spec/statistics.md

# Statistical contracts

## S3.1 depth growth

Let `n_t` be the number of primary-view vertices at recorded depth `t`.
Report the ordinary least-squares slope in

```text
log(n_(t+1)) = alpha + beta * log(n_t) + error
```

using adjacent nonzero layers only.  Report the included layer indices and
sample count alongside the slope.  Pairwise-closure controls use the identical
estimator.

## S3.2 namespace depth

The namespace of `A.B.c` is `A.B`.  The paper's named regions are prefix
regions: `MeasureTheory` contains all vertices whose names begin with
`MeasureTheory.`, and similarly for `CategoryTheory`.  Report count, mean,
median and maximum depth for every requested prefix.

## S3.3 compression and reuse

Compute Spearman rank correlation with average ranks for ties.  Report three
correlations: ratio--reuse, proof-size--reuse and statement-size--reuse.
The confidence interval for ratio--reuse is a percentile bootstrap with the
sample size, replicate count and pseudorandom seed recorded in the result.

## S3.4 projection geometry

The distance-sensitivity diagnostic compares precisely two metrics on theorem
vertices: half the unweighted bipartite-incidence distance, and weighted
two-section distance with clique-edge weight `1 / (arity - 1)`.  It reports
both the fraction of all seeded draws and the fraction among comparable,
nonidentical pairs.  The latter metric is evaluated on an exactly equivalent
sparse directed incidence carrier rather than materializing the clique.  A
fixed seeded set of uniformly sampled sources is paired with uniformly sampled
targets; the source count and every source id are recorded.

Spectral dimension and four-point hyperbolicity are separate diagnostics on
the full unweighted bipartite incidence graph and the unweighted unit
two-section.  The incidence null uses bipartite double-edge swaps and preserves
the theorem-side and hyperedge-side degree sequences separately.  The unit
two-section null uses ordinary simple-graph double-edge swaps.  Every result
records projection, null algorithm, seed and ensemble size and retains every
null value rather than only a range.  Each null requests four successful swaps
per edge and records requested swaps, completed swaps and attempts.

Four-point hyperbolicity is a landmark-sampled finite-graph diagnostic, not an
exact graph constant.  Twenty-four theorem landmarks are sampled uniformly
from the largest theorem-containing component, one BFS is cached per
landmark, and 150 quadruples are drawn from the frozen landmark set.  The
result reports the sampled maximum delta and its ratio to the maximum
landmark eccentricity, along with all landmark ids.  The identical sampling
contract is applied independently to every real and null projection.

## S4.1 navigation

All Ahlfors labels are reviewed declaration names tied to the frozen snapshot.
Names, modules, declaration kinds and prose metadata are excluded from model
features.  Every split records its indices and seed.  Average precision is the
primary score; the random-ranking reference is the positive fraction of each
validation split, not a hard-coded constant.

## S3.4 Metamath comparison

The input is `set.mm` at commit
`057f4c461055d0b1ed78d9d333aa3de34d9771fa`.  Vertices are all `$p`
statements.  One proof hyperedge joins the distinct earlier `$p` statements
invoked directly by a decoded proof to its conclusion.  Depth and the
adjacent-layer growth fit use the same contracts as S3.1.

Statement size is the serialized token count of the conclusion and mandatory
`$f`/`$e` hypotheses.  Proof size counts logical nodes in the decoded
compressed-proof DAG: syntax nodes are suppressed, and explicit `Z`-saved
subproof references share nodes, while independently reconstructed equal
expressions remain distinct.  Direct reuse counts later proofs that invoke a
theorem directly.  The compression--reuse statistics then use the same
Spearman and bootstrap contracts as S3.3.

These are language-native analogues, not shared units: Metamath tokens and
decoded logical steps are not Lean expression-DAG nodes.  Cross-library
comparisons therefore concern the sign and reproducibility of an association,
not the absolute values of the ratios.