Calibration dossier: noisy differences around a network
Build and revise one calibration dossier for a network of relative-reading sensors. Recover what can be identified, reconcile an inconsistent loop using declared nonuniform weights, verify the solution by genuinely different methods, and defend what the result does not establish. Keep the original measurements unchanged while separating fitting, later inspection and a changed-network transfer. This is a public supported portfolio, not automatic mastery, an unseen examination or a certification that any physical sensors are accurate.
Planning estimate: 16 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.
What do the sensors determine?
Each arrow reports the potential at its head minus the potential at its tail. Before calculating, predict whether these readings can determine all four node values.
Draw the loop sum and identify which node shifts leave every predicted reading unchanged. Keep this prediction in your report before opening worked solutions.
Written data: 1→2 has reading 2 and weight 1; 2→3 has reading 1 and weight 4; 3→1 has reading −2 and weight 1. Node 4 has no measurement. Weights specify the fitting objective; they do not by themselves establish sensor accuracy.
Preparation and working agreement
Work over R. A vertex carries an unknown scalar potential p, measured in one common calibration unit. An oriented edge i→j reports the difference pⱼ−pᵢ. The primary graph has vertices 1,2,3,4 and edges e₁:1→2, e₂:2→3, e₃:3→1; vertex 4 is isolated. Synthetic recorded readings are d=(2,1,−2). Their positive weights are w=(1,4,1). Minimize J=Σᵢ wᵢ(predictionᵢ−dᵢ)². The weights are declared penalties, not empirically estimated inverse variances; no confidence interval or sensor independence follows from them.
Use column vectors. For a real vector v, its Euclidean norm is ||v||=√(Σᵢvᵢ²), the square root of the sum of squared coordinates. A gauge is a choice of an arbitrary potential origin on each disconnected component: adding the same constant throughout that component changes no within-component difference. The edge-by-vertex difference matrix has rows (−1,1,0,0),(0,−1,1,0),(1,0,−1,0). Fix p₁=0 and p₄=0 as separate gauge conventions, not extra observations; write θ=(p₂,p₃). Then D has rows (1,0),(−1,1),(0,−1). Let W=diag(1,4,1), C=W^(1/2)D and c=W^(1/2)d. Thus C has rows (1,0),(−2,2),(0,−1), c=(2,2,−2), and J=||Cθ−c||².
Submit a network sketch with arrow directions, a matrix-and-units contract, exact computation tables, proof-based certificates, a sensitivity record, a revision history and a final defended report. A supported noncoding route uses handwritten or typed exact tables and independent scalar checks. An implementation route additionally supplies reproducible code and tests. Both routes must justify the general claims; successful execution or matching a public answer alone is not independent understanding.
The later inspection reading is not part of the primary fit: an additional 1→3 device reports 3 with a declared acceptance tolerance 1/6. Freeze the primary fit before calculating this discrepancy. This small synthetic inspection is not a random test sample or a statistical accuracy estimate. Once inspected, record its exposure and do not call it unseen again after a revision.
The workbench saves written text, not uploaded files or executed code. Each milestone draft is bounded at 8,000 characters. Keep concise evidence summaries and reference clearly named calculations, diagrams or code files retained separately; do not imply those external artifacts were uploaded or automatically verified.
Retain dated first predictions, exact intermediate values, failed hypotheses, source/assistance disclosures and actual review comments. If nobody independently reviews the dossier, mark review pending rather than invent feedback. Suggested hours are an author estimate. The transfer is public too; disclose solution exposure when presenting it as evidence.
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.
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0 of 6 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Define identifiability and the complete observation space
Start the dossier with a labeled graph and derive the full 3×4 difference matrix from its arrows. Prove its kernel and image, then give every potential vector consistent with an arbitrary exact reading vector y. Explain why setting only p₁=0 leaves an unidentified variable. Test the supplied data d for exact consistency and give a left-null obstruction certificate. Keep this exact-consistency decision separate from the later least-squares fit.
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.
Fit the declared loss and preserve its residual certificate
Derive the weighted normal equations by expanding J(θ+h), and solve them exactly. Record raw predicted readings, raw residuals, weighted residuals and J. Prove unique global minimality for every perturbation h. Then fit the same data with all weights equal to 1 and compare both fitted answers under the original weighted objective. Explain why comparing unlabelled residual sums from different objectives would be misleading.
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.
Audit the fit through QR and a cycle-only argument
Construct a thin QR factorization of C with positive diagonal in R and solve Rθ=Qᵀc. Independently derive the optimal raw residual on an arbitrary single directed cycle with positive edge weights wᵢ and readings dᵢ: use only the loop sum and weighted Cauchy–Schwarz, not the normal equations. Compare the two certificates with the previous result. Supply independent exact checks; if coding, do not let both verification routes call the same solver.
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.
Quantify what measurement changes and residuals can certify
Keep the graph and positive weights fixed. Derive the exact coefficient change when the first reading alone changes by δ. Then prove the general bound ||Δθ||≤||W^(1/2)Δd|| for refitting any changed readings in this primary model, and show its constant is sharp. Derive an equation-residual bound for an approximate coefficient vector using G, and explain why neither bound is a statistical confidence statement. Record two changed numerical trials, predictions made before calculation and the checks 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.
Freeze the fit, inspect it, then connect the isolated station
Freeze the primary fit and evaluate the separate 1→3 inspection reading 3 with tolerance 1/6. Record acceptance or failure and what may legitimately be concluded. Next add a bridge sensor 3→4 reporting 3 with weight 2, retain the three original observations and weights, and use only gauge p₁=0. Derive the complete new weighted minimizer. Explain why retaining the old convention p₄=0 as a constraint would change the estimation problem rather than merely fix a gauge. State a reproducible revision plan that does not relabel exposed inspection data as unseen.
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.
Assemble and defend the complete calibration dossier
Assemble one coherent final report linking the original network, complete exact solution theory, fitted estimates, QR and cycle verification, sensitivity and new-information results. Include a first conjecture that was revised, actual source/assistance disclosures, and a defense of the three claims: “gauge fixing is extra data,” “the smallest residual proves the sensors are correct,” and “a new bridge makes every old discrepancy disappear.” Ask a reviewer to challenge one proof or computation if available; otherwise record a self-review with independent review pending. Reference separately retained exact tables or runnable tests, summarize their checks in the text draft, and state precisely what evidence has and has not been supplied.
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.
Targeted lesson returns
Define identifiability and the complete observation space
