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Executable Science local pilot drafts
Enumerate all binomial outcomes at n=20 and p=0.05, then compare exact interval-coverage calculations with a fixed-seed simulation.
Investigate Do nominal 95% intervals achieve that coverage in this prespecified small-sample setting?
Executable Science local pilot drafts
Provide a small bisection implementation, a valid cubic-root example, an invalid-bracket control, and a high-precision Newton comparison.
Investigate Does the solver preserve a shrinking root bracket and reject the specified invalid input?
Executable Science local pilot drafts
Enumerate every balanced assignment for two groups of four observations and compare the exact two-sided randomization p-value with a sampled estimate.
Investigate Can a Monte Carlo permutation implementation be checked against a fully enumerated finite reference?
Executable Science local pilot drafts
Resample a fixed twenty-value empirical distribution and compare bootstrap moments with their conditional analytical values.
Investigate How do observed bootstrap moments compare with their conditional expectation and variance?
Executable Science local pilot drafts
Use rational arithmetic to compare composite quadrature errors for x^4 on [0,1] across five nested grid resolutions.
Investigate What convergence pattern appears without floating-point rounding for these two quadrature rules?
Executable Science local pilot drafts
Investigate four step sizes for y'=-10y using the known amplification factor 1-10h, including stable, boundary, and growing discrete trajectories.
Investigate When does a decaying continuous solution produce a non-decaying numerical trajectory?
Executable Science local pilot drafts
Compare unadjusted and Bonferroni thresholds on fixed-size families of synthetic uniform null p-values, with analytical familywise probabilities.
Investigate How does applying twenty tests change false-positive risk under a declared independent-null model?
Executable Science local pilot drafts
Sweep sixteen step sizes for forward and central differentiation of exp(x) at zero, exposing truncation and rounding behavior against the analytical derivative.
Investigate How does derivative error change across a fixed grid of shrinking step sizes?
Executable Science local pilot drafts
Measure loss of precision in sqrt(x+1)-sqrt(x) at x=10^16 and compare an algebraically rationalized expression with a high-precision reference.
Investigate How much error is removed by changing the expression while preserving its exact-arithmetic meaning?
Executable Science local pilot drafts
Compare residuals near simple and double roots at the same candidate locations to illustrate why a small residual can have different interpretations.
Investigate How does a multiple root change the relationship between residual magnitude and distance to the known root?
Executable Science local pilot drafts
Provide Dijkstra and Bellman–Ford implementations with hand-enumerated reference graphs, explicit negative-edge rejection, and a negative-cycle control.
Investigate Do the implementations respect their input assumptions and return the expected distances on the supplied controls?
Executable Science local pilot drafts
Compare naive exponentiation with a shifted log-sum-exp calculation and a 70-digit Decimal reference on three fixed large inputs.
Investigate Can a shifted calculation retain an accurate finite result when direct exponentiation overflows?
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