§ 01Abstract
The Black Hole Index (BHI) measures structural lock-in for digital platforms — the degree to which a user, organisation, or counterparty cannot leave a system without proportional cost. We define a single dimensionless quantity B as the ratio of total capture surface to total escape surface, computed from eleven independently-scored parameters across capture, escape, and extended dimensions. This document specifies the scoring rubric, the V3.1 aggregation function, and the validation regime against external migration-cost data.
§ 02Axiom & framing
The instrument rests on a single axiom: structural lock-in is asymmetric capture relative to substitution. A platform with high capture surface but high substitution availability is not locked-in; a platform with modest capture and zero substitution is. The B-index makes this ratio explicit and bounded, so that platforms across radically different sectors (a stablecoin, an IDE assistant, a national grid operator) can be placed on the same scale.
BHI is intentionally not a quality, ethics, or pricing measure. A platform can clear B = 3 because its lock-in is structural, useful, and welcome — or because it is exploitative. The instrument reports the structural fact; interpretation belongs to the reader.
§ 03Eleven parameters
Each parameter is scored on a 0-to-10 ordinal rubric with anchored definitions at 0, 3, 5, 7, and 10. The rubric is published in full in the appendix to the paper.
| # | Parameter | Group | Anchor at 10 |
|---|---|---|---|
| 01 | Data | Capture | All user-generated state lives inside the platform with no equivalent export. |
| 02 | Memory | Capture | Persistent, system-side context that compounds with use and cannot be reconstructed elsewhere. |
| 03 | Action | Capture | Platform executes consequential actions on behalf of the user across multiple external surfaces. |
| 04 | Process | Capture | Operating procedures of the user/org are written around the platform's primitives. |
| 05 | Network | Capture | Direct two-sided network effects with no portable identity layer. |
| 06 | Closeness | Capture | Daily, intimate, multi-modal use; the platform has surface area against habit. |
| 07 | Portability | Escape | Full one-click export of all state to a competing system, including history and configuration. |
| 08 | Substitutability | Escape | Multiple commodity-grade alternatives exist at parity feature, parity price. |
| 09 | Human Fallback | Escape | The work the platform performs can be done by a human in reasonable time at reasonable cost. |
| 10 | Org Depth | Extended | Platform is embedded across >10 functions of the user organisation; removal requires structural change. |
| 11 | Momentum | Extended | Lock-in is increasing year-over-year; the gradient ∂B/∂t is positive and material. |
§ 04Aggregation function (V3.1)
All eleven parameters are normalised to [0, 1] by dividing by 10. The four core capture variables (D, M, A, P) are aggregated using a modified Constant Elasticity of Substitution (CES) function with elasticity σ = 2. The three escape variables (Po, Sb, Hf) use the same CES form. Closeness (C), Network (N), Org Depth (O), and Momentum (T) act as multiplicative amplifiers.
§ 05Zone thresholds
The continuous B is bucketed into five heuristic bands. The threshold at B = 1 is definitional: the point where capture equals escape under the declared denominator. Whether B = 1 corresponds to an observable behavioural transition in platform switching is an empirical question not yet tested.
- Useful Tool · B < 0.4 — heuristic band; label only.
- Growing Gravity · 0.4 ≤ B < 0.7 — heuristic band; label only.
- Transition Zone · 0.7 ≤ B < 1.3 — heuristic band; label only (the Altman Z-Score reference is an analogy, not a derivation).
- Event Horizon · 1.3 ≤ B < 2.5 — heuristic band; label only.
- Black Hole · B ≥ 2.5 — heuristic band; label only.
Corrected 13 September 2026. Until this date each band carried a one-line description of what membership meant for a user — “system is genuinely optional”, “leaving is expensive and gets more expensive over time”, “leaving requires organisational restructuring”. Those descriptions are withdrawn as empirical statements. Tested against the Q3-2026 release they are contradicted by the rows’ own scores: three Event Horizon platforms (ChatGPT, Claude, Grok) score substitutability and human fallback at 7 or above, which the rubric anchors describe as near drop-in replacement and minor inconvenience; six Black Hole platforms score every escape parameter at 5 or above; four Black Hole platforms score organisational depth at 4 or below. B is a ratio, so a high value can come from very high capture with moderate escape as well as from hard escape, and a band cannot say which. What survives is the numeric thresholds as heuristic cut-points and the names as labels. The paper’s §3.5 table prints the withdrawn descriptions and is not re-issued; the corrigendum beside it records the change.
None of the four numeric boundaries has a published derivation, and only one of them has any support in the distribution. B = 1 is the single derived value in the scheme — the point at which capture equals escape — and it is not itself a boundary. Tested against the Q3 2026 release (n = 184): 0.4 falls in a gap wider than 96% of the gaps between adjacent platforms, next to the only trough in the density of log B. The other three do not. 0.7 sits at the 72nd percentile of gaps, 1.3 at the median — that is, at nothing — and 2.5, the boundary carrying the heaviest label, in a gap wider than only 27% of them, separating two platforms 0.02 apart. 73% of the universe sits in one zone, and a data-driven five-class split of the same values places its breaks at 0.44, 1.51, 3.16 and 6.05 instead. Whether these lines correspond to anything observable is an open question, and it is recorded as one rather than settled by assertion. The analysis is reproducible: scripts/zone-boundary-analysis.py, against the canonical specification.
§ 06Universe construction
The universe is the top 184 platforms by economic relevance, drawn from eighteen sectors. Inclusion criteria, in order: (i) market cap or comparable economic footprint, (ii) user/transaction volume, (iii) sector representativeness, (iv) data availability for all eleven parameters. These criteria are the author’s own and are applied by the author: the universe is not externally determined. A sentence stood here until 8 September 2026 saying that borderline cases are noted in the appendix. That was not true — no appendix records them — and it is removed rather than quietly replaced.
How far this universe corresponds to external designation was measured for the first time on 8 September 2026, and the answer is: barely. Six designation regimes have been admitted under a discovery rule frozen before the search — the EU Digital Markets Act, the US Financial Stability Oversight Council, the FSB’s global systemically important banks, the CMA’s strategic market status, Germany’s section 19a, and HM Treasury’s critical third parties. Between them they designate 72 objects. Of those, 10 appear here as a row for that service; 10 are unresolved, because a row bears the designated group’s name and no row records what it measures; and 52 are absent — including 21 whose parent or affiliate is in the universe, which under the identity rule does not count as coverage. It is reported those three ways and never as one coverage figure. The full census, its sources and the reasoning are in the public research record.
The list is not a ranking of importance — it is a fixed evaluation universe, kept stable across releases so that quarter-on-quarter trajectories are comparable.
18 sectors · 184 platforms
§ 07Scoring protocol
Each platform is scored by a single evaluator against the published anchored rubrics. The protocol requires:
- Anchored evidence. Every parameter score is to be backed by a citation — a public filing, an API/export specification, a published migration case, or an admin-console screenshot.
- Written rationale. Each score is to carry a prose justification explaining the evidence and the mapping to the rubric anchor.
- Reproducibility. Scores, rationales and computed B values are to be published together.
- Inter-rater reliability. Independent multi-evaluator scoring is the primary validation target (ICC > 0.75). This remains pending — see the validation report.
Points 1 to 3 above state what the protocol requires. Until today they were written in the present tense, which described a practice this project has not yet met for its published universe. The true state:
The 184 platforms in the current release carry 2024 parameter scores between them. Their citations and per-parameter rationales are not published, and for all but one platform they are not recorded in the released dataset either. Established 13 September 2026: platform-level rationales were written for the 100 platforms of the Q1-2026 release and published until the 4 June 2026 migration to the database, which carried none of them over; the 84 platforms added in Q2-2026 never had one. Whether the 100 are restored is a pending decision recorded in the validation register. What is recorded is the eleven scores per platform and, for most score changes made through the quarterly process, the reason for that change. So a reader can recompute B from the parameters, but cannot see the evidence behind a standing level.
Corrected 8 September 2026. Until today this paragraph said a reason was recorded for every score change and that a reader could see why a score moved. Both halves were false, and they were measured rather than estimated. The measured split, as corrected on 13 September 2026: there are 42 observed change events across the three observed quarterly vintages (8 in the Q2-2026 release, 34 in Q3-2026), and all 42 carry a recorded reason. The count first published here on 8 September — 68 events, 26 without a reason, all 26 in the Q1-2026 transition — treated the Q4-2025 rows of the quarterly history as observed. They are not: they are a modelled back-projection from Q1-2026, so the 26 differences between the modelled Q4-2025 vintage and the first observed vintage are not score changes and never had a reason to record. And no public page or endpoint exposed any reason at all: they existed only behind the admin route, so a reader could not in fact see why any score moved. The reasons are now published on /api/platform/<id>/history with a reasonRecorded flag, so both what is recorded and what is missing are visible. Eight reasons were also filed under the wrong quarter — their own text named a different one — and have been re-filed.
Where the protocol is met in full is in the determinations prepared for regulators — the cloud infrastructure determination, the JFTC mobile-core determination and the integrated-stack determination — each of which records, for every parameter, the score, its evidence tier, the searches run, the sources retrieved, the primary source and the measurement provenance. Closing the same gap for the published universe is tracked, with its status and the date it was last assessed, in the validation register.
Added 8 September 2026. Six attempts have since been made to evidence a standing score in this published universe, four of them on targets drawn by lot: Slack and Ethereum on portability, BYD on momentum, Johnson & Johnson on process centrality, Cursor on data depth, and Coinbase on closeness. All six returned INDETERMINATE, for six different reasons, and none refutes the score it examined.In each case the rubric required something the published record does not contain — which product the score measures, against which comparator, in whose working day, on which dimension, at two dates, or for which user. So the count of published scores here that a reader can trace to evidence is still zero, and the obstacle turned out to be the record rather than the availability of sources: the Coinbase attempt rested on audited SEC filings and failed anyway. Each determination is published in full in the research record.
§ 08Release cadence
Quarterly. Major version bumps (V1 → V2 → V3) revise the aggregation function; minor bumps (V3.0 → V3.1) revise rubric anchors or universe composition. Each release is documented in the changelog, which names that release’s notable score changes — not every one of them; the per-change reasons, where they exist, are published at /api/platform/<id>/history. Historical scores are reissued under each new function so trajectories are not retroactively warped.
§ 09Limitations
BHI is a structural measure, not a behavioural one. It does not capture: (i) willingness to pay the migration cost, (ii) sector-specific regulatory dynamics, (iii) macro-cyclical effects on substitutability. The instrument is most useful when read alongside, not instead of, sector domain knowledge.
§ 10Changelog
- V3.1 · Q2 2026 — Org-depth multiplier introduced; momentum normalised; ε-floor formalised.
- V3.0 · Q3 2025 — Eleven-parameter rubric; capture/escape/extended split.
- V2.x · 2024 — Eight-parameter rubric; first public release.
- V1.x · 2023 — Six-parameter pilot; internal use only.