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)

Paper IV introduced the network coupling of household discharge capacity through the kinship network and showed that atomic tax extraction destroys household discharge capacity super-multiplicatively. This paper formalises the structural reason. We prove the Network-Correlation Theorem: tax extraction is the correlated-shock case across the kinship network — a uniform proportional shock to every node simultaneously — and the network’s contingent discharge function is structurally designed to buffer idiosyncratic shocks, not correlated ones. The theorem identifies why network amplification of tax-induced deficit is not a parametric feature but a categorical one: the architecture of kinship discharge is foreclosed permanently rather than temporarily under continuous correlated extraction. The paper includes the Categorical Worst-Case Theorem which establishes that tax is the categorical worst case among extraction structures by structural form (continuous, uniform, fixed-fraction, every period, every node reduced by the same fraction), not merely by magnitude.

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.

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