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Connections check: model, units and error

Use this as the first 45–60 minutes of the existing 4–6-hour integration review, not another project or additional workload. Reuse the graph investigation, rotation experiment and calculus appendix. The semester remains 520–750 hours. Existing proofs, numerical labs and inference gates remain required.

Attempt before consulting the calibration​

  1. For the triangle-plus-tail fixture (01,12,02,23), list the common-neighbour count of every edge. Derive the estimator expectation and variance for k independent uniform draws with replacement. Predict the change from k to 4k before computing. Explain why a single observed run need not improve.
  2. Your sampling implementation draws only from the three triangle edges but still multiplies by m/3 with m=4. Compute its expectation. Can extra samples or higher floating-point precision repair the error? Name the assumption violated.
  3. A waiting-time model has density lambda * exp(-lambda*t) for nonnegative t, where t is seconds and lambda is inverse seconds. Convert to milliseconds without changing the physical distribution. Explain the units of density and its integral. Then separate the error of a finite-window numerical probability into numerical integration, excluded tail, and mismatch between the assumed model and actual observations.
  4. Reuse your rotation experiment: explain why adjacency remains exact while computed distances need tolerance. Give a scale-aware comparison criterion, distinguish accumulated rounding from a wrong transformation, and identify what evidence could falsify your explanation.

Write predictions, derivations, observed results and a repair in separate fields. Do not manufacture a measured result for the paper-only cases. Existing experiment outputs suffice; rerun only an affected check.

Calibration — open after the attempt​

For task 1, edge counts are 1,1,1,0; mean is 3/4 and population variance is 3/16. The estimate has expectation 1 and variance 16*(3/16)/(9*k) = 1/(3*k). Quadrupling k quarters variance and halves standard deviation, under the stated independence model. These are distributional statements, not monotonic guarantees for individual runs.

For task 2 every sampled count is 1, so expectation is 4/3. The bias is 1/3, regardless of k. More samples reduce variability, not this selection bias. This is a sampling-contract/model failure, not floating-point rounding. Correct the sampling population or derive a justified weighted estimator; do not silently retain the old unbiasedness proof.

For task 3, set u=1000t and lambda_ms=lambda/1000. The new density is lambda_ms * exp(-lambda_ms*u) per millisecond. Probability is dimensionless. For a window ending at T seconds, the excluded model tail is exp(-lambda*T). A quadrature bound concerns the chosen model on the retained interval; it cannot certify that real waiting times are exponential. Compare held-out observations and dependence assumptions separately. A tolerance with units of seconds cannot be added directly to a probability bound.

For task 4, adjacency is combinatorial and must not be reconstructed from rounded positions. A possible distance test uses abs(new-old) <= atol + rtol*abs(old) with atol in the coordinate's distance unit and rtol dimensionless. Choose tolerances from scale, operations and evidence; do not increase them just to pass. Compare a direct rotation to repeated rotations and an independent distance computation. Passing one tolerance does not prove the transformation correct for every input.

Passing and repair​

Score each task using the semester 0–3 scale; all four need at least 2. Record these as subchecks of the existing probability, numerical-judgment and calculus project categories, not a ninth category or new exam. A correct formula with incompatible units, hidden sampling bias or conflated model/numerical error requires repair. Revisit the relevant existing module, then use a fresh graph or rate/unit conversion without the calibration. Reuse one fresh connection in the existing 7–14-day delayed review. Independent reviewer and actual timing remain pending.