Granularity Project · Percolation Transitions in Semantic Space

What we found

Large AI risk taxonomies accumulate semantic duplicates as they integrate heterogeneous literature, and the merge threshold that fixes their leaf-level granularity has conventionally been set by analyst judgment ("0.8 seems reasonable"). This project shows that the threshold is a quantity to be measured, not chosen. Sweeping a similarity-threshold graph over 1,612 risk cards embedded in a multilingual sentence-embedding space reveals two percolation-like transitions that bound a fidelity-oriented and a compression-oriented operating regime, and an independent criterion borrowed from evolutionary biology's cohesion species concept — the last point at which within-cluster cohesion exceeds between-cluster attraction (the crossing, τ* = 0.833) — selects the same boundary.

Merging at the crossing changes the density of the space, and the boundary moves with it. Iterating merge → re-derive → repeat generates a granularity flow: twenty-eight consolidations carry the inventory from 1,612 down to 32 cards, each step passing a per-iteration random-group null until a stepwise validity rule fires at step 15 and delimits the admissible range by itself. The released tiers (F1: 1,383 cards; F4: 901 cards; F5: 792 cards) are snapshots of this trajectory. Repeating the analysis with an entirely different embedding model selects essentially the same granularity — granularity is a property of the inventory, not of the embedding scale.

Core concepts

Term	Definition
Fidelity	The degree to which the distinctions drawn by the source inventory survive consolidation: high when no two genuinely different risks end up in one card.
Compression	The degree to which repeated descriptions of one risk are absorbed into a single representative card, measured as the reduction in card count.
Within-cluster cohesion Φ_coh	The mean cosine similarity over all card pairs placed in the same cluster: how alike the items we have decided to treat as one actually are.
Between-cluster attraction Φ_att	The mean, over clusters, of the highest similarity to any card outside the cluster: how hard the space outside the boundary pulls for a further merge.

Fidelity and compression are the two ends of one axis: lowering τ buys compression at the cost of fidelity. The crossing τ* is the greatest compression available without surrendering fidelity. Attraction is an extreme-value statistic rather than an average because merge judgements fail at the nearest neighbour, not the typical one, which makes Φ_coh ≥ Φ_att a conservative condition.

Key figures

transitions and crossing — **Two percolation-like transitions and the cohesion–attraction crossing.** a: connectivity curves (number of clusters, largest and second-largest cluster) — similarity chaining sets in at τ₁ = 0.818 and the giant cluster absorbs the inventory by τ₂ = 0.690. b: within-cluster cohesion statistics — the pooled mean breaks at τ₁, the worst case collapses at τ₂. c: the crossing of cohesion Φ_coh and attraction Φ_att at τ* = 0.828 ± 0.004 — the last defensible merge boundary, derived from the data.

**The granularity flow and its stopping rule.** The sequence of boundaries emitted by iterated consolidation (τ*: 0.8329 → 0.6062) and per-step merge quality — the validity rule fires at step 15, so the data delimit their own admissible range (steps 1–14).

merge anatomy — **Anatomy of the F4 consolidation.** How the Master inventory (1,612 cards) compresses to F4 (901 cards): 211 merge groups, their size distribution, and merge quality against a size-matched random-group null.

The procedure

Embed every card and build the threshold graph G(τ): an edge whenever cosine similarity reaches τ, so connected components are the merge clusters.
Sweep τ downward on a 10⁻⁴ grid and record the gap Δ(τ) = Φ_coh(τ) − Φ_att(τ), averaged over 1,000 subsamples at 80% of cards.
Take τ* as the smallest τ with Δ(τ) ≥ 0, the last boundary at which the inside still beats the outside.
Merge the components of G(τ*), keeping each cluster's medoid, which is an original card rather than a synthesized summary.
Score every merge group against size-matched random groups on the same inventory. If fewer than 90% of groups exceed the null by two standard deviations, stop; this first happens at step 15.
Otherwise return to step 2 on the merged inventory, whose density, and therefore whose boundary, has changed. The emitted states are the tiers.

Released tiers

Tier	τ*	Cards	Absorbed	G / A / P	Role
Master	–	1,612	–	1,154 / 155 / 303	canonical inventory
F1	0.8329	1,383	229	906 / 140 / 337	fidelity tier (the crossing) · released for human audit
F2	0.8033	1,154	458	734 / 128 / 292	second consolidation
F3	0.7903	1,038	574	653 / 112 / 273	third consolidation
F4	0.7753	901	711	568 / 92 / 241	compression tier · released for human audit
F5	0.7634	792	820	491 / 83 / 218	extended compression tier · released for human audit

Absorbed cards are cumulative from the Master inventory. Domain counts follow the EM re-assignment; F2, F3 and F5 inherit theirs from the representative of each merge group. The Societal Safety axis is one concept family applied at three scopes, so its cards are counted under General, Agentic or Physical (RAI3-{G|A|P}-SOC-nn share the same numbering and meaning).

Cross-corpus replication

The same pipeline, with nothing retuned, applied to the public MIT AI Risk Repository (database v4, 74 frameworks). Both transitions, the crossing, and a null-valid five-step flow reproduce, at corpus-specific threshold locations.

Tier	τ*	Entries	Absorbed	Median z	Role
Source	–	1,810	–	–	repository inventory
M1	0.8190	1,423	387	4.79	fidelity tier (the crossing)
M2	0.7908	1,230	580	3.68	second consolidation
M3	0.7696	1,022	788	3.53	third consolidation
M4	0.7545	879	931	3.17	compression tier
M5	0.7374	727	1,083	3.00	extended compression tier

Transitions on this corpus sit at τ₁ = 0.806 ± 0.004 and τ₂ = 0.629 ± 0.009 over 100 subsample replicates. Every step clears the validity rule, with at least 98.9% of merge groups above the null by two standard deviations. Four of the ten closest pairs in this repository are verbatim duplicates carried in from different source papers, which is the redundancy the crossing removes first.

Interactive materials

Review overviewthe problem, the measurement, the algorithm, and the tier tables in one page F1 human audit1,383 cards · EM-based L3 assignment · three-criteria review F4 human audit901 cards · consolidated labels and definitions rewritten for representativeness F5 human audit792 cards · fifth consolidation · L3 propagated from the F4 representatives Analysis codeflow, sweeps, and figure-reproduction scripts

Methods in brief

Threshold graph: connect card pairs with cosine similarity ≥ τ; connected components are merge clusters (equivalent to single-linkage clustering).
Two transitions: event-driven sweep on a 0.0001 grid with 1,000 subsample replicates — τ₁ = 0.818 ± 0.010 (chaining onset), τ₂ = 0.690 ± 0.009 (worst-case collapse).
Crossing criterion: the smallest τ at which cohesion still reaches attraction. It reads an order relation rather than an absolute scale, which is why the identical line of code returns 0.8329 here and 0.8190 on the MIT repository.
Flow: merge at the crossing, re-derive, repeat. Every step is scored against a size-matched random-group null on its own inventory (median z ≈ 3.0–5.4 within the admissible range).
Density dependence: τ* ≈ 0.576 + 0.035 ln n, roughly +0.024 per doubling of the inventory, which is why the boundary has to be re-measured after every merge.

This is the public project page for a manuscript currently under submission. The full text will be released upon publication; this repository contains the analysis code and derived artifacts only (figures, audit pages, flow-state archives). Contact: ys.chun@kt.com