitly
- A boundary plot and a 250-word interpretation

### Scoring criteria

- Dimensionally valid diagnostics: 7 points
- Reproducible sweep and uncertainty treatment: 7 points
- Cautious boundary interpretation: 6 points

### Solution outline

- Compare detector and switching timescales with the curvature timescale.
- Normalize the quantum source to the background curvature scale.
- Classify points by the first violated assumption; do not infer new physics from approximation failure.

## 3. Discriminate state preparation from detector artifact

**Task type:** design

Design a blinded comparison that could distinguish two state prescriptions while controlling switching transients, finite sampling, and detector calibration. Include a conventional explanation capable of producing the same apparent response.

### Deliverables

- A preregistered primary contrast and null model
- Calibration, sham, and state-label blinding plan
- Stop rules and a replication package manifest

### Scoring criteria

- Identifiable primary contrast: 7 points
- Controls address leading artifacts: 7 points
- Decision and release rules are prespecified: 6 points

### Solution outline

- Vary state preparation independently of detector timing wherever possible.
- Use injected signals and sham schedules to test the response pipeline.
- Require stress-energy and detector predictions to agree within the declared model before a state claim advances.

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