A Delta-State Finite Automaton Framework for High-Energy Collatz Corridors: Exhaustive Symbolic Enumeration and Profinite State Compression
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2026
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| _version_ | 1866901367837163520 |
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| author | Moon, Kyung-Up |
| author_facet | Moon, Kyung-Up |
| contents | <p>We introduce a Δ-state compression framework for the accelerated Collatz <br>dynamics. High-energy valuation words w = (a₁,...,aₖ) with energy <br>B(w) = k·log₂3 − Σaᵢ ≥ 10 are analyzed via two complementary engines.</p> <p>CPSE (Corridor-exit-compensation symbolic engine) exhaustively enumerates <br>the complete finite domains W(30,{1,2},10) and W(30,{1,...,6},10), covering <br>1,510,808 and 13,681,233 valuation words respectively. In both cases, every <br>word resolves into convergence to 1 or bounded-delay compensation <br>(v₂(3n+1) ≥ 3) with zero unresolved words and maximum compensation delay <br>at most 53.</p> <p>PCDS (Profinite constrained dynamical system) identifies the corrected <br>finite-state descriptor</p> <p> q*(w) = (u, Δ(w) mod 2^(A(w)+m))</p> <p>where u is a fixed-length suffix and Δ(w) satisfies the linear recurrence <br>Δₖ₊₁ = 3Δₖ + 2^Aₖ. Experiments confirm that exit residues factor through <br>q*(w): words sharing the same state reach the same exit residue modulo 2^m. <br>Apparent exceptions at low lifting precision (≤ 42 bits) vanish completely <br>at ≥ 64 bits, confirming the factorization is a genuine structural property.</p> <p>This work reduces the Collatz escape problem to a profinite Δ-state safety <br>condition and identifies the remaining open problem: establishing Δ-state <br>factorization in full generality.</p> <p>This paper does not prove the Collatz conjecture. It is Paper B of a <br>four-part series on Collatz obstruction theory.</p> <p>Series:<br>Paper A: https://doi.org/10.5281/zenodo.20068553<br>Paper C: https://doi.org/10.5281/zenodo.20068757<br>Paper D: https://doi.org/10.5281/zenodo.20068845</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20068640 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Delta-State Finite Automaton Framework for High-Energy Collatz Corridors: Exhaustive Symbolic Enumeration and Profinite State Compression Moon, Kyung-Up Collatz conjecture 3x+1 problem valuation dynamics obstruction framework 2-adic integers <p>We introduce a Δ-state compression framework for the accelerated Collatz <br>dynamics. High-energy valuation words w = (a₁,...,aₖ) with energy <br>B(w) = k·log₂3 − Σaᵢ ≥ 10 are analyzed via two complementary engines.</p> <p>CPSE (Corridor-exit-compensation symbolic engine) exhaustively enumerates <br>the complete finite domains W(30,{1,2},10) and W(30,{1,...,6},10), covering <br>1,510,808 and 13,681,233 valuation words respectively. In both cases, every <br>word resolves into convergence to 1 or bounded-delay compensation <br>(v₂(3n+1) ≥ 3) with zero unresolved words and maximum compensation delay <br>at most 53.</p> <p>PCDS (Profinite constrained dynamical system) identifies the corrected <br>finite-state descriptor</p> <p> q*(w) = (u, Δ(w) mod 2^(A(w)+m))</p> <p>where u is a fixed-length suffix and Δ(w) satisfies the linear recurrence <br>Δₖ₊₁ = 3Δₖ + 2^Aₖ. Experiments confirm that exit residues factor through <br>q*(w): words sharing the same state reach the same exit residue modulo 2^m. <br>Apparent exceptions at low lifting precision (≤ 42 bits) vanish completely <br>at ≥ 64 bits, confirming the factorization is a genuine structural property.</p> <p>This work reduces the Collatz escape problem to a profinite Δ-state safety <br>condition and identifies the remaining open problem: establishing Δ-state <br>factorization in full generality.</p> <p>This paper does not prove the Collatz conjecture. It is Paper B of a <br>four-part series on Collatz obstruction theory.</p> <p>Series:<br>Paper A: https://doi.org/10.5281/zenodo.20068553<br>Paper C: https://doi.org/10.5281/zenodo.20068757<br>Paper D: https://doi.org/10.5281/zenodo.20068845</p> |
| title | A Delta-State Finite Automaton Framework for High-Energy Collatz Corridors: Exhaustive Symbolic Enumeration and Profinite State Compression |
| topic | Collatz conjecture 3x+1 problem valuation dynamics obstruction framework 2-adic integers |
| url | https://doi.org/10.5281/zenodo.20068640 |