Distribution and statistical assertions

Purpose and scope

Separates analytic distribution closeness from a statistical claim supported by finite samples. Similar percentages can have different uncertainty and acceptance rules.

Core design rules

• Source must specify analytic or sampled; no mode is inferred.
• Analytic distributions use total variation distance; sampled claims use Hoeffding bounds and a frozen Bonferroni family.
• A sampled family declares a seed, 1..4096 shots, common alpha and explicit error budget before execution.

Example from the original

This example illustrates the design recorded in the original. It is not by itself a claim of executable or stable support; check required options and the current version.

@seed(17);
@shots(4096);

test "bell distribution" {
  let q: QReg<2> = qreg[2];
  H(q[0]);
  CNOT(q[0], q[1]);

  assert distribution(q[0..1]) analytic ~= {
    "00": 0.5,
    "01": 0.0,
    "10": 0.0,
    "11": 0.5
  } tvd <= 1e-9;

  assert probability(q[0..1] in {"00", "11"}) sampled ~= 1.0
    abs_error <= 0.05 alpha 0.01 family "bell";
}

Limits and interpretation

• A sampled estimate is not an exact probability; an analytic value is not measurement evidence with a confidence level.
• The result is not device certification, tomography or a quantum-advantage claim.

Status and implementation boundary

The source is proposed and requires a separate review record before implementation. Display metric, uncertainty rule, family size and limitations together when interpreting a result.