Formal Stability Proof of the Absolute(0) Stability Framework v3.0.1
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2026
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| author | Morrison, Kemar Armando |
| author_facet | Morrison, Kemar Armando |
| contents | <p>We present a formal stability proof for the Absolute(0) Stability Framework version 3.0.1 (AS3.0.1), a</p> <p>real-time governance engine for autonomous systems. The central contribution is ASV3-FORMAL</p> <p>compliance: six targeted fixes (FIX-A through FIX-F) that elevate the framework from an engineering stability</p> <p>heuristic to a mathematically certifiable invariant system. We define an explicit admissible set A as a closed</p> <p>ball in Rd, construct a smooth barrier function B(x) = ||x||2/R2</p> <p>− 1, prove that the projection operator Proj■ is</p> <p>non-expansive (Lipschitz-1), and establish forward-invariance of A under the complete governance pipeline.</p> <p>Six formal propositions cover admissible set geometry, projection non-expansiveness, barrier descent, IDIG</p> <p>convergence, budget-tightening monotonicity, and composite pipeline certification. Empirical self-tests over</p> <p>200 steps across all four defence domains confirm zero invariant violations and persistent GREEN/BLUE</p> <p>phase dominance.</p> <p>Keywords: Lyapunov stability; barrier functions; invariant sets; formal governance; autonomous systems; spectral</p> <p>control; projection operators; non-expansive mappings; real-time certification</p> <p> </p> <p>Main Contributions</p> <p>1. Introduction of an explicit admissible set A with barrier function B(x) satisfying B(x) ≤ 0 iff x ∈ A,</p> <p>replacing the implicit stability region of prior versions.</p> <p>2. Proof that the radial projection Proj■ is non-expansive (Lipschitz constant 1), providing a contractivity</p> <p>certificate for the governance map.</p> <p>3. Derivation of the AGGL descent lemma: when δ > 0, the gradient correction u■ reduces B(x + u■)</p> <p>relative to B(x) at each step.</p> <p>4. IDIG convergence theorem: the gradient flow is energy-decreasing and admits a closed-form</p> <p>fixed-point characterisation.</p> <p>5. Budget-tightening monotonicity: the admissible radius R(t) is non-increasing under the SIR-triggered</p> <p>tightening schedule.6. Composite pipeline certification: the full governance map G = Proj■ ■ LHAR ■ GLSM ■ IDIG ■ AMPO</p> <p>■ AGGL is forward-invariant.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19296319 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Formal Stability Proof of the Absolute(0) Stability Framework v3.0.1 Morrison, Kemar Armando <p>We present a formal stability proof for the Absolute(0) Stability Framework version 3.0.1 (AS3.0.1), a</p> <p>real-time governance engine for autonomous systems. The central contribution is ASV3-FORMAL</p> <p>compliance: six targeted fixes (FIX-A through FIX-F) that elevate the framework from an engineering stability</p> <p>heuristic to a mathematically certifiable invariant system. We define an explicit admissible set A as a closed</p> <p>ball in Rd, construct a smooth barrier function B(x) = ||x||2/R2</p> <p>− 1, prove that the projection operator Proj■ is</p> <p>non-expansive (Lipschitz-1), and establish forward-invariance of A under the complete governance pipeline.</p> <p>Six formal propositions cover admissible set geometry, projection non-expansiveness, barrier descent, IDIG</p> <p>convergence, budget-tightening monotonicity, and composite pipeline certification. Empirical self-tests over</p> <p>200 steps across all four defence domains confirm zero invariant violations and persistent GREEN/BLUE</p> <p>phase dominance.</p> <p>Keywords: Lyapunov stability; barrier functions; invariant sets; formal governance; autonomous systems; spectral</p> <p>control; projection operators; non-expansive mappings; real-time certification</p> <p> </p> <p>Main Contributions</p> <p>1. Introduction of an explicit admissible set A with barrier function B(x) satisfying B(x) ≤ 0 iff x ∈ A,</p> <p>replacing the implicit stability region of prior versions.</p> <p>2. Proof that the radial projection Proj■ is non-expansive (Lipschitz constant 1), providing a contractivity</p> <p>certificate for the governance map.</p> <p>3. Derivation of the AGGL descent lemma: when δ > 0, the gradient correction u■ reduces B(x + u■)</p> <p>relative to B(x) at each step.</p> <p>4. IDIG convergence theorem: the gradient flow is energy-decreasing and admits a closed-form</p> <p>fixed-point characterisation.</p> <p>5. Budget-tightening monotonicity: the admissible radius R(t) is non-increasing under the SIR-triggered</p> <p>tightening schedule.6. Composite pipeline certification: the full governance map G = Proj■ ■ LHAR ■ GLSM ■ IDIG ■ AMPO</p> <p>■ AGGL is forward-invariant.</p> |
| title | Formal Stability Proof of the Absolute(0) Stability Framework v3.0.1 |
| url | https://doi.org/10.5281/zenodo.19296319 |