ĀRU INTELLIGENCE / INWARD MATHEMATICS™Back to the lab

Exact mathematics / interactive research

Several readings can move. Does the result hold?

Set independent bounds for 3–5 readings and a fixed target or target interval. Find an attainable tie, the exact worst-case squared-error lead, or a complete map of how both change across targets. Ask where a chosen lead is guaranteed, then check and share the exact answer.

01 / Define an experiment

What might each reading become?

Reading 1
Reading 2
Reading 3
Reading 4
Reading 5

Use integers, fractions such as -3/2, or decimals up to six places. Each interval includes both endpoints. Leave “Target through” empty for one fixed target; fill it to certify every target in the closed range.

Edit the example bounds or generate its receipt.

What will the proof say?

The two estimators are recomputed across every real configuration inside your chosen ranges. If an exact tie can occur, you receive readings and a target that attain it. Otherwise the receipt gives the strict winner, the exact smallest squared-error lead, and readings and target where that minimum occurs.

To map the whole target range, enter a strictly higher “Target through” value and choose “Map every target.” The map finds every tie target and each exact linear piece of the worst-case lead, with readings that attain it. Then choose a required lead to find the exact target regions that guarantee it.

The calculation runs on this device. The target and ranges are assumptions you choose; the page does not know whether they represent real measurements. The target may vary independently of the readings.

See the exact proof and limits

02 / Independent check

Inspect the receipt


The verifier recomputes the claim without importing the constructor. You can also generate or check receipts offline using the documented Python and Node commands.

Verified result

Choose a certificate or load an example.

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All-reading guarantee or witnessNo certificate checked yet.
Exact mean minus median range—
Exact estimator-midpoint range—
SHA-256 of checked JSON bytes—

What this establishes

A conditional result across infinitely many configurations

The exact finite calculation covers every real reading and target inside the supplied intervals. A strict v3 verdict includes the exact attained minimum squared-error lead; legacy v1/v2 receipts retain their conservative bound. A v4 map can answer a required-lead query and export exact target regions linked to the checked source JSON. A tie result includes attainable readings and target. The receipt does not authenticate measurements, assign probabilities, or establish which estimator is appropriate for a real problem.

Explore the standalone, verified target-sweep report · Read the proof, limits, and reproduction commands