Lesson 25 of 26Plan a cumulative investigation with a feasible fallback
Specify evidence before collecting a favorable result.
Try
Choose one coherent investigation route: a modeling question with observations, a proof-centered question about a mathematical family, or a numerical-method comparison with an independent reference. These routes can overlap, but a proposal should not promise every technique at once.
A concrete modeling proposal might study whether total-stock observations can distinguish two loss channels. A proof proposal might derive a spanning-tree formula for a graph family. A numerical proposal might compare heat solvers against exact sine modes under equal work budgets.
Predict
Which proposal can still produce a substantial result if its preferred conjecture fails? Identify a fallback that preserves a meaningful question rather than replacing the project with a summary of definitions.
Experiment
For the loss-channel project, use M′=−(a+b)M with known M₀. Total-stock data identify s=a+b. If channel A accumulated loss Lₐ is also observed at positive time, then r=Lₐ/(M₀−M) identifies a/s, giving a=rs and b=(1−r)s.
| Milestone | Observable deliverable |
|---|
| Question and source review | Parameter domain, observation plan, verified references |
| Mathematical mechanism | Proof of same-sum ambiguity and added-channel recovery |
| Controlled investigation | Synthetic data, noise sensitivity and a frozen evaluation case |
| Report and review | Reproducible artifacts, limitations, response to critique |
The formula fails when total loss is zero. Include that boundary case in the proposal’s evidence plan. Under noise, ratio instability near zero loss may be a central finding rather than an inconvenience to omit.
Understand
A complete proposal specifies the question, motivation, prerequisites, sources, assumptions, methods, data permissions, computational bounds, milestones and acceptance criteria. Each criterion should connect to a claim: conservation checks support the solver, a rank argument supports identifiability, and held-out observations support a defined predictive comparison. None of these alone establishes the entire investigation.
For a proof route, define the object family and quantified conjecture. Plan exact small cases, an attempted general lemma and adversarial examples. If the conjecture fails, a classified obstruction or a corrected theorem can be a legitimate result. Do not promise novelty before investigating existing work.
For a numerical route, choose an exact special case or independently verified reference, state error norms and refinement sequences, and account for computational work. Comparing methods at the same step size may not mean equal work. Distinguish model error from discretization error and record when a method fails or exceeds its budget.
For an empirical route, unavailable data may require a documented scope change. Synthetic data can study recovery under known assumptions, but cannot replace real validation while retaining the same empirical claim. Write this fallback before results so it does not become a hidden post hoc change.
Apply
Submit a proposal with four evidence-based milestones and one substantive fallback. Include at least one unfamiliar evaluation or proof challenge that is not copied from a worked lesson. Ask a human reviewer to assess feasibility, mathematical depth and evidence alignment. The proposal remains unapproved until that review actually occurs; this lesson supplies preparation and tasks, not an approval mechanism.
Continue
The final lesson assembles a defended investigation. Completion requires an actual argument and evidence package; a draft’s length or the presence of saved text cannot certify research quality.
