# Negative result (a contribution): rank-based coverage objectives fail for rare-lesion retention

## Claim

Effective-rank / coding-rate "coverage" objectives — RankMe, coding rate (MCR2-style) — are
**structurally mismatched** to retaining rare, small-region pathology under token pruning. Using
them as the pruning objective is worse than simply ranking tokens by lesion-subspace membership.

## Mechanism (the transferable part)

A rank-based coverage functional `C(S) = effrank(P_L Z_S)` is maximized by a retained set that
**diversely spans** the lesion subspace's directions. The decisive property of a small lesion is
that it is **rare** — a **few** high-membership tokens out of ~196. A *set*-level rank/coverage
objective is insensitive to such a cluster: a handful of tokens cannot materially raise the retained
set's effective rank, so the objective spends budget on abundant background directions and drops the
lesion. This is a **rarity** mechanism, not low internal geometry — measured at the operating layer,
lesion tokens are *not* low-rank relative to background (pooled effective rank 339 vs 307;
participation ratio 18.9 vs 13.9; `research_v4/lesion_spectrum.json`). Concentration, not spanning,
is what rare-pathology retention needs.

Formally: rank coverage rewards the *entropy of the retained set's singular spectrum*; lesion
retention rewards *mass on the top membership tokens*. These objectives diverge whenever the
critical signal is a **rare** cluster — of any internal rank. (The synthetic closed-form law of the
companion paper isolates a second, distinct route — a genuinely low-rank signal, gap `(m-r)/m`; real
lesions fail via rarity, not low rank.)

## Three independent lines of evidence (same verdict)

1. **Ablation (decisive).** At matched budget, the coverage-floor pruner retains 0.22 vs 0.82
   (budget 0.25) and 0.46 vs 0.98 (budget 0.5) of small lesions vs membership top-k; the
   difference CI excludes 0. The floor does not under-help — it actively hurts. (`ablation_floor.json`)
2. **Faithfulness (principled Gate 2).** Under the random-pruning protocol, coverage-drop predicts
   lesion-detection-drop no better than attention-drop: ρ 0.480 vs 0.479, difference CI includes 0.
   Coverage is not a superior proxy. (`gate_2_baseline.json`)
3. **Adaptive budget (Gate 4).** The "difficulty-adaptive budget" never materializes: aggregate
   coverage C* is the same on lesion-positive and -negative slices (250.4 vs 247.2), because a
   1–3 patch lesion cannot move an aggregate over ~196 tokens. (`gate_4_block3.json`)

All three reduce to one fact: **aggregate rank-coverage is blind to the few tokens that carry a
small lesion**, even though those tokens are individually highly localizable (Gate 1, AUROC 0.87).

## Closed-form law (S1, confirmed by controlled synthetic)

A controlled synthetic isolates the mechanism and yields a closed form (`research_v2/s1_crossover.json`).
Inject a signal of effective rank r across `m` tokens; select to a budget by a spanning objective
(farthest-point / effective-rank) vs a concentration objective (top-energy):

- **spanning retention(r) = min(r, m) / m**, **concentration retention = 1**,
- **gap(r) = max(0, (m − r) / m)**, **crossover r\* = m**.

A spanning objective retains a signal only in proportion to its rank; it matches concentration
only when the signal is *fully diverse* (rank = token count). Rare pathology is maximally
concentrated (r ≈ 1–3 patches, m small), so rank-based coverage is maximally mismatched there —
quantitatively reproducing the covtoken ablation (floor 0.22 vs membership 0.82 ≈ gap (m−1)/m for
the dominant 1-patch lesions). The verdict is no longer anecdotal: it is a closed-form prediction.

## Why this matters beyond this paper

RankMe-flavored objectives are an increasingly common, tempting choice for medical SSL
representation quality and for "coverage"-style regularizers. This result is a warning with a
mechanism: **for rare-pathology tasks, prefer concentration objectives (energy / membership mass)
over rank/spanning objectives.** A negative result with a transferable mechanism is citable;
"our dual didn't converge" is not. This is the former.

## What survives

The label-free lesion **subspace** (the geometry that produces membership) is intact and is the
contribution. The failure is specifically the **rank-coverage functional** built on top of it and
the constrained-optimization machinery that optimized it. Replacing the objective with the
membership/energy quantity recovers the result — but then the "constraint + dual" adds nothing
over a top-k rule, so it is dropped honestly rather than dressed up.