What does this research mean for the field?

The difficulty in lifting terminal cycles for quadratic maps arises specifically from a codimension-one restriction where requiring the marked point to be the critical point collapses an explicit q-point family into finite Frobenius-controlled algebraic data. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To evaluate whether lifting obstructions for terminal cycles in quadratic families disappear when the marked point is decoupled from the critical point.

September 2, 2026Open Access

Freeing the marked point: A codimension-one localisation for terminal cycles of marked quadratic maps.

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Authors

SGSylvain Gefffroy

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Implication

Theoretical analysis reveals that freeing marked points resolves cycle obstructions in quadratic maps, highlighting how critical constraints collapse rational curves into rigid algebraic sets.

Key Points

To evaluate whether lifting obstructions for terminal cycles in quadratic families disappear when the marked point is decoupled from the critical point.
Reparametrized the quadratic family x² + c over finite fields by placing the marked point directly on a fixed point to render the multiplier a free coordinate.
Constructed a one-dimensional affine return map on the fiber above the cycle and derived a signed landing gate polynomial of degree 2ⁿ⁻¹ − 1 at depth n to monitor critical orbit arrival.
Freeing the marked point yielded closed-form witnesses without even terminal period factors for every odd prime, forming a rational curve with q points across the finite field.
Restricting the marked point to the critical line collapsed the continuous rational curve into a finite algebraic set governed by Frobenius action.
Demonstrated that expanding the search to the general three-parameter family x² + ax + b with marked point 0 is an affine change of coordinates that encounters the exact same codimension-one obstruction.

Cite This Study

Sylvain Gefffroy (2026) studied this question.

synapsesocial.com/papers/6a97e318c562ede874ec7b91 https://doi.org/10.5281/zenodo.22212275

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Implication

Theoretical analysis reveals that freeing marked points resolves cycle obstructions in quadratic maps, highlighting how critical constraints collapse rational curves into rigid algebraic sets.

Freeing the marked point: A codimension-one localisation for terminal cycles of marked quadratic maps.

Discussion

Member takes

Implication

Key Points

Cite This Study

Freeing the marked point: A codimension-one localisation for terminal cycles of marked quadratic maps.

Discussion

Member takes

Implication

Key Points

Cite This Study

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