Written 2026-11-06, before any board run of the joint test, and committed before it. Not edited
after the data.

WHAT IS TESTED. The sample kernel's joint law, not only its per-cell frequencies: a sampler that drew
each cell from its right marginal, independently, would pass every per-cell test. Four cases, each a
workload small enough that the exact law of a chain's final state can be enumerated over all 3^n
states (src/exact.rs, `exact::cases`):

  ring 8, fixed T = 1.5, 4 sweeps                     200,000 chains   6,561 states
  ring 8, the job's anneal from T = 2.5, 10 sweeps    200,000 chains   6,561 states
  dense 6, fixed T = 4, 2 sweeps                      200,000 chains     729 states
  sparse 10, the job's anneal from T = 1.5, 8 sweeps  400,000 chains  59,049 states

Chain r of a case has the seed chain_seed(2027, r). (Seed 2026 was used while writing the test, on
the model; 2027 has not been run anywhere before this file was committed.) Every chain runs on the
board through the AXI link's sequencer, and on the CPU reference.

THE EXACT LAW. A probability vector over every state, pushed through each cell update of each sweep
with the three probabilities a/2^53, (b-a)/2^53 and 1 - b/2^53 from the reference's cut points
(a, b) = cuts(h, cost, beta), then through the synchronous sweep at temperature 0 where the job's
anneal ends with one. It assumes ideal uniform 53-bit draws; nothing is sampled.

STATISTICS AND THE RULE, per case:
  1. Pearson chi-square and the G-test over every state expected at least 5 times, the rest pooled
     into one bin: p >= 0.001 for both.
  2. Every cell's three marginal frequencies (z scores) and every pair of cells' nine joint
     frequencies: Bonferroni p >= 0.001 for each family.
  3. Every chain's final state on the board equals the CPU reference's.
  4. Control: the same histogram against the law with the exact marginals and independent cells
     must be rejected, chi-square p < 1e-6, while its cell family still passes. This shows the test
     sees the joint structure that per-cell statistics cannot.
A case passes if 1-4 hold. Sixteen p values are taken at 0.001 each; the chance that a correct
kernel fails one by chance is about 1.6%.

WHAT IT DOES NOT SHOW. Because the board is bit-exact (rule 3), its histogram is the reference's:
the test holds the shared kernel (the reference's algorithm and its counter-based draws) to exact
enumeration on the board's own outputs; it is not an independent sample of a different machine.
Board-level, one board.