§ 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, 5, 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.

(1)

CES capture core with Milgrom-Roberts complementarity correction. SynergyBoost = 1 + 0.3 × (σ_K + σ_O) / 2, where σ_K = √(d × m) (knowledge synergy) and σ_O = √(a × p) (operational synergy).

(2)

Total capture. Closeness (c) is the base multiplier; network effects (n), organisational depth (o), and momentum (t) are multiplicative amplifiers with coefficients α = 0.5, ω = 0.5, τ = 0.3.

(3)

CES escape core — the same functional form applied to the three escape variables.

(4)

Endogenous escape suppression. As CaptureBase (= CaptureCore before SynergyBoost) deepens, the effective contribution of escape to the denominator decreases. The floor at 0.3 prevents escape from becoming negligible. δ = 0.35 is a working parameter grounded in the status quo bias literature (Samuelson & Zeckhauser, 1988).

(5)

B-index — bounded below by ε = 0.1 in the denominator to prevent runaway divergence and to bound B at a finite maximum (B_max ≈ 38).

[1]The six structural constants (α, ω, τ, δ, synergy coefficient, ε) are theoretically grounded working parameters, not empirically calibrated. Sensitivity analysis in §5 of the paper demonstrates ranking robustness to ±30% variation. See the validation report for details.

§ 05Zone thresholds

The continuous B is bucketed into five interpretive zones. The threshold at B = 1 is definitional: the point where capture equals escape. Whether B = 1 corresponds to an observable behavioural transition in platform switching is an empirical question not yet tested.

Useful Tool · B < 0.4 — system is genuinely optional; users can leave with minimal cost.
Growing Gravity · 0.4 ≤ B < 0.7 — dependency forming; switching feasible but increasingly inconvenient.
Transition Zone · 0.7 ≤ B < 1.3 — gray zone (cf. Altman Z-Score methodology); outcome depends on trajectory.
Event Horizon · 1.3 ≤ B < 2.5 — leaving is expensive and gets more expensive over time.
Black Hole · B ≥ 2.5 — structural lock-in; leaving requires organisational restructuring.

§ 06Universe construction

The Q2 2026 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. Borderline cases are noted in the appendix.

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

AI Platforms · 10Crypto Assets · 10Crypto Infrastructure · 8Social Media · 15Big Tech & Semiconductors · 12SaaS & Cloud · 24Banks & Financial Markets · 15Fintech & Payments · 13Gaming & Entertainment · 11E-Commerce · 12Pharma & Biotech · 11Critical Infrastructure · 11Telecom & Connectivity · 8Cybersecurity & Identity · 5Automotive & Mobility · 6Education, Research & Professional Information · 8Healthcare Information Systems · 2Industrial Software · 3

Sector boundaries are the same across V1, V2, and V3.1 to preserve longitudinal comparability.

§ 07Scoring protocol

Each platform is scored by a single evaluator using the published anchored rubrics, with written rationale for each parameter score. The protocol enforces:

Anchored evidence. Every parameter score must 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 includes a prose justification explaining the evidence and the mapping to the rubric anchor.
Reproducibility. All scores, rationales, and computed B values are published at blackholeindex.com/rankings.
Inter-rater reliability. Independent multi-evaluator scoring is the primary validation target (ICC > 0.75). This remains pending — see the validation report.

§ 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. Every numerical change is documented in the changelog, and 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.

Methodology