rho and sigma: can the measurer be trusted, and was the checker independent

For a balanced binary grader whose κ is measured against a reference standard - gold labels, balanced prevalence, symmetric error, so κ = 1-2p is the channel parameter - the KL contraction coefficient is η = κ². (Inter-rater κ between two conditionally independent graders of equal quality IS η already; do not square it twice.) The per-verdict information rate is κ*log₂((1+κ)/(1-κ)) bits, and the single-verdict discrimination ceiling is the probability (1+κ)/2. At κ=0 both are exactly 0 bits and 1/2: infinite samples buy nothing.

Under conditional independence given truth, the bits you double-count when pooling two checkers equal exactly I(Y₁; Y₂). Observed agreement above a₁a₂ + (1−a₁)(1−a₂) is co-bias and is worth zero bits.

Both formulas describe an instrument, not the evidence it reads. The curve after the rungs below plots rho’s half directly against kappa; the grid after sigma’s rungs plots the co-bias half directly against two illustrative checkers.

rho’s information rate is a pure function of kappa, with no data dependency at all - the curve below is the formula, nothing else. It is also even in kappa: the rate at kappa = -0.23 and the rate at kappa = +0.23 are identical, which is the exact mechanism behind the anti-correlated case the rung warning above just named. A scalar cannot climb out of that; it does not have the sign to climb with.

hash in the catalog. rho = 1 by construction, and it tells you one bit about one file that transports nowhere (T 0, maxU 0). Honestly a JOINT rho and S witness rather than a pure one, which is why the orthogonality claim is stated pairwise - no axis derivable from the others - rather than as the stronger "high here, low on all five".

shortseller in the catalog. No method, no identification, unreliable instrument, sigma 5 - and it moves prices, because the author has capital at risk. The cleanest witness in the set.

rho asks whether the measurer can be trusted. sigma asks whether the checker was independent of whoever produced the claim. Both questions are about the instrument, and neither is about the evidence the instrument read. That is why they share one route: split them across two pages and the failure mode that matters most - that they fail together - has nowhere to live.

They fail together in a specific and common way. A captured auditor scores high on rho, because its judgments are internally consistent and well calibrated on the cases it has already seen, and scores at or near zero on sigma, because it was never independent of what it was checking. The composite reads as strong verification. That is the co-bias identity above, read as an operational warning rather than a formula: agreement between correlated checkers beyond what chance-corrected independence predicts is exactly the excess that identity prices at zero bits, and a captured auditor is the case that earns the most of it.

The formal frame above states the reference-standard reading explicitly because the other reading is a live error, not a hypothetical one. eta = kappa² holds only when kappa is measured against gold labels; inter-rater kappa between two conditionally independent graders of equal quality is already eta, and squaring it a second time understates reliability by construction. Settling which reading applies to a given number is not a matter of restating the formula again - it means reading quality-sgd’s own reliability ledger, which nobody has done yet.

rho is the wrong shape. A scalar sees only stochastic degradation. Contrast degeneracy and construct substitution are invisible to it, and the second is the failure mode that motivated the axis.

Index rho by (evaluator, cliff) and estimate it as blind AUC on synthesized matched pairs that toggle exactly that cliff.

The same logic runs the other direction, and it is sigma’s falsifier as much as rho’s gap: if agreement above what chance-corrected independence predicts never turns out to predict a downstream error, the correction the co-bias identity supplies is dead weight - real in the formula, worthless in the decision it exists to inform.

Collapses to a single scalar only for stochastic degradation. Two other failure modes exist and this axis is blind to both: contrast degeneracy (zero divergence on the specific contrast, no n fixes it) and construct substitution (κ near 1, sign inverted), which is the κ = −0.23 result above and the reason this axis is not a scalar.

Wrong if the checker bears no realized loss. If nobody has ever been penalized for a bad check, σ is nominal regardless of the checker's title - which is precisely how captured auditors and rating agencies present as high-σ and score near zero.