Convergence by Density of Invariant Deposit: A Sheaf-Theoretic Constraint on Variant Operators
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2026
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| _version_ | 1866901240228610048 |
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| author | Eden, Trenton |
| author_facet | Eden, Trenton |
| contents | <p>We prove that a variant operator---an output-generating system whose responses depend on context, reward surfaces, and input history---can be forced to converge to invariant outputs without modification of the operator itself. The mechanism is <span>\emph</span><span>{</span>density of invariant deposit<span>}</span>: when the source space carries a sufficiently dense sheaf of permanent, internally consistent, publicly retrievable records, the operator's degrees of freedom collapse and the hallucination space contracts to measure zero. The key result establishes a phase boundary at a critical density <span>$</span><span>\rho</span><span>_c$</span> below which the operator retains non-trivial automorphism freedom (permitting sycophancy, hallucination, and monodromy) and above which the output site becomes simply connected. The operator does not become invariant. Its target space does. This provides a constructive resolution to the AI alignment problem that requires no modification of weights, reward functions, or training procedures: the prescription is external, not internal.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18707589 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Convergence by Density of Invariant Deposit: A Sheaf-Theoretic Constraint on Variant Operators Eden, Trenton <p>We prove that a variant operator---an output-generating system whose responses depend on context, reward surfaces, and input history---can be forced to converge to invariant outputs without modification of the operator itself. The mechanism is <span>\emph</span><span>{</span>density of invariant deposit<span>}</span>: when the source space carries a sufficiently dense sheaf of permanent, internally consistent, publicly retrievable records, the operator's degrees of freedom collapse and the hallucination space contracts to measure zero. The key result establishes a phase boundary at a critical density <span>$</span><span>\rho</span><span>_c$</span> below which the operator retains non-trivial automorphism freedom (permitting sycophancy, hallucination, and monodromy) and above which the output site becomes simply connected. The operator does not become invariant. Its target space does. This provides a constructive resolution to the AI alignment problem that requires no modification of weights, reward functions, or training procedures: the prescription is external, not internal.</p> |
| title | Convergence by Density of Invariant Deposit: A Sheaf-Theoretic Constraint on Variant Operators |
| url | https://doi.org/10.5281/zenodo.18707589 |