# Repeated reference readings and shared calibration uncertainty

Original synthetic, completed-line comparison with 512 independently calibrated trials, 128 target positions per profile and eight fixed reference positions. (a) Standard uncertainty of one profile's mean absolute height for 8, 32 and 128 total reference readings; connecting lines guide the eye between the three archived counts. The dotted line is the analytic infinite-read limit of 0.302076 nm under the common-calibration model. (b) Coverage of nominal 95% mean-height intervals over the same 512 paired trials. Error bars are the archived approximate 95% Monte Carlo intervals. At 128 reads, coverage is 94.53125% for the correct model and 23.2421875% for wrong independence. (c) Covariance of eight averaged reference values after subtracting the shared calibration observation, at 16 reads per position. Both maps use the same linear scale in nm²; cell values follow the source plotting algebra and frozen input variances. (d) All 512 archived mean-height errors at 128 reads, displayed as density in fixed 0.1 nm bins. The teal and orange dashed boundaries show symmetric error spans for the two reported nominal 95% mean-height intervals, with half-widths of approximately 0.593 and 0.091 nm. They use the same observations and point estimates. Calibration errors differ between trials but are shared within each trial. The known-zero control assumes no calibration error. This is a numerical metrology example, not physical AFM calibration or a universal uncertainty floor.

## Panel guide

- **(a) An uncertainty floor remains.** Repeated reads reduce independent reference noise. A shared calibration contribution remains.
- **(b) False precision loses coverage.** The wrong covariance makes intervals narrower without changing the acquired observations or point estimates.
- **(c) Shared error is correlated.** One calibration error reaches all eight averaged reference values, producing nonzero off-diagonal covariance.
- **(d) Same errors, different intervals.** The archived error distribution is unchanged. The wrong nominal 95% span excludes most trial truths.

## Interpretation and limits

- This is an AFM-inspired linear inverse-problem model, not a physical model of an AFM tip, scanner or feedback loop.
- No measured specimen, traceable calibration certificate, physical instrument-performance validation or production recipe is represented.
- All noise levels, reference geometry, covariance laws and fixed effects are assumed; covariance parameters were not learned from real data.
- The 512 experiments have independently generated calibration errors. They are not 512 profiles sharing a single real certificate; averaging the latter would not remove its common error.
- Repeat counts describe ideal observations at fixed positions. They do not model additional elapsed time, drift during repeats, thermal stability or acquisition cost.
- Correct and wrong uncertainty models share the same point estimates. The lower reported uncertainty is false precision, not an improvement in the acquired data.
- Other units in the AFM series use different models, measurands and denominators. Their RMSE values must not be combined into one performance ranking.
- Numerical verification does not establish physical predictive accuracy, publication novelty or acceptance.

## Metric definitions

- **Mean-height standard uncertainty:** Model standard deviation for the estimate of the mean absolute height over all 128 target positions of one profile. Units: nm. It is not pointwise RMSE, roughness, or uncertainty of the average of all 512 experiments.
- **Interval coverage:** Fraction of 512 independent synthetic calibration experiments whose nominal 95% mean-height interval contains that experiment's true profile mean. It is not simultaneous coverage of the 128 individual heights.
- **Monte Carlo interval:** Approximate 95% interval describing finite-trial uncertainty in the reported coverage fraction, taken directly from the archived metric records. It does not add instrument uncertainty or imply an exact binomial interval.
- **Reference readings:** Eight fixed reference positions multiplied by 1, 4 or 16 ideal readings per position: 8, 32 or 128 readings in total. The same 512 trial identities are paired across the three counts.
- **Reference covariance:** Covariance among the eight averaged reference values after subtracting their shared calibration observation, at 16 reads per position. Units: nm².
- **Error histogram:** Distribution of the 512 archived profile mean-height errors at 128 reference readings, shown in fixed 0.1 nm bins as density in 1/nm. All source errors are included. No fitted distribution is added.
- **Analytic infinite-read limit:** A mathematical limit of the already verified linear model, combining the 0.3 nm calibration component and 0.035355 nm target component. It is not an acquired infinite dataset or a universal AFM performance limit.

## Source context

- [NIST: Combining uncertainty components](https://physics.nist.gov/cuu/Uncertainty/combination.html): Conceptual source on uncertainty propagation with covariance; no numerical inputs or figures copied.
- [NIST TN 1297: Combined standard uncertainty](https://www.nist.gov/pml/nist-technical-note-1297/nist-tn-1297-5-combined-standard-uncertainty): Conceptual source on uncertainty of corrections; no compliance or certificate claim.

All numerical results and figures are original. The source study read these official pages on 2026-09-30; this derivative does not claim a new full-publication literature review. Exact canonical source hashes and panel mappings are in source-manifest.json.