What does this research mean for the field?

Bounded-depth collision data in polynomial dynamical families is arithmetically too sparse to certify universal modular statements, revealing a fundamental depth-degree obstruction. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate how much arithmetic information can be reconstructed uniformly across all primes from finite-depth dynamical collisions in marked polynomial families.

September 2, 2026Open Access

A depth obstruction for terminal cycles of marked polynomial families over finite fields.

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Authors

SGSylvain Gefffroy

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Implication

Theoretical analysis reveals depth obstructions in marked polynomial families, indicating fundamental limits when reconstructing arithmetic information from collision traces.

Key Points

Investigate how much arithmetic information can be reconstructed uniformly across all primes from finite-depth dynamical collisions in marked polynomial families.
Analyzed the average number of modular roots in collision portraits by identifying Frobenius fixed points using the Chebotarev density theorem and Burnside's lemma.
Determined Galois group cycle structures and derangements for period-3 packets via modular factorizations, Jordan's theorem, and discriminant field comparisons.
Demonstrated that an irreducible portrait packet contributes an average of only one root modulo primes despite exponential raw algebraic degree growth, showing severe arithmetic sparsity.
Established full symmetric Galois groups of S₆ and S₁₂ for two period-3 quadratic family packets, yielding an exact simultaneous failure set density of approximately 13.54%.
Identified a depth–degree obstruction wherein dynamical depth naturally scales as a square root while algebraic tools restrict control to a logarithmic window.

Cite This Study

Sylvain Gefffroy (2026) studied this question.

synapsesocial.com/papers/6a97e2d9c562ede874ec73d6 https://doi.org/10.5281/zenodo.22210186

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Implication

Theoretical analysis reveals depth obstructions in marked polynomial families, indicating fundamental limits when reconstructing arithmetic information from collision traces.

A depth obstruction for terminal cycles of marked polynomial families over finite fields.

Discussion

Member takes

Implication

Key Points

Cite This Study

A depth obstruction for terminal cycles of marked polynomial families over finite fields.

Discussion

Member takes

Implication

Key Points

Cite This Study

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