Public companion · Not peer reviewed

Can a few echoes reveal a hidden arithmetic shape?

Change how three primes distribute weight across their exponent levels, then watch a small set of harmonic readings separate—or confuse—the resulting hidden profiles.

The idea

An arithmetic object heard as a chord

The primes 2, 3, and 5 act like three instruments. Each shape control changes how strongly its four exponent levels contribute. A reading combines all 64 components into one complex echo.

Shape below 0.5
low exponents dominate
Shape at 0.5
four levels are equally weighted
Shape above 0.5
high exponents dominate

Interactive illustration — synthetic declared model

Arithmetic Observability Lab

Normalized four-level factors · primes 2, 3, 5 · exact AO V three-reading schedule

Projected ambiguous cells—reading distance ≤ 0.020
Empirical secant floor—sampled lower ratio, shape gap ≥ 0.08
Closest separated competitor—reading-space distance
Confusability slice Candidate prime-2 and prime-3 shapes; candidate prime-5 shape is optimized on a grid.
Confusability heatmap Minimum harmonic reading distance over the hidden prime-5 shape, plotted against candidate prime-2 and prime-3 shapes.
within pair cutoff source and nearest competitor
Observed separation envelope Smallest reading-space distance found for each sampled compact shape distance.
Observed separation envelope A sampled lower envelope of harmonic reading distance as compact shape distance increases.
sampled lower envelope pair cutoff

How to read it

Confusability moves when the hidden shape or the instrument changes

Shape is an exponent profile

The sliders do not alter the primes. They move probability among the four allowed exponent levels of each prime.

Orange means possible overlap

Hatched cells lie within the selected pair cutoff after the hidden candidate prime-5 shape is allowed to compensate.

More readings cross-cut valleys

One echo can leave broad near-collisions. Independent phase-separated readings observe different directions and reduce the surviving confusable region.

Controlled physical analogy

A toy spectrum built from prime logarithms

An engineered four-level quantum system can assign energy ℏΩ(a log 2 + b log 3 + c log 5) to the state |a,b,c⟩. Its survival amplitude then reproduces the lab's harmonic reading after a time rescaling. In that toy model, shape controls occupation bias and the envelope becomes an instrument-resolution curve.

Boundary: this is an exact mathematical embedding, not evidence that elementary particles naturally have prime-logarithmic energy levels. The sampled envelope is not an energy spectrum.

Experimental agent interface

A small read-only WebMCP pilot

Compatible browser agents may inspect the declared model, compare one bounded pair of exponent profiles, and retrieve the evidence boundary through three typed tools. The tools do not change this page, sweep the full grid, upload data, navigate, persist state, contact a server, or start paid computation.

Progressive enhancement: the pilot registers only when the browser exposes the experimental document.modelContext API. The ordinary laboratory remains complete without WebMCP. Read the machine-readable lab manifest.