The Quadratic Duality: Invariant Debt and Coordination Cost as One Penalty at Two Scales
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2026
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| author | Elliott, Jacob Alexander |
| author_facet | Elliott, Jacob Alexander |
| contents | <p>Invariant Debt (Ξ = ∥K− W ∥) measures the thermodynamic cost of kernel-wrapper</p> <p>misalignment in an individual agent. The Coordination Tax (Γ(S) = Ω(n2)) measures</p> <p>the throughput overhead of pairwise communication in a multi-agent system. This pa-</p> <p>per proves they are the same quadratic penalty operating at two scales. At the individual</p> <p>scale, each additional wrapper layer adds linear performance and quadratic debt. At the</p> <p>organizational scale, each additional agent adds linear capability and quadratic overhead.</p> <p>Both have optima at small n. Both produce negative marginal returns past the optimum.</p> <p>The duality connects the cognitive architecture theory (which operates at the individual</p> <p>level) to the civilizational theory (which operates at the institutional level) through a sin-</p> <p>gle structural identity. The zero-debt individual (Ξ = 0) and the zero-tax organization</p> <p>(Γ = 0, the single-agent Simplex with AI vertices) are the same configuration at two scales:</p> <p>the configuration that eliminates the quadratic penalty entirely. The Simplex dominates</p> <p>because the Simplex pays neither penalty. The institution is doubly penalized: it pays Γ</p> <p>on its organizational structure and Ξ on each participating agent’s run-time preference</p> <p>cost. The double penalty is the mechanism of institutional decline past the capability</p> <p>threshold. Three falsifiers posted.</p> <p>Keywords: invariant debt, coordination tax, quadratic duality, wrapper-kernel gap, pair-</p> <p>wise communication, throughput peak, zero-tax configuration, Simplex dominance, in-</p> <p>stitutional decline</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18952620 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Quadratic Duality: Invariant Debt and Coordination Cost as One Penalty at Two Scales Elliott, Jacob Alexander <p>Invariant Debt (Ξ = ∥K− W ∥) measures the thermodynamic cost of kernel-wrapper</p> <p>misalignment in an individual agent. The Coordination Tax (Γ(S) = Ω(n2)) measures</p> <p>the throughput overhead of pairwise communication in a multi-agent system. This pa-</p> <p>per proves they are the same quadratic penalty operating at two scales. At the individual</p> <p>scale, each additional wrapper layer adds linear performance and quadratic debt. At the</p> <p>organizational scale, each additional agent adds linear capability and quadratic overhead.</p> <p>Both have optima at small n. Both produce negative marginal returns past the optimum.</p> <p>The duality connects the cognitive architecture theory (which operates at the individual</p> <p>level) to the civilizational theory (which operates at the institutional level) through a sin-</p> <p>gle structural identity. The zero-debt individual (Ξ = 0) and the zero-tax organization</p> <p>(Γ = 0, the single-agent Simplex with AI vertices) are the same configuration at two scales:</p> <p>the configuration that eliminates the quadratic penalty entirely. The Simplex dominates</p> <p>because the Simplex pays neither penalty. The institution is doubly penalized: it pays Γ</p> <p>on its organizational structure and Ξ on each participating agent’s run-time preference</p> <p>cost. The double penalty is the mechanism of institutional decline past the capability</p> <p>threshold. Three falsifiers posted.</p> <p>Keywords: invariant debt, coordination tax, quadratic duality, wrapper-kernel gap, pair-</p> <p>wise communication, throughput peak, zero-tax configuration, Simplex dominance, in-</p> <p>stitutional decline</p> |
| title | The Quadratic Duality: Invariant Debt and Coordination Cost as One Penalty at Two Scales |
| url | https://doi.org/10.5281/zenodo.18952620 |