Separate diffraction scale degeneracy from an unused prediction

1. Establish the optical mechanism

The original paraxial diffraction packet gives positions 6.4,12.4,18.4 mm at orders 1,2,3. Its forward law is x=c+m g λ/100, with wavelength λ in nm, gain g dimensionless and the conversion factor fixed by the stated geometry. Fitting a line estimates the product gλ, not wavelength and gain separately.

For equally weighted orders 1,2,3, slope is (x3−x1)/2=6 mm/order and intercept is mean(x)−2slope=0.4 mm. An independent gain anchor g with hard interval g±0.01 gives λ=600/g. With each fitted position bounded by ±e, the slope error is at most e; opposite endpoint signs attain that bound.

The wavelength interval is [100(6−e)/(g+0.01),100(6+e)/(g−0.01)]. A common gain uncertainty cannot be reduced simply by averaging more orders under that same calibration. These are deterministic hard bounds, not standard errors or confidence intervals.

Keep the fourth-order record 24.6±e mm unused until the fit is frozen. Its predicted center is 24.4 mm. The linear prediction has reading weights −2/3,1/3,4/3, so the hard prediction error is (7/3)e. Comparing intervals permits a center discrepancy up to (10/3)e. The gain anchor changes inferred wavelength but cancels from this same-coordinate predicted position; independent calibration and unused validation constrain different aspects of the model.

2. Work the original evidence

At g=1 and e=0.04 mm, wavelength is 600 nm with hard bounds about 590.099 to 610.101 nm. The unused prediction is 24.4±0.093333 mm, while the unused record is 24.6±0.04 mm. The intervals are disjoint because 0.2 exceeds 0.133333.

3. Predict and investigate

Can a better gain anchor fix a disagreement in a held-out position predicted directly in the original coordinate system?

Fit only the three assigned orders, derive the prediction weights and propagate bounds without assuming independent random errors. Freeze the unused prediction before comparing the fourth record and distinguish a failed model check from a unique diagnosis of its cause.

Primary mechanism context. Checked 31 October 2026. Original prose, records and diagrams; no copied source exercises or physical light-source operation. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.