# Corrigendum 1 to *The Black Hole Index: A Structural Lock-In Measurement Framework for Digital Platforms* (SSRN 6555158; served at /paper.md)

**Dated 2026-09-13. Corrigendum 1 to the Black Hole Index paper.** The paper is a published artefact with a DOI and is not re-issued. This corrigendum lists, sentence by sentence, the statements in it that later measurement contradicted, with the evidence and the corrected wording. Every number below was recomputed from the paper's own 100-platform cross-section (identical to the Q1-2026 vintage in the quarterly record) or from the live 184-platform release, as stated. No score, formula, constant or ranking in the paper changes.

| # | location | published wording | what was measured | corrected wording |
|---|---|---|---|---|
| 1 | Abstract, l.17 | "generating the first systematic structural lock-in ranking of the global platform economy" | searches (2026-09-13) found no published cross-sector ranking of named platforms by a formula-based structural lock-in score; adjacent instruments are country-level (Digital Dependence Index; Global Digital Platform Power Index) or relationship-level (cloud vendor lock-in frameworks; experimental labour-market lock-in). Absence from search is not proof of absence. The universe is a curated 100 (now 184) that includes utilities, pharmaceutical companies, protocols and two archetypes | "generating what is, to the author's knowledge, the first published cross-sector ranking of named platforms by a structural lock-in score" — a novelty claim, not a validity claim |
| 2 | §2.3, l.57 | "the same 11 parameters and rubrics apply to an AI assistant, a semiconductor manufacturer, and a payment network" | the anchors for Memory, Action and Closeness are written for AI assistants ("creates PRs", "learned work style", "first interface opened each morning") and were applied to grid operators, a clearing house and foundries at 8–9; the project's own evidence-first determinations returned 0 determinate scores of 8 attempts on published rows, and recorded Closeness, Human Fallback and Momentum as not scorable at entity level for all four cloud providers | "the same 11 parameters were applied across sectors; whether the rubric anchors, written for software platforms, apply as written to infrastructure, manufacturing and payment networks is not established, and later evidence-first scoring found three parameters unscorable at entity level without a declared reference subject" |
| 3 | §3.5 table, l.231–237 | zone descriptions ("System is genuinely optional …", "Leaving is expensive and gets more expensive over time", "Structural lock-in. Leaving requires organizational restructuring") | on the Q3-2026 release: three Event Horizon platforms (ChatGPT, Claude, Grok) score substitutability and human fallback ≥ 7; six Black Hole platforms score every escape parameter ≥ 5; four Black Hole platforms score organisational depth ≤ 4; the four numeric boundaries have no published derivation; 9.5 % of platforms change band under ±1 scoring noise | the thresholds are heuristic cut-points and the names are labels; the descriptions are withdrawn as empirical statements |
| 4 | §3.7, l.247 | "differences in B smaller than 0.5 should be interpreted with caution" | under the paper's own ±1 scoring-noise benchmark (live release, 1,000 replicates), 80 % intervals of B are wider than 0.5 for 94.6 % of platforms, wider than 1.0 for 77.2 %, median width 1.71 (WeChat 12.9–25.0) | "the resolution of B is set by scoring noise, not by a fixed constant; under a ±1 benchmark the median platform's 80 % interval is about 1.7 B-units wide and scales with B, so adjacent ranks and differences below that width should not be read as distinctions" |
| 5 | §4.1, l.257 | "with written rationale for each parameter score … All scores, rationales, and computed B values are published at blackholeindex.com/rankings" | platform-level rationales (one paragraph per platform, no per-parameter citation) were written for the 100 platforms and published until the 4 June 2026 database migration, which did not carry them over; the 84 platforms added in Q2-2026 never had one; on 2026-09-13 the served dataset carries one | "with a written platform-level rationale for each platform; per-parameter citations were not recorded. The rationales were published with the Q1-2026 release; they were not carried into the database at the June 2026 migration and are not currently served" |
| 6 | §5.3, l.400 | "The top-5 and bottom-5 platforms are invariant across all coefficient combinations" | §5.3 does not name the six; the paper's constant list elsewhere includes λ, a §6 dynamic-model rate that does not enter B. Over the five constants that do enter the cross-sectional B (α, ω, τ, δ, ε; ±30 %, 243 combinations) on the paper's own 100: the top-5 set changes in 3 combinations and the bottom-5 set in 83, including a single move of α or ω to ×0.7 (Perplexity ↔ Lido); with the synergy coefficient added as a sixth (729 combinations): 9 and 249; rank correlation with the baseline stays above 0.998 throughout | "Neither the top-5 nor the bottom-5 set is invariant across all ±30 % coefficient combinations — over the five constants that enter B the top five change in 3 of 243 combinations and the bottom five in about a third, including single-coefficient moves — while rank correlation with the baseline stays above 0.998" |
| 7 | §5.3, l.400 | "The EDA duopoly, TSMC, and ASML remain the highest-B platforms; Dogecoin, Cardano, and XRP remain the lowest" | in the paper's own Appendix B, Cardano is ranked 90 and XRP 91 of 100; the lowest five are Dogecoin, Chat LLM (Generic), OpenSea, Disney+, Perplexity | "Dogecoin, Chat LLM (Generic) and OpenSea remain the lowest" |
| 8 | §5.4, l.409 | "Rankings are unaffected — the ordering is invariant to ε within this range" | on the paper's 100: ε = 0.05 changes 47 of 100 ranks (37 pairwise inversions); ε = 0.20 changes 52 ranks (55 inversions) and the top-5 set (ASML out, Amazon in); rank correlation with the baseline stays ≥ 0.998 | "Rank correlation with the baseline stays above 0.998 across the range, but individual ranks are not invariant: about half of all ranks change at either end of the range and the top-5 set changes at ε = 0.20. Absolute B is conditional on ε (its share of the denominator is 25 % at the median platform and 32–60 % for the eight highest platforms of this cross-section; 47–60 % on the Q3-2026 release)" |
| 9 | §5.4, l.407–408 | "ε = 0.05: B_max increases by ~40 % … ε = 0.20: B_max decreases by ~25 %" | recomputed on the paper's 100: +42.2 % and −37.2 % for the EDA duopoly | "+42 % … −37 %" |
| 10 | §7.4, l.537 | "ρ ≈ 0.77 (p < 0.001, t = 4.13, df = 12)" | t = 4.13 at df = 12 gives a two-sided p of 0.0014 (one-sided 0.0007); on the current release the live cohort gives ρ = 0.785, two-sided p = 0.0009, permutation p = 0.0012, bootstrap 95 % interval 0.41–0.96 | "ρ ≈ 0.77 (two-sided p ≈ 0.001; permutation p ≈ 0.001; bootstrap 95 % interval approximately 0.4–0.96)" |
| 11 | §7.4, l.537 | "Amazon Prime (~95 %)" in the cohort | the retention figure is Prime's CIRP annual renewal rate; the scored object is the Amazon ecosystem row (no Prime row exists) — an object mismatch | "Amazon (ecosystem row; the retention figure is Prime's renewal rate — an object mismatch that weakens this observation)" |
| 12 | §7.4, l.537 | "NVIDIA CUDA (~98 %)" as an annual retention rate | the underlying metric is AI-framework share (~92 %), stored as 98; it is not a retention rate | remove NVIDIA from the retention cohort, or state the metric as share |
| 13 | §7.4, l.537 | "Synopsys/Cadence (~99 %), Visa (~99.9 %), SWIFT (~99.9 %)" as measured rates | these three values are characterisations ("structural", "zero voluntary departures", "zero voluntary exits since 1973"), not measured annual rates; without the four asserted rows the cohort is n = 10, ρ = 0.74, p = 0.015 | state them as asserted values and report the sensitivity |
| 14 | §7.4, l.539 | "Rank-order alignment reaches ρ = 1.0 for … e-commerce (eBay < Shopee < Coupang < MercadoLibre < Amazon Prime)" | the served metric values for those five are of three different types (user trajectory, growth, retention); Spearman between B and those values is 0.87, not 1.0, on both the paper's cross-section and the current release (the five B values are identical in both); the listed order is the B ordering | withdraw the e-commerce ρ = 1.0 claim; the streaming ordering (Disney+ < Netflix < Spotify) stands on three documented annual rates |
| 15 | §7.4, l.547 | "BHI predicts a retention floor, not a ceiling … No high-BHI platform shows weak retention" | the cohort is concurrent, not predictive; among the 91 pairs, the higher-B platform has the lower retention in 16 of the 88 comparable pairs (three pairs tie on retention) — e.g. Spotify 2.31 / 96.5 % above Apple 4.60 / 89 %; the live page already states "it is not a prediction" | "In this concurrent cohort, retention is positively associated with B; the association is not a prediction, and a monotone floor is violated in 16 of the 88 comparable platform pairs" |
| 16 | §7.4 (implicit) | the association is offered as support for the formula | on the same cohort, escape-only forms and single parameters correlate at least as strongly (x alone 0.87, the escape side alone 0.84–0.87 depending on the form, a constant-free ratio 0.85 vs B 0.79), with every paired difference including zero | "this cohort does not distinguish the full formula from its escape half or from a single parameter; it cannot serve as evidence for the formula's complexity" |
| 17 | Appendix B, l.896 | "All scores, written rationales, and interactive exploration available at blackholeindex.com/rankings" | as item 5 | "All scores and interactive exploration are available at blackholeindex.com/rankings; the platform-level rationales published with the Q1-2026 release are not currently served" |

**What is not changed by this corrigendum:** the formula, the constants, the 100-platform scores in Appendix B (identical to the Q1-2026 vintage), the sensitivity-to-coefficients finding on rank correlation (§5.5 is confirmed by a global variance-based design: the worst of 4,096 constant samples still ranks at ρ 0.998), and the paper's declared limitations (§7.2).

**Provenance of this corrigendum:** the measurements are recorded in the project's audits C4–C6 (2026-09-13) and reproduced independently by three separate implementations; the underlying files are `public/bhi-spec.json`, `scripts/audit-env/input-snapshot.json` and the Q1-2026 vintage in the quarterly record.