Anchors — where a gate gets its authority
An anchor is the part of a check your implementation did not produce. It is what separates a gate from a changelog.
Without one, the strongest thing a check can say is "this run agrees with the last run," which is true of a codebase that has been quietly wrong since the first commit.
Ordered strongest first. Take the highest one available; they compose, and a gate with two anchors of different kinds is much harder to fool than one with two of the same kind.
1. A published constant
Someone measured or proved it, in a paper or a standard, and printed the digits.
Questions that find one. What field has been studying this for decades? What is the textbook version of my quantity? Does my problem have a named constant attached to it?
Watch the range. Published tables stop somewhere. Max's 1960 quantizer table stops at 5 bits; a check built on it is structurally incapable of noticing an error at 8 bits, and will report PASS in exactly the tone it uses for the rates it covers. Record the covered range as a coverage limit.
Watch the precision. A hand-computed 1960 table has a last digit that may be worse than your float64 fixed point. When your converged value disagrees with a published one by a fraction of a percent, the table is sometimes the one that is wrong — but say so with evidence (convergence residual, stability across iteration counts, a uniqueness argument), not by widening the tolerance until the test goes green.
2. A closed form
An exact expression you can evaluate independently of the machinery under test — an integral, a recurrence, a combinatorial count.
This is the strongest anchor available for anything with a tractable special case, because it does not merely constrain the answer, it is the answer.
Pattern that works well: find a degenerate configuration where a complex implementation must reduce to a simple formula. Lift a scalar quantizer's levels into an m-dimensional product grid, and nearest-neighbour assignment decomposes per coordinate — so a KD-tree measurement and a closed-form integral must agree to sampling error. That agreement exonerates the instrument, which converts "one of these two things is broken" into "this one thing is broken."
Always exonerate the instrument before blaming the subject.
3. A conservation law or invariant
Something that must hold regardless of the answer: a total that is preserved, a norm left unchanged by an orthogonal map, a round-trip that must return the input, an operation applied twice returning to the byte-identical original.
Cheap, and unusually good at catching the class of bug that produces plausible-looking output. An inverse-transform round-trip that agrees to 1e-7 rules out a large family of indexing and sign errors in one line.
Also invariant-shaped: monotonicity that theory requires (quality improves with more bits, error shrinks with more samples), and orderings that must hold between arms.
4. A theoretical bound
A quantity that cannot be exceeded — an information-theoretic limit, a complexity lower bound, a physical constraint.
Bounds are one-sided, so they are best used as the far side of a bracket: better than the baseline, and not better than the bound. A result that beats its own theoretical limit is not a triumph, it is a bug, and this is the check that catches it.
Bonus signal: watch the ratio-to-bound as a parameter sweeps. If theory says it should approach a constant, a smooth monotone approach is corroboration that many independent parts are right at once.
5. An independent implementation
A second implementation of the same computation — a reference library, a slow brute-force version, a different language.
Weaker than it looks. Two implementations agree on a false result whenever they share a modelling assumption, and if the same author or the same model wrote both, they share more than they appear to. Vary the assumption and the author, not just the code. A deliberately naive O(n²) brute force is usually a better second implementation than a clever one, because it shares less.
6. A known-answer fixture
An input whose output you know for reasons outside the code: a worked example from a textbook, a case computed by hand, a case with an answer fixed by symmetry.
Small ones are fine. The value is that the answer's provenance is external.
When nothing published exists
Some quantities have no literature. Options, in order:
Construct a degenerate case. Set a parameter to a value where the answer becomes obvious — zero noise, one dimension, an identity transform, a single-element input. Assert the obvious answer.
Use a differential. Absolute correctness may be unavailable while a difference is provable: this input must score higher than that one; adding data must not make the fit worse; the treated arm must beat the control. Differential anchors survive a great deal of implementation drift.
Manufacture a ground truth. Generate synthetic data with a known answer baked in, then check the pipeline recovers it. Guard against the pipeline recovering it by construction — plant a case where the answer is deliberately not what the pipeline would assume.
Bound it from both sides with two crude methods that are wrong in opposite directions. Neither is the answer; together they are a bracket.
An anchor is also where a magnitude comes from
Anchors are usually discussed as sources of correctness — is this value right? They are equally the only source of practical significance: is a difference big enough to care about?
A threshold derived from measurement noise answers "is this effect real". It cannot answer "is this effect worth having", and the two diverge badly when the things being compared are similar: a paired estimator's standard error shrinks as the arms converge, so a 3-sigma margin can approach zero and admit an effect three orders of magnitude below anything that matters.
So when a check exists to certify that something helps, it needs two floors:
- a statistical floor from measured noise — the effect is not sampling luck;
- a practical floor from an anchor — the effect is at least as large as <published method / theoretical gain / the smallest difference that would change a decision>.
State which anchor supplies the second one. If none does, say so as a coverage limit: "this check certifies the effect is real, not that it is useful."
Anchor hygiene
- Write the source next to the number.
0.0716821 # Conway & Sloane, normalised second moment of E8. An unattributed constant becomes a golden value within a release, and nobody can tell whether it is authority or history. - Never derive an anchor from the code under test, including "I ran it once and it looked right."
- Version-stamp cached artifacts. If an anchor or a fitted object is cached, key the cache on the method as well as the problem, and delete on mismatch. A cache keyed only on the problem silently serves values produced by code you have since changed.
- State the covered range every time. It is the single most common way an anchored gate turns out to have been blind.