Published April 28, 2026 | Version v2
Preprint Open

Obligation Density: A General Law of Fragility in Financial and Complex Systems

Authors/Creators

Description

We introduce obligation density (OD) as a general diagnostic for systemic fragility. Modern risk analysis focuses on the level of obligations — debt, liabilities, guarantees — but ignores their concentration relative to the survivable capacity of the system carrying them. We define Ω(t) as the ratio of total claims M(t) to effective capacity C(t)·S(t), where S(t) synthesises interest efficiency, fiscal alignment, and maturity structure into a survivability index. The fragility gradient F(t) = d/dt ln Ω(t) decomposes into claim-growth and capacity-growth rates; instability emerges when claim-growth exceeds capacity-growth in sustained form and Ω approaches a system-specific threshold. A key theorem establishes that modern accounting systematically underestimates OD because true claims exceed recorded claims: contingent, derivative, and legal obligations accumulate off balance sheets. The under-measurement is not an accounting failure correctable by improved instruments; it is the macro-financial manifestation of a deeper constitutional structure formalised in the companion Foundation Paper. We apply the framework to the United States sovereign (2010–2025), constructing a proxy index from public data, and find structural deterioration over the period.

Notes

v2.0 — 28 April 2026

This version applies three changes to the v1 deposit. (1) An explicit
forward-reference paragraph is added to the Conclusion section, naming the
Capital-Preserving Redistribution class characterised in OD_06 as the
constructive design space within which the macro-financial diagnosis
admits structural resolution. (2) Four bibliography entries previously
attributed to "Broomhead Sovereign Conglomerate" are corrected to
"Broomhead Sovereign Private Trust" — the correct attribution per the
trust's standing instruction. The "Conglomerate" attribution was a
factual error on the v1 deposit. (3) The AI/LLM disclosure block is
updated to the 2026-environment-appropriate form, which acknowledges
hallucination risk specifically (fabricated citations, miscoded notation,
plausible-sounding but incorrect attributions) and provides a correction
path for readers identifying any such artefact. The mathematical content
of v1 — the central law, the fragility gradient, the US sovereign
diagnostic, the Stage 2 / Ω* ≈ 1.70 calibration — is unchanged.

Notes

The Obligation Density Programme is a seventeen-paper deposit batch establishing a structural-mathematical framework for systemic fragility in financial, legal, and complex systems. It defines obligation density Ω(t) = M(t)/(C(t)·S(t)) — total claims over effective survivable capacity — and proves that systematic under-measurement of Ω(t) by mainstream instruments is the macro-financial manifestation of a deeper constitutional structure: the Witness Principle, that the public side of any administrative system has zero standing on the private ledger. The bundle moves from macro entry and constitutional foundation through household, firm, and cross-scale instances, the General Witness Theorem across combinatorial, legal, and economic domains, and the Privacy/Property normative anchors. The framework is structural, not policy-prescriptive.

Notes

This record is part of the seventeen-paper Obligation Density Programme deposit batch. The full bundle map is below; the temporal and structural ordering of the programme is externally verifiable via the DOI records listed.

OD_00 — Obligation Density (DOI: 10.5281/zenodo.18615905) — THIS PAPER

OD_01 — The Witness Principle and the Witness-Extraction Asymmetry (DOI: 10.5281/zenodo.18383361)

OD_02 — The STAR-OD Theorem (DOI: 10.5281/zenodo.19228716)

OD_03 — Population Discharge Distribution T(t) (DOI: 10.5281/zenodo.18383365)

OD_04 — The Tax Extraction Asymmetry Theorem (DOI: 10.5281/zenodo.19228708)

OD_05 — The Network Discharge Theorem and the Temporal Compounding of Tax-Induced Obligation Deficit (DOI: 10.5281/zenodo.19809921)

OD_06 — Capital-Preserving Redistribution (DOI: 10.5281/zenodo.19809925)

OD_07 — The Network-Correlation Theorem (DOI: 10.5281/zenodo.19809927)

OD_08 — Empirical Specification for the Obligation Density Framework (DOI: 10.5281/zenodo.19809935)

OD_09 — The Capital Formation Witness Problem (DOI: 10.5281/zenodo.19809943)

OD_10 — The Cross-Scale Witness Principle (DOI: 10.5281/zenodo.19809968)

OD_11 — The General Witness Theorem (DOI: 10.5281/zenodo.19809992)

OD_12 — Privacy as Constitutional Foundation (DOI: 10.5281/zenodo.19810000)

OD_13 — Cross-Domain Postscript (DOI: 10.5281/zenodo.19810008)

OD_14 — Property as Constitutional Foundation (DOI: 10.5281/zenodo.19810020)

OD_15 — Empirical Test Log (DOI: 10.5281/zenodo.19810022)

OD_16 — The Distributed-Consensus Witness Problem (DOI: 10.5281/zenodo.19810025)

External anchors (already published, separately deposited):

Local Witnessed Descent and the Nonlocality of Strong Regularity (DOI: 10.5281/zenodo.18408373)

Deterministic Descent via Witness Principles in Discrete Global–Local Systems (DOI: 10.5281/zenodo.18344559)

 

Other

Trust and Licensing Notice

This work is issued as a trust-marked scientific publication under the Broomhead Sovereign Private Trust (field anchor: geoffreybroomhead.eth).

All intellectual property rights are vested in the Broomhead Sovereign Private Trust. This document is governed by the Broomhead Sovereign Private Trust Custodial License v1.0.

For licensing or implementation inquiries, contact: recursive.broomhead@proton.me

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Related works

Is supplemented by
Preprint: 10.5281/zenodo.18408373 (DOI)
Preprint: 10.5281/zenodo.18344559 (DOI)
Preprint: 10.5281/zenodo.18383359 (DOI)
Preprint: 10.5281/zenodo.19491055 (DOI)