Use a withheld spatial ratio without erasing gain ambiguity

Define the source and observer

A model can fit one field reading by adjusting its charge even if its geometry is wrong. Here a uniform finite rod of half-length L is compared with a point-source hypothesis. Fit the point-source amplitude using only the first field at distance 1 m and withhold the second at 2 m to test that point hypothesis. A finite rod with both unknown length and amplitude cannot be fitted from that first reading alone. In a second stage, use both readings to calibrate the rod, then reserve a fresh third field at 3 m for validation. An unknown common positive sensor gain multiplies both readings.

Derive before approximating

For the rod, the ratio R=E2/E1 equals √(1+L²)/[2√(4+L²)], independent of total charge and common gain. Squaring and rearranging gives L²=(16R²−1)/(1−4R²). This inversion requires 1/4 < R < 1/2. A point source predicts ratio 1/4 regardless of its fitted charge. Within the restricted rod family, the ratio is monotone and identifies L; it still does not identify absolute charge without gain calibration.

Check evidence and the domain

With independent bounded reading errors ±ε, form the smallest compatible ratio (E2−ε)/(E1+ε) and largest (E2+ε)/(E1−ε), then apply the monotone inverse if both remain in its domain. The registered ε=0.02 N/C corners do remain inside. These are deterministic compatible intervals, not confidence levels. Other source geometries or spatially varying sensor errors could mimic the ratio, so the exercise does not prove a unique real charge distribution. The second field is test data for the first-fit point model but calibration data for the unknown-length rod; it cannot simultaneously be called fresh validation of the fitted rod.

Worked original case

For L=1 m and unit gain, E1=18/√2≈12.727922 N/C and E2=9/√5≈4.024922 N/C. Their ratio is 1/√10≈0.316228 and the inverse recovers L=1 m. A point model fitted to E1 predicts only E1/4≈3.181981 N/C at the withheld location. After calibrating the rod with both readings, predict its fresh third field at 3 m: 6/√10≈1.897367 N/C. The supplied third record agrees, but does not uniquely prove the source family. Multiplying all readings by 1.05 leaves the inferred L unchanged while multiplying fitted charge-times-gain and the third prediction by 1.05.

Investigation

Freeze the point-model prediction before revealing the second reading. Then explicitly relabel the first two readings as rod-calibration data, infer L and Q times gain, and freeze a third-distance prediction before revealing the record at 3 m. Compare common gains 0.9,1,1.1, propagate the stated reading bounds and propose an independent amplitude reference. Do not count the same second reading as fresh validation of the rod it helped fit.

Changed-case practice

Sources and boundaries

Original derivations and numerical packets checked 30 October 2026 against the primary explanation. The electrostatic field construction also uses continuous source integration. All records are synthetic; no electrical apparatus, high-voltage handling or practical certification is requested. Public feedback is not independent mastery; retain first predictions, alternative models and criterion-linked critique in the separate course evidence workspace.