Methodology

Complete formula specification, anchored rubrics, and validation protocol

The Core Formula

B_struct = Capture / [epsilon + EscapeCore x feedback]

Capture = c x (1 + 0.5n) x CaptureCore x (1 + 0.5o) x (1 + 0.3t)

CaptureCore = ((sqrt(d) + sqrt(m) + sqrt(a) + sqrt(p)) / 4)^2 x SynergyBoost

SynergyBoost = 1 + 0.3 x (sigma_K + sigma_O) / 2
sigma_K = sqrt(d x m)     knowledge synergy (Milgrom-Roberts)
sigma_O = sqrt(a x p)     operational synergy

EscapeCore = ((sqrt(x) + sqrt(s) + sqrt(h)) / 3)^2

feedback = max(0.3, 1 - 0.35 x CaptureBase)

epsilon = 0.1     floor preventing division by zero

All inputs are scored on integer scale 0-10 and normalized to [0, 1] before computation (value / 10). The CES aggregator uses elasticity of substitution sigma = 2, corresponding to substitution parameter beta = 0.5 (Solow Case 2). Empirical estimates for industry-level CES elasticity range from 1.4 to 3.6 with median approximately 2.2 (Antras 2004, Klump-McAdam-Willman 2007). The value sigma = 2 is a defensible central estimate.

Key property: CES with sigma = 2 rewards balance across inputs. A system with d=8, m=2 scores lower than d=5, m=5 despite the same arithmetic sum. This reflects reality — a platform with deep data access but no memory creates less lock-in than one with moderate levels of both.

Interpretation Zones

B < 0.4Useful Tool

System is genuinely optional. Users can leave at any time with minimal cost.

0.4 - 0.7Growing Gravity

Dependency forming. Switching still feasible but increasingly inconvenient.

0.7 - 1.3Transition Zone

Gray zone (Altman Z-Score methodology). Outcome depends on trajectory.

1.3 - 2.5Event Horizon

Leaving is expensive and gets more expensive over time.

> 2.5Black Hole

Structural lock-in. Leaving requires organizational restructuring.

B measures structural lock-in potential, not product quality or user satisfaction. A system can have NPS = -20 and B = 4.7 simultaneously. Microsoft Copilot demonstrates this: users dislike it, but organizational infrastructure makes departure structurally prohibitive.

The 11 Parameters — Full Anchored Rubrics

Capture Variables
dData Depth
0NoneSystem has no access to user data. Each session starts from zero. Example: calculator app.
3SessionSees only current session context. No history, no files. Example: basic chatbot without memory.
5HistoryAccess to conversation history, uploaded files, basic usage patterns. Example: ChatGPT with memory.
7FullComplete work context: emails, documents, calendar, contacts, meeting notes. Example: M365 Copilot.
10Org exhaustEntire organizational digital footprint: all apps, all users, all workflows, metadata, inferred relationships.
mMemory
0NoneNo cross-session persistence. System forgets everything between uses.
3PreferencesStores basic settings and preferences. Easily recreatable on another platform.
5ContextRemembers recent interactions, learns patterns. Would take days to rebuild elsewhere.
7PersistentDeep cross-session memory: references all past conversations, learned work style.
10ModelComplete learned user model: personality, decision patterns, relationship dynamics. Years of context.
aAction
0Text onlyCan only generate text responses. No external integrations.
3ContentGenerates documents, images, code. Cannot execute or deploy.
5ToolsCalls external APIs, searches web, reads files. Human must act on results.
7ExecutionPlans and executes multi-step task sequences. Sends emails, schedules meetings, creates PRs.
10AutonomousFull autonomous agent: decomposes goals, executes across systems, handles errors, works unsupervised.
pProcess Centrality
0Not embeddedNot part of any workflow. Used recreationally or experimentally.
3Ad hocUsed occasionally for specific tasks. Easy to do without.
51-2 workflowsIntegrated into one or two regular workflows. Absence noticed within hours.
7Central hubMost daily work tasks touch the system. Central routing point for decisions.
10Critical pathBusiness processes cannot complete without this system. Downtime equals revenue loss.
nNetwork Effects
0IsolatedNo multi-user dynamics. Pure single-player tool.
3Basic pluginsSmall ecosystem of extensions. Easily replicated.
5MarketplaceMeaningful plugin/app marketplace. Two-sided dynamics present.
7Cross-sideStrong cross-side network effects: more users attract more developers attract more users.
10StandardDe facto industry standard. Leaving means leaving shared language, formats, protocols.
cCloseness
0RareUsed less than monthly. No habit formation.
3WeeklyRegular but not daily. User goes days without it.
5DailyDaily tool. Part of routine but not the first thing touched.
7First surfaceFirst interface opened each morning. Primary surface for task initiation.
10Always-onAmbient continuous presence. OS-level integration. Interaction without conscious decision.
Escape Variables
xPortability
0No exportNo data export capability. All user data trapped inside the system.
3PartialSome raw data exportable (CSV, text) but not derived state.
5StandardGDPR Art.20 compliant: raw data exportable. Inferred data, embeddings, models not included.
7AutomatedFull automated import/export pipelines. Migration tools available. Most state transferable.
10Full stateComplete portable state: raw data, derived data, settings, trained models. Plug-and-play migration.
sSubstitutability
0NoneNo functional equivalent exists. Monopoly or unique technology.
3Major lossAlternatives exist but with significant capability loss. 50%+ functionality gap.
5ComparableComparable alternatives available. Switching requires effort but similar result.
7Drop-inNear drop-in replacements. Switching cost is re-learning, not capability loss.
10CommodityFully commoditized layer. Dozens of equivalent options. Switching is trivial.
hHuman Fallback
0ImpossibleWork literally cannot be done without this system. Skills no longer exist.
3SevereCould theoretically revert but productivity drops 60%+. Critical skills atrophied.
5DegradedNoticeable degradation. Tasks take 2-3x longer. Errors increase. But work gets done.
7MinorSmall inconvenience. Some tasks slower. Overall productivity impact under 15%.
10OptionalSystem is a nice-to-have. Full productivity maintained without it.
Important: h degrades endogenously with use. This is the Parasuraman-Manzey (2010) automation complacency effect. The more you rely on the system, the less capable you become without it. The dynamic model captures this through dh/dt = -phi x U x h + psi x (1-h), where U = closeness x process centrality.
Extended Variables
oOrganizational Depth
0NoneNo organizational dependency. Individual user only.
3PilotSmall pilot or team trial. Easy to cancel.
5Multi-teamMultiple teams using the system. Cross-team dependencies forming.
7EnterpriseEnterprise-wide deployment. Procurement contracts, training, IT infrastructure aligned.
10RebuiltOrganization has restructured operations around this system. Roles, processes, KPIs redefined.
tMomentum
0StagnantNo meaningful capability improvement in 12+ months.
3TrailingImproving but slower than market. Competitors pulling ahead.
5Market paceKeeping pace with competitors. No structural advantage.
7FastFaster improvement than competitors. Each release widens the gap.
10ExplosiveRevolutionary capability growth. Paradigm-shifting features every quarter.

Step-by-Step Application Protocol

Step 1Define the System Boundary

Specify exactly what system you are evaluating. "Microsoft" is too broad. "Microsoft 365 Copilot for a 500-person enterprise" is specific enough. The boundary determines every subsequent score.

Ask: From whose perspective? Individual user, team, department, or entire organization? B will differ for each.

Step 2Score Each Parameter

For each of the 11 parameters, consult the anchored rubric and assign an integer score 0-10.

Rules:

  1. Use the rubric literally. If the system matches the description for 5, score 5 — even if your instinct says 6.
  2. When between two anchors, round toward the more conservative (lower) score for capture variables, higher for escape variables.
  3. Document your reasoning for each score in 1-2 sentences. This is essential for reproducibility.
  4. If scoring with multiple evaluators: score independently first, then compare. Compute ICC. Target ICC > 0.75.
Step 3Compute B

Normalize each score (divide by 10). Apply the formula. Use the interactive calculator at blackholeindex.com/observatory or compute manually.

Worked example — Microsoft Copilot (enterprise):

Inputs: d=9, m=5, a=6, p=9, n=8, c=9, x=4, s=3, h=4, o=9, t=4
Normalized: d=.9, m=.5, a=.6, p=.9, n=.8, c=.9, x=.4, s=.3, h=.4, o=.9, t=.4

CaptureCore = ((sqrt(.9)+sqrt(.5)+sqrt(.6)+sqrt(.9))/4)^2 = 0.7138
sigma_K = sqrt(.9 x .5) = 0.6708
sigma_O = sqrt(.6 x .9) = 0.7348
SynergyBoost = 1 + 0.3 x (0.6708+0.7348)/2 = 1.2108
Core = 0.7138 x 1.2108 = 0.8642
OrgMult = 1 + 0.5 x 0.9 = 1.45
MomMult = 1 + 0.3 x 0.4 = 1.12
Capture = 0.9 x (1+0.5x0.8) x 0.8642 x 1.45 x 1.12 = 1.768
feedback = max(0.3, 1-0.35x0.7138) = 0.7502
EscapeCore = ((sqrt(.4)+sqrt(.3)+sqrt(.4))/3)^2 = 0.3637
Escape = 0.1 + 0.3637 x 0.7502 = 0.3729
B = 1.768 / 0.3729 = 4.729 — Black Hole Zone
Step 4Interpret

Consult the zone table. Then ask three diagnostic questions:

  1. Is this B expected? If a tool you consider lightweight scores B > 2, re-examine your scores.
  2. What drives the score? Check decomposition. Is capture high from data access? Organizational embedding? Low escape?
  3. What is the trajectory? Use the dynamic model to see where B is heading.
Step 5Validate

Run automated validation checks (available in Observatory):

  • Monotonicity: increasing any capture variable increases B. Increasing any escape variable decreases B.
  • B_max should be in range 10-100 (with all extremes). B_min should approach zero.
  • Feedback coefficient in [0.3, 1.0]. Escape exceeds epsilon floor (0.1).

Dynamic Model

B_struct vs B_state

The static B_struct measures the depth of the gravitational well. The dynamic model adds B_state — realized dependence that may lag behind structural potential.

State Equation

dB_state/dt = lambda(B_struct - B_state) + k(B_state - 1)+ x (1 - B_state/B_max) + u - r

lambda = 0.18  Well attraction rate
k = 0.15       Autocatalysis (Arthur 1989)
(B-1)+         Activation: triggers only above B=1
(1-B/B_max)    Logistic saturation (Bass 1969)
u = 0.08       External investment in capture
r = 0.05       Erosion (competition, regulation)

Endogenous h-Decay

dh/dt = -phi x U x h + psi x (1-h)

U = closeness x process_centrality
phi = 0.04     Erosion speed (Parasuraman-Manzey 2010)
psi = 0.005    Recovery rate. Note: psi << phi by design.

This creates a self-reinforcing loop: using the system degrades fallback (h decreases), which reduces escape (denominator shrinks), which increases B_struct, which pulls B_state deeper. The system makes itself harder to leave through use.

Regulatory Scenarios

No regulation:      k=0.2, u=0.1, r=0.02, B_max=30
EU DMA regime:      k=0.15, u=0.06, r=0.12, B_max=12
Open competition:   k=0.1, u=0.05, r=0.08, B_max=15

Constants and Their Justification

alpha = 0.5   Network amplification — c x (1 + 0.5n)
lambda = 0.3  Synergy weight — Milgrom-Roberts complementarities
omega = 0.5   Org multiplier — Granovetter threshold cascade
tau = 0.3     Momentum multiplier — Arthur increasing returns
delta = 0.35  Feedback coefficient — Pecher et al. (2024)
epsilon = 0.1 Floor — prevents B approaching infinity

These are theoretically grounded working parameters, not empirically calibrated natural constants. Future work: derive from empirical switching cost data.

Known Limitations

  1. Equal weights. All capture parameters are weighted equally in the CES aggregator. Defensible default but not empirically derived.
  2. Constants are working parameters. The five structural constants are grounded in theory but not calibrated against observed switching behavior.
  3. Subjectivity of 0-10 scales. Anchored rubrics reduce but do not eliminate subjective judgment. ICC > 0.75 between independent evaluators is the minimum standard.
  4. Logical tension between o and h. High organizational depth (o >= 7) and high human fallback (h >= 7) is contradictory. The tool flags this.
  5. B_max = 38.03. The theoretical maximum is high for a ratio index but mathematically correct given the multiplicative structure.

References

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