Senior investigation: distinguish mechanisms behind the same observations
Develop, test and defend a bounded mathematical investigation over several work sessions. The default project asks which measurements distinguish two loss channels when total stock follows the same decay law. All supplied data are synthetic and all guidance is public. You may instead pursue the supported theorem project or propose a comparably scoped question with a reviewer. Saved milestones record learner-authored evidence; they do not authenticate data, assess proof quality, or establish an independently reviewed capstone. A final manuscript, reproducible artifacts and a human defense/revision review are separate evidence requirements.
Planning estimate: 30 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
Review ordinary differential equations, exponential functions, elementary estimation, uncertainty and mathematical proof. Use exact algebra alongside numerical checks; programming is helpful but a transparent spreadsheet or fully specified hand calculation is acceptable.
Default synthetic apparatus: initial stock M(0)=24 mg, no inflow, two irreversible loss channels A and B, time in hours. Let λ>0 have units h⁻¹ and let q(t) be the dimensionless fraction of the instantaneous total loss going to A. Equations are M′=−λM, A′=q(t)λM, B′=(1−q(t))λM, with A(0)=B(0)=0 and 0≤q(t)≤1. M, A and B are amounts in mg. The apparatus has no empirical calibration claim.
Self-contained pilot data table (exact teaching values): time t [h] | stock M [mg] | cumulative A [mg]; 0 | 24 | 0; 1 | 12 | not measured; 2 | 6 | 6. Do not treat an unmeasured entry as zero. Candidate C uses an unknown constant share q. Candidate S uses unknown shares u for 0≤t≤1 and v for t>1; the change time is fixed in advance at one hour.
Public evaluation table, provided for later comparison but already exposed to readers: t=1, A=4 mg; t=3, M=3 mg and A=7 mg. These values can illustrate a frozen-analysis workflow but are not a secure held-out test. For independent assessment, a reviewer must supply a genuinely new scenario and record prior exposure.
Supported theorem alternative: characterize observational equivalence of channel allocations, uniqueness from cumulative channel measurements, and stability of parameter recovery for this model class. Replace empirical-validation claims with precise hypotheses, proofs, sharp counterexamples and independently checked computational examples. Do not relabel numerical agreement as a theorem or this scaffold as a novel result.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
Loading saved investigation…
0 of 8 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Frame the investigation
Write a one-page proposal identifying your chosen question, audience, observable, admissible model class and a falsifiable claim. For the default project ask whether stock-only measurements can determine channel allocation and which additional measurement distinguishes C from S. State the decision that an answer would support without inventing a practical safety threshold. Specify artifacts, review checkpoints and stopping criteria. Include a source ledger distinguishing supplied assumptions, externally checked mathematical results and your own claims. For the theorem alternative, state a proposed identifiability theorem with quantified hypotheses and an explicit counterexample target. Obtain or request human feedback on scope; record what has actually been reviewed.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive competing mechanisms
Derive both competing models, their units, conservation law and physically admissible domains. Nondimensionalize using M0 and λ⁻¹, keeping the switch location explicit. Derive the exact recurrence between equally spaced stock observations and contrast it with an Euler approximation. Give a possible particle-level stochastic interpretation, clearly distinguishing an expected amount from one realized path. For a self-chosen alternative, provide two explicit mechanisms and an analogous derivation. The theorem route must prove conservation and positivity and state whether its conclusions concern deterministic trajectories or stochastic observations.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Solve the inverse problem
Fit the default pilot data exactly and characterize every admissible switched-share pair consistent with them, rather than reporting a single fitted pair. Exhibit two distinct mechanisms with the same pilot observations, and prove that even an entire stock curve cannot distinguish channel shares. Compute predictions at t=1 and t=3 for one constant and one switched candidate. Explain how noisy observations would change the fitting problem and how residuals differ from parameter uncertainty. The theorem route must generalize the equivalence construction to arbitrary positive M0, λ and switch time, including exceptional zero-loss cases.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Design a discriminating measurement
Design a minimal additional measurement under the constraint that only one new channel measurement can accompany the pilot. Measuring A(1) requires a prospective design, a suitable archived sample, or a reproducible reset/repeat experiment: it cannot be collected retrospectively from a run already at t=2. State which route you use and justify stable parameters, initial conditions and calibration across repeats. Compare measuring stock again, measuring A(1), and measuring cumulative B(2). Show which choices increase identifiability and which are redundant through conservation. Derive the observation matrix for (u,v), its rank and an explicit inverse when appropriate. Specify measurement precision, calibration, provenance and replication for an actual experiment without pretending the exact pilot table supplies an error distribution. The theorem alternative must prove necessity and sufficiency of its proposed observation conditions and give a failed design.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Verify results and bound uncertainty
Verify your derivations independently and quantify a stated uncertainty scenario. For the default case, assume λ and stock readings are exact but each of A(1)=4 and A(2)=6 has an unknown additive error bounded in magnitude by 0.2 mg. Derive conservative intervals for u and v and explain dependence between the recovered estimates. Then derive the first-order sensitivity of λ=log(M0/M1) to perturbing M1 with M0 fixed. Compare this local approximation with an exact interval calculation for M1 in [11.8,12.2]. Specify reproducible arithmetic or code, boundary tests, mass-balance tests and a convergence check if using a numerical solver. The theorem route must distinguish proved error bounds from computed illustrations.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare models and evidence
Write and freeze a comparison rule before using the public evaluation rows. Record their prior exposure honestly. Compare the two specific pilot-fitting candidates against A(1)=4, A(3)=7 and M(3)=3, including residuals. Discuss what changes if the switched family is refitted after observing A(1), and why family-level selection differs from rejecting one chosen switched parameter pair. State a falsification criterion and a possible unmodeled mechanism. For the theorem route, replace empirical validation with a stress test of theorem hypotheses using at least two sharp counterexamples or limiting cases; neither route may call the public rows an untouched independent examination.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prepare the reproducible manuscript
Submit a reproducible manuscript and artifact manifest. Include an abstract with a bounded claim, model definitions and units, proofs or derivations, the observation table with missing entries labeled, fitting and design methods, uncertainty analysis, comparison results, limitations and sources actually checked. Provide a versioned calculation artifact with instructions sufficient for a reviewer to regenerate every reported number or a fully specified hand-calculation appendix. Distinguish original contributions from this public scaffold. Include one annotated figure or table whose caption explains what is fixed, inferred and unmeasured. The theorem route must supply a complete proof spine and identify every external theorem used.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Defend and revise the investigation
Prepare a short oral defense and a revision dossier. Answer: Why does more stock sampling not solve channel identifiability? Which assumption makes the two-channel inverse calculation unique? What would change if the switch time were unknown? What conclusion survives measurement error? Ask a human reviewer to challenge one proof and one modeling decision; record the questions, your responses, and a revised claim or calculation where needed. If a reviewer is unavailable, label the exercise self-review and keep independent review outstanding. Include a before/after revision with an explanation of which evidence changed the conclusion. Do not equate completion of these fields with acceptance of a capstone.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
