Piece 01Median on the interior
[0, 979/20]
Exact smallest lead: G(t) = (-58/5)t + (15631/25)
Attained at readings [48, 49, 51, 51, 85].
Exact target landscape
Where does the worst case change?
For every real target in [0, 80], this report maps the smallest attainable squared-error difference between Mean and Median as independent readings move within their bounds.
Tie attainable for every target in [533/10, 543/10].
At target 533/10, readings [48, 49, 50, 51, 85] attain a tie. The full tie interval follows from the continuity of the estimator midpoint, or from an equal-estimator reading when applicable.
01 / The full curve
Exact breakpoints. Attained margins.
The connecting lines are exact between displayed rational breakpoints. Screen geometry is approximate; the receipt and fraction labels are exact.
02 / Inverse guarantee
Where is a required lead guaranteed?
Choose a positive minimum squared-error lead. The exact intervals include their endpoints, and a boundary witness shows when the guarantee is tight. The target and readings remain independent.
How far can the inspected target move?
Use the target inspected above and this required lead. The exact answer gives left and right reach, symmetric tolerance, and cutoff witnesses. If the lead fails, permitted readings and both squared errors prove it.
Choose a target above and calculate its tolerance.
03 / Attaining readings
Every strict piece
A piece's interior has the stated strict winner. A closed endpoint that touches the tie band has gap zero. At other shared endpoints, adjacent formulas agree.
Piece 02Median on the interior
[979/20, 533/10]
Exact smallest lead: G(t) = (-66/5)t + (17589/25)
Attained at readings [48, 49, 50, 51, 85].
Piece 03Mean on the interior
[543/10, 286/5]
Exact smallest lead: G(t) = (66/5)t + (-17919/25)
Attained at readings [49, 50, 51, 52, 86].
Piece 04Mean on the interior
[286/5, 80]
Exact smallest lead: G(t) = (58/5)t + (-15631/25)
Attained at readings [48, 49, 51, 51, 85].
04 / Evidence and limits
Replay the exact claim
Input assumptions
Reading intervals: [48, 49], [49, 50], [50, 51], [51, 52], [85, 86].
Target interval: [0, 80]. Each reading and the target vary independently. These bounds are stipulated; the report cannot establish how they were measured or which estimator is appropriate.
Independent checker
The report was rendered only after a separate BigInt verifier recomputed the v4 claim from the certificate. This file embeds the original JSON bytes and verifier and checks again when opened. Its SHA-256 fingerprint identifies matching bytes; it does not authenticate their author.
d192f94eb904f91a627589fc6aab23bdff44a7d6dea240cd9160c04966cc5b86
Replay from the repository root: node tools/verify-joint-reading-box.cjs joint-reading-box-sweep-v4.json. The exact-envelope argument and its limits are in the proof notes. Lean proves the generic affine endpoint step; candidate completeness and receipt parsing remain outside that formal proof.