feat(metrics): publish margin of error on every delta — #2495, #2524

The metrics card had one volume threshold doing two jobs.
recMetricsLowVolume = 20 is a DISPLAY floor — below that a skip rate is
anecdote — but the card then presented deltas as though it were also a
DECISION floor. Those differ by an order of magnitude: detecting the ~13pp
differences that matter needs ~133 plays per arm for 80% power at a=0.05.

So Discover's taste-matched (59 plays) and random-unheard (70) both rendered
as full-confidence rows with a bold delta beside them, and that comparison
sits at p ~ 0.06. The card said "signal"; the arithmetic said "maybe". It
produced a recommendation the data didn't support, and any reader with the
same numbers would have made the same call.

Deltas now carry a 95% margin of error and a `distinguishable` flag, computed
server-side so both clients read the same arithmetic instead of each
re-deriving it. Skip rate is a two-proportion difference; completion is
Welch, which needs a variance — hence completion_sqsum in the query. It is
the sum of squares rather than stddev_samp on purpose: raw source rows are
merged into surface families in Go, and sums of squares combine across groups
exactly whereas standard deviations cannot.

recMetricsLowVolume is untouched. "Too thin to show" and "too thin to act on"
are different questions.

Web renders an indistinguishable delta as dimmed and prefixed "≈", with the
range on hover and a legend explaining the glyph. Colour is withheld unless
the delta clears its margin — colouring noise red is what made the old card
misleading. Breakdown rows go through the same path; those are the thinnest
samples on screen and where the old card misled most.

Also fixes the admin trends view, which had the same problem worse: its
"Latest skip"/"Latest completion" columns are one WEEK while the adjacent
Plays column is the whole window. I misread exactly that and briefly
concluded Deep cuts was the worst surface, from ~17 plays in a single week —
over 180 days it is one of the best. Headers now name their period and the
skip cell carries that week's play count.

#2524: resolveArtist now recognises a duplicate-MBID unique violation as the
expected condition it is, matching resolveAlbum. Two rows mapping to one
MusicBrainz artist is a merge candidate, not a fault; without the branch it
logged a generic warning plus a Postgres ERROR line on every scan, which
teaches an operator to ignore database errors.

internal/api/metrics_significance_test.go

@@ -0,0 +1,152 @@
+package api
+
+import (
+	"math"
+	"testing"
+)
+
+func TestProportionDelta_ReproducesTheDiscoverCase(t *testing.T) {
+	// The comparison that motivated #2495: Discover taste-matched (59 plays,
+	// 15.3% skip) vs random-unheard (70 plays, 28.6%). A 13.3pp gap that the old
+	// card rendered as a confident coloured number, sitting at p ≈ 0.06.
+	d := proportionDelta(0.153, 59, 0.286, 70)
+	if d == nil {
+		t.Fatal("expected a delta for two real samples")
+	}
+	if math.Abs(d.DeltaPP-(-13.3)) > 0.1 {
+		t.Errorf("DeltaPP = %.2f, want ≈ -13.3", d.DeltaPP)
+	}
+	// This is the assertion the whole task exists for: at these sample sizes the
+	// margin swallows the difference.
+	if d.Distinguishable {
+		t.Errorf("13.3pp on n=59/70 reported as distinguishable (margin %.2f) — "+
+			"this is exactly the false confidence #2495 set out to remove", d.MarginPP)
+	}
+	if d.MarginPP <= 13.3 {
+		t.Errorf("MarginPP = %.2f, expected it to exceed the 13.3pp delta", d.MarginPP)
+	}
+}
+
+// Same effect size, ~10x the volume: now it is real. Proves the flag tracks
+// sample size rather than just the size of the gap.
+func TestProportionDelta_SameGapBecomesDistinguishableWithVolume(t *testing.T) {
+	d := proportionDelta(0.153, 600, 0.286, 700)
+	if d == nil {
+		t.Fatal("expected a delta")
+	}
+	if !d.Distinguishable {
+		t.Errorf("13.3pp on n=600/700 should be distinguishable (margin %.2f)", d.MarginPP)
+	}
+}
+
+func TestProportionDelta_SignAndDirection(t *testing.T) {
+	// Surface skips MORE than baseline -> positive delta (worse for skip rate).
+	worse := proportionDelta(0.40, 500, 0.25, 500)
+	if worse == nil || worse.DeltaPP <= 0 {
+		t.Fatalf("expected a positive delta, got %+v", worse)
+	}
+	better := proportionDelta(0.10, 500, 0.25, 500)
+	if better == nil || better.DeltaPP >= 0 {
+		t.Fatalf("expected a negative delta, got %+v", better)
+	}
+}
+
+func TestProportionDelta_EmptySamples(t *testing.T) {
+	if d := proportionDelta(0.2, 0, 0.3, 100); d != nil {
+		t.Errorf("n1=0 produced a delta: %+v", d)
+	}
+	if d := proportionDelta(0.2, 100, 0.3, 0); d != nil {
+		t.Errorf("n2=0 produced a delta: %+v", d)
+	}
+}
+
+// Two degenerate rates have zero standard error, which would report a margin of
+// 0 and therefore "distinguishable" for a delta of exactly 0. Reporting nothing
+// is the honest answer.
+func TestProportionDelta_DegenerateRates(t *testing.T) {
+	if d := proportionDelta(0, 50, 0, 50); d != nil {
+		t.Errorf("both rates 0 produced a delta: %+v", d)
+	}
+	if d := proportionDelta(1, 50, 1, 50); d != nil {
+		t.Errorf("both rates 1 produced a delta: %+v", d)
+	}
+	// One degenerate side is still informative — the other side carries variance.
+	if d := proportionDelta(0, 200, 0.3, 200); d == nil {
+		t.Error("one degenerate rate should still yield a delta")
+	}
+}
+
+func TestMeanDelta(t *testing.T) {
+	// Completion is bimodal, so ~0.16 variance (sd ≈ 0.4) is realistic.
+	const v = 0.16
+	thin := meanDelta(0.82, v, 59, 0.54, v, 70)
+	if thin == nil {
+		t.Fatal("expected a delta")
+	}
+	if math.Abs(thin.DeltaPP-28.0) > 0.1 {
+		t.Errorf("DeltaPP = %.2f, want ≈ 28.0", thin.DeltaPP)
+	}
+	// 28pp is large enough to survive even a wide margin at this n.
+	if !thin.Distinguishable {
+		t.Errorf("28pp on n=59/70 with sd 0.4 should be distinguishable (margin %.2f)", thin.MarginPP)
+	}
+	// A small completion gap at the same volume should not be.
+	small := meanDelta(0.56, v, 59, 0.54, v, 70)
+	if small == nil {
+		t.Fatal("expected a delta")
+	}
+	if small.Distinguishable {
+		t.Errorf("2pp on n=59/70 reported as distinguishable (margin %.2f)", small.MarginPP)
+	}
+}
+
+// A sample variance needs at least two observations per side.
+func TestMeanDelta_NeedsTwoObservations(t *testing.T) {
+	if d := meanDelta(0.8, 0.1, 1, 0.5, 0.1, 100); d != nil {
+		t.Errorf("n1=1 produced a delta: %+v", d)
+	}
+	if d := meanDelta(0.8, 0.1, 100, 0.5, 0.1, 1); d != nil {
+		t.Errorf("n2=1 produced a delta: %+v", d)
+	}
+}
+
+func TestMeanDelta_ZeroVarianceBothSides(t *testing.T) {
+	if d := meanDelta(0.8, 0, 50, 0.5, 0, 50); d != nil {
+		t.Errorf("zero variance on both sides produced a delta: %+v", d)
+	}
+}
+
+func TestSampleVariance(t *testing.T) {
+	// Observations 0, 1: mean 0.5, sample variance 0.5.
+	if got := sampleVariance(1.0, 1.0, 2); math.Abs(got-0.5) > 1e-9 {
+		t.Errorf("sampleVariance = %v, want 0.5", got)
+	}
+	// Identical observations -> zero variance, and must not go negative through
+	// floating-point cancellation.
+	if got := sampleVariance(4.0, 4.0, 4); got != 0 {
+		t.Errorf("identical observations gave variance %v, want 0", got)
+	}
+	if got := sampleVariance(0, 0, 1); got != 0 {
+		t.Errorf("n=1 gave variance %v, want 0", got)
+	}
+}
+
+// Clamping matters: a negative variance would become NaN in the square root and
+// propagate into the JSON as a null-ish number.
+func TestSampleVariance_NeverNegative(t *testing.T) {
+	// sqSum slightly below sum²/n, as cancellation can produce.
+	if got := sampleVariance(10.0, 24.999999999, 4); got < 0 {
+		t.Errorf("variance went negative: %v", got)
+	}
+}
+
+func TestNewDelta_BoundaryCountsAsDistinguishable(t *testing.T) {
+	d := newDelta(5.0, 5.0)
+	if !d.Distinguishable {
+		t.Error("a delta exactly equal to its margin should count as distinguishable")
+	}
+	d = newDelta(4.999, 5.0)
+	if d.Distinguishable {
+		t.Error("a delta just inside its margin should not count as distinguishable")
+	}
+}