# Spectral-response exclusion challenge

Turn a proposed zero-point-field inertia mechanism into normalized coefficients, bounded predictions, and precision null tests.

## Instructor edition

## 1. Reconstruct the spectral inertia coefficient

**Task type:** derivation

Starting from a declared spectral density and material response function, derive the effective inertial coefficient, state the cutoff or convergence condition, and test the expression in narrow-band and vanishing-response limits.

### Deliverables

- Normalized integral with dimensions
- Declared response and cutoff assumptions
- Two analytic or numerical limiting cases

### Scoring criteria

- Integral and coefficient construction: 8 points
- Convergence and dimensions: 7 points
- Limiting cases: 5 points

### Solution outline

- Keep the spectral density, coupling, and response normalization distinct.
- Show whether the result converges or depends on an external cutoff.
- A fitted coefficient alone does not establish the microscopic interpretation.

## 2. Turn predictions into an exclusion map

**Task type:** analysis

Sweep response center, bandwidth, cutoff, and experimental precision. Compare predicted composition or boundary-condition contrasts with calibrated bounds and identify excluded, testable, and presently unconstrained regions.

### Deliverables

- At least 36 parameter records
- Exclusion map with uncertainty propagation
- A list of assumptions controlling each boundary

### Scoring criteria

- Prediction-bound comparison: 7 points
- Uncertainty and threshold handling: 7 points
- Transparent assumption dependence: 6 points

### Solution outline

- Use the same observable definition for prediction and experimental bound.
- Propagate calibration uncertainty before assigning exclusion.
- Label regions unconstrained when the apparatus lacks sensitivity; absence of exclusion is not support.

## 3. Blind a composition-and-boundary null test

**Task type:** design

Design a precision experiment that independently modulates composition and electromagnetic boundary conditions. Control thermal, electrostatic, magnetic, vibration, and analysis-selection artifacts.

### Deliverables

- Factorial schedule with concealed condition labels
- Artifact budget and injected-signal qualification
- Primary estimator, stop rule, and replication trigger

### Scoring criteria

- Orthogonal modulation and identifiability: 7 points
- Artifact controls and metrology: 7 points
- Blinded decision protocol: 6 points

### Solution outline

- Use randomized blocks so drift cannot mimic a condition effect.
- Qualify the apparatus with hidden injections spanning the target scale.
- Require the predicted pattern across both factors, not a single residual, before advancing the mechanism.

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