What does this research mean for the field?

For every integer greater than 1, a finite initial family of polynomials can universally reconstruct the entire ideal lattice of the residue ring Z modulo n, and this complete global reconstruction is equivalent to finite local separation conditions. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine whether the ideal lattice of a residue ring modulo n can be fully reconstructed using dynamic collisions generated by polynomial orbits.

August 29, 2026Open Access

Remanent Tomography of Multiplicative Structures by Dynamic Collisions: Universal Reconstruction and Local–Global Separation

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To determine whether the ideal lattice of a residue ring modulo n can be fully reconstructed using dynamic collisions generated by polynomial orbits.
Analyzed critical orbits generated by the quadratic transport family Fc(x) = x^2 + c modulo n.
Constructed collision ideals from differences between orbit terms and closed them under addition and intersection to generate sublattices.
Proved a universal reconstruction theorem showing a finite initial family of polynomials reconstructs the entire ideal lattice for every integer n > 1.
Identified admissible collisions between initial points and first iterates that directly yield prime-power divisors to generate all divisor ideals.
Established a local–global separation theorem demonstrating that complete reconstruction is equivalent to local oriented collision witnesses separating non-forced levels.

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Implication

Theoretical analysis demonstrates universal reconstruction of ideal lattices in residue rings modulo n, showing global lattice recovery reduces to finite local separation conditions.

Cite This Study

Sylvain Gefffroy (2026) studied this question.

synapsesocial.com/papers/6a9299ac8e5d7d1fc0c11b61 https://doi.org/10.5281/zenodo.22133200

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Implication

Theoretical analysis demonstrates universal reconstruction of ideal lattices in residue rings modulo n, showing global lattice recovery reduces to finite local separation conditions.

Remanent Tomography of Multiplicative Structures by Dynamic Collisions: Universal Reconstruction and Local–Global Separation

Key Points

Discussion

Implication

Cite This Study

Remanent Tomography of Multiplicative Structures by Dynamic Collisions: Universal Reconstruction and Local–Global Separation

Key Points

Discussion

Implication

Cite This Study