The Principle of Geometric Balance: Unconditional Canonical Localization, the Impossibility of Model-Free Globalization, and PGB in Structured Regimes
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2025
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| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>The Principle of Geometric Balance (PGB) posits: asymptotically, a cooperative game<br>has a non-empty core if and only if it contains no Globally Canonically Antagonistic Structure (G–CAS). This paper provides a complete and unified formal development of the necessity direction (Empty Core ⇒ G–CAS), merging and reconciling earlier routes.<br>(I) Route C1 and the Generator. We derive the exact triangular decomposition of<br>any size-symmetric Bondareva–Shapley (B–S) functional into aggregated canonical testers bbk. We then prove unconditionally that the associated coefficients γk are nonnegative (the Generator Property), via an LP extreme-point reduction and exact combinatorial identities (a Beta-function identity and subset-of-a-subset factorization). This settles the generator gap and implies that any global violation yields a positive canonical deficit in some size k (an L–GCAS).<br>(II) Canonical localization. Using a quantitative canonicalization lemma (via Erd˝os–<br>Rado canonical Ramsey), we upgrade a positive expectation margin to an explicit canonical witness P with Fk(P ) > 0.<br>(III) Minimax orbit smoothing and finite reductions. We formalize orbit averag-<br>ing, the size-symmetric subspace, exact formulas for layer-uniform witnesses, and a conic Carath´eodory reduction that yields few-size canonical certificates.<br>(IV) Limits of globalization and structured regimes. We prove a model-free No-<br>Go theorem: internal dividends of P cannot universally bound cross-capacity S(P ; N \ P ). Consequently, L–to–G globalization requires structure. We then fully prove PGB necessity in two regimes providing such structure: (a) network-generated games with rapid attenuation (Strong Localization) and (b) bounded cross-capacity with a magnitude-aware tradeoff.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17596085 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Principle of Geometric Balance: Unconditional Canonical Localization, the Impossibility of Model-Free Globalization, and PGB in Structured Regimes Fathi, Kevin Cooperative Game Theory Core Stability Principle of Geometric Balance (PGB) Bondareva-Shapley Theorem Harsanyi Dividends Canonical Localization L-GCAS (Local G-CAS) G-CAS (Globally Canonically Antagonistic Structure) Aggregated Canonical Testers Canonical Ramsey Theorem Erdős-Rado Theorem Network-Generated Games (NGGs) Cross-Coalition Synergy <p>The Principle of Geometric Balance (PGB) posits: asymptotically, a cooperative game<br>has a non-empty core if and only if it contains no Globally Canonically Antagonistic Structure (G–CAS). This paper provides a complete and unified formal development of the necessity direction (Empty Core ⇒ G–CAS), merging and reconciling earlier routes.<br>(I) Route C1 and the Generator. We derive the exact triangular decomposition of<br>any size-symmetric Bondareva–Shapley (B–S) functional into aggregated canonical testers bbk. We then prove unconditionally that the associated coefficients γk are nonnegative (the Generator Property), via an LP extreme-point reduction and exact combinatorial identities (a Beta-function identity and subset-of-a-subset factorization). This settles the generator gap and implies that any global violation yields a positive canonical deficit in some size k (an L–GCAS).<br>(II) Canonical localization. Using a quantitative canonicalization lemma (via Erd˝os–<br>Rado canonical Ramsey), we upgrade a positive expectation margin to an explicit canonical witness P with Fk(P ) > 0.<br>(III) Minimax orbit smoothing and finite reductions. We formalize orbit averag-<br>ing, the size-symmetric subspace, exact formulas for layer-uniform witnesses, and a conic Carath´eodory reduction that yields few-size canonical certificates.<br>(IV) Limits of globalization and structured regimes. We prove a model-free No-<br>Go theorem: internal dividends of P cannot universally bound cross-capacity S(P ; N \ P ). Consequently, L–to–G globalization requires structure. We then fully prove PGB necessity in two regimes providing such structure: (a) network-generated games with rapid attenuation (Strong Localization) and (b) bounded cross-capacity with a magnitude-aware tradeoff.</p> |
| title | The Principle of Geometric Balance: Unconditional Canonical Localization, the Impossibility of Model-Free Globalization, and PGB in Structured Regimes |
| topic | Cooperative Game Theory Core Stability Principle of Geometric Balance (PGB) Bondareva-Shapley Theorem Harsanyi Dividends Canonical Localization L-GCAS (Local G-CAS) G-CAS (Globally Canonically Antagonistic Structure) Aggregated Canonical Testers Canonical Ramsey Theorem Erdős-Rado Theorem Network-Generated Games (NGGs) Cross-Coalition Synergy |
| url | https://doi.org/10.5281/zenodo.17596085 |