# Inverse-geometry stage-gate review

Move from a desired geometry to source requirements, numerical credibility, surrogate tests, and explicit stop rules.

## Instructor edition

## 1. Solve the inverse source problem

**Task type:** derivation

Choose a bounded target metric, compute its curvature invariants and implied source tensor, then identify conservation, energy-condition, and regularity constraints. State which requirements are mathematical and which lack a demonstrated physical source.

### Deliverables

- Metric, connection method, and invariant set
- Implied source components in a declared frame
- Constraint and feasibility-status table

### Scoring criteria

- Geometry-to-source calculation: 8 points
- Consistency checks: 7 points
- Physical status labels: 5 points

### Solution outline

- Verify coordinate-independent invariants before interpreting coordinate components.
- Check covariant source conservation and singular limits.
- Keep mathematical source requirements separate from material realizability.

## 2. Build the gate-constrained design map

**Task type:** analysis

Sweep at least two geometry parameters and score invariant regularity, source magnitude, numerical convergence, null-model discrimination, and replication readiness in sequence. A failed early gate must prevent later readiness claims.

### Deliverables

- At least 30 parameter records
- Ordered gate map with uncertainty
- Sensitivity analysis for thresholds and mesh resolution

### Scoring criteria

- Gate metrics are reproducible: 7 points
- Sequential logic is enforced: 7 points
- Numerical and threshold sensitivity: 6 points

### Solution outline

- Evaluate invariant and source gates before experimental analogies.
- Demonstrate convergence across resolution changes rather than reporting one solver output.
- A later attractive metric cannot rescue a failed prerequisite gate.

## 3. Design a surrogate metric experiment

**Task type:** design

Propose an analogue or surrogate experiment tied to one calculable metric signature. Define what it can test, what it cannot establish about gravity, and the artifact that would most plausibly mimic the signal.

### Deliverables

- Forward model linking control to observable
- Matched null and dominant-artifact controls
- Stage gates, stop rules, and replication artifact list

### Scoring criteria

- Analogy has a precise mathematical target: 7 points
- Limits and artifacts are explicit: 7 points
- Advancement rules are prespecified: 6 points

### Solution outline

- Validate the control-to-observable transfer function before testing the target condition.
- Use a null system with the same readout chain.
- Report a surrogate result as a test of the analogue model, not direct production of spacetime curvature.

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