ATLASREALITY CALCULUS Enter the lab
03 / THE SCIENTIFIC LABORATORY

Change a parameter.
Keep the distinction.

Interactive mathematics and explicitly synthetic examples. These demonstrations are reproducible; they are not physical experiments.

Optical equivalence Synthetic uncertainty State transitions

Two expressions.
One prediction.

MATHEMATICAL IDENTITY
Comparing identical functions cannot detect a novel physical law.

EXPERIMENT 01 / MATHEMATICS ONLY
f(θ) = cos²(θ)
g(θ) = [1 + cos(2θ)] / 2
f(θ)0.500000
g(θ)0.500000

0° to 180°, at 0.05° intervals. Numerical results appear here.

Identical normalized optical responsesBoth functions equal 1 at zero and 180 degrees, and 0 at 90 degrees. They overlap everywhere. The selected angle and values are also available in text.1.00.50.00°90°180°NORMALIZED RESPONSE

Cyan: both functions · Violet: selected angle
The analytic curves coincide at every angle.

Why the curves are identical

The double-angle identity gives cos(2θ) = 2cos²(θ) − 1. Adding 1 and dividing by 2 yields [1 + cos(2θ)]/2 = cos²(θ) for every real θ.

The analytic identity is the proof. Small differences in a numerical sweep are floating-point roundoff. Agreement between these expressions provides no distinctive RC-specific prediction.

Download the dependency-free numerical check

Noise is not a discovery.

SYNTHETIC / FIXED SEED 42
Generated samples around an arbitrary mean of 1. No sensor measurements.

Sampling illustration

Each setting uses a deterministic normal-noise sequence with the same fixed seed. Increasing the sample count retains the earlier samples.

The interval is x̄ ± 1.96s/√n, a normal-reference approximation assuming independent, identically distributed samples. It is not a prediction interval and does not include model error or systematic bias.

Synthetic sample values and mean intervalGenerated sample values plotted by sample index. A horizontal reference marks the generator mean of 1. A violet band marks the approximate 95 percent interval for the sample mean, not the spread of individual samples. Text results also provide the mean and interval.SAMPLE INDEX

Cyan: generated samples · Dashed: generator mean 1
Violet line/band: sample mean and approximate 95% mean interval

Enable JavaScript to generate the synthetic sample summary.

The vertical scale is fixed from −3 to 5 so different noise settings remain comparable. Out-of-range values, if any, are counted in the text summary.

Generator and interpretation

A 32-bit linear congruential generator (multiplier 1664525, increment 1013904223) with seed 42 supplies uniform numbers. The Box–Muller transform produces normal deviates; samples are 1 + σz.

A single displayed interval does not establish repeated-sampling coverage. Changing n can move the sample mean; the interval width need not decrease at every step because s is estimated. Zero noise gives identical samples and a zero-width interval.

Inspect reproduction instructions and limitations before interpreting the visualization.