What the Devil's Game teaches us about the prime machine: A geometric reading, a conjecture, and a call for action
Year: 2026
Type: preprint
Abstract: What the Devil’s Game teaches us about the prime machine: A geometric reading, a conjecture, and a call for action Can a purely logical puzzle act as a structural stress test for the geometry of numbers? This paper runs Freudenthal's classical Sum-and-Product Puzzle (The Impossible Puzzle or Devil's Game) inside the deterministic prime machine, uncovering a profound conservation law mapped onto the integer square. By lifting the classical puzzle's upper bound (HI) and exhaustively enumerating all solutions up to HI = 99,999, the text isolates 18 unique integer pairs (x,y). Strikingly, every single solution obeys a rigid structural geometry: one component is strictly odd, while the other is always divisible by 4. This paper proves that this distribution is a direct manifestation of the prime machine's core symmetries, translating abstract group actions into a visible record on the two-dimensional lattice. 🏛️ The Geometric Intersections Proved Inside: 1. Two Blocking Geometries (Section 2.3): Demonstrates that the puzzle's logical constraints map natively to two intersection fields on the integer square \(Q^{=}\): Sam acts rigidly on a 1D line, Paul operates on a hyperbola, and their unique survival points force a 2-adic asymmetry on every solution. 2. The Parity Invariant Law (Lemma 6): Proves that the machine's anchor stage involution σ₁ is strictly parity-preserving. Parity is established not as a heuristic tendency, but as a global, unconditioned invariant of the machine's orbit structure. 3. Decoding the Outliers (Section 7): Identifies and structurally justifies the only two anomalous entries within the 18 solutions: (404, 503), which carries a hidden odd prime (404 = 2² ⋅ 101), and (16, 111), which utilizes a composite odd component (111 = 3 ⋅ 37). These outliers expose a finer, rich combinatorial layer embedded within the higher-order orbit hypercubes \(Q_{k-1}\). 🌐 Central Conjecture & Open Call for Action (GPU / Parallel Computing) The paper condenses these computational and structural observations into a rigorous multi-part framework (Conjecture 7), positing that: The ν₂ ≥ 2 parity asymmetry holds unconditionally for all space. The share of pure-pattern solutions (where the even component is a pure power of 2 and the odd component is prime) tends asymptotically to 1. The absolute density of solutions vanishes completely as HI → ∞. The Computational Ceiling Faced: Testing this boundary past HI = 99,999 scales at \(\mathcal{O}(HI^2 \log HI)\), requiring days at HI = 10⁶ and months at HI = 10⁸ on single-core setups. The text concludes with an explicit Call for Action, inviting researchers with parallelized architectures or GPU implementations to extend the enumeration to HI = 10⁶, 10⁷, 10⁸ and test the limits of the machine's structural bounds. File Content: Full peer-review ready PDF containing complete geometric translations of logic statements, the full 18-pair prime-factorization matrix up to HI = 99,999, exact 2-adic valuation distribution tables, and the open-source reference parameters for the verification architecture. Keywords: Sum-and-Product puzzle, Impossible Puzzle, Freudenthal, prime machine, orbit minimum, 2-adic valuation, hyperbola intersection, logic games, parallel computing, Goldbach conjecture, Sophie-Germain primes.
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164880215
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