A mathematics preprint proposes a cohomology-sensitive refinement of Hamiltonian Floer persistence, which organizes filtered Floer data as an action threshold changes. In a constructed genus-two example, sufficiently small normalized autonomous Hamiltonians have identical ordinary Floer barcodes and all Schwarz selectors, yet their principal cap barcodes differ by a positive amount and their Hamiltonian-conjugacy orbits are separated by at least the same amount.
This is a theorem-based methods paper, not an experiment. Wenmin Gong’s version-one arXiv preprint is dated 20 August 2026; it studies filtered Hamiltonian Floer modules and constructed mathematical examples, with no sampled participants or empirical observations.
The information ordinary barcodes leave out
An ordinary barcode is a compact record of how filtered pieces persist across action levels. The paper augments that record with the effects of cohomology cap actions. For a homogeneous ideal J, a depth-ℓ cap layer gathers outputs from every length-ℓ word of these actions generated by J. The full flag uses the positive-degree ideal, while a principal depth-one ideal recovers a single cap-operator image.
The resulting displacement quantity depends on the external symplectic map only through its induced graded-ring action on cohomology. It is unchanged by Hamiltonian conjugacy and remains constant across symplectic isotopy classes.
A stable comparison
To compare two Hamiltonians, the paper uses filtered continuation maps that can be chosen to respect both the cap actions and the persistence maps. In practical terms, the two filtered systems are δ(H,K)-interleaved: they can be matched after a controlled shift set by the continuation distance.
For normalized Hamiltonians, the persistent cap-length and cap-profile barcode distances are stable under continuation and no larger than the Hofer norm of the Hamiltonian change. These are theorem-level bounds, not statistical uncertainty intervals.
The cap profile also lower-bounds the Hofer pseudodistance between Hamiltonian-conjugacy orbits. A positive profile therefore certifies a minimum separation, but not the exact distance.
The genus-two test
In the constructed genus-two case, the principal cap-barcode distance is written τ c0 > 0 for sufficiently small τ, and the orbit distance is at least τ c0. The result does not provide a numerical value for c0.
Because the ordinary barcodes and all Schwarz selectors are identical for this pair, the cap profile supplies a distinction those summaries do not. The paper presents this as a constructed example, not as evidence that ordinary barcodes or Schwarz selectors fail in every setting.
A profile with several depths
The preprint also defines selectors at successive cap depths. For normalized non-degenerate Hamiltonians, they form a monotone spectral hierarchy, with the extremal Schwarz selectors at its endpoints; the depth-k selector is at least the depth-zero selector plus k times the minimum positive action gap.
In a weighted torus Morse model, explicit formulas for the selectors and their widths recover successive contributions from the ordered weights.
A separate cyclic extension adds a multiplicity-sensitive spread. Under the paper’s additional assumptions on α-atoroidality, coefficient-field characteristic and roots of unity, the spread changes by no more than p times the lifted Hofer distance. It vanishes for full p-th powers and supplies a 1/p-scaled lower bound on distance to that set.
Where the method stops
One higher-layer result is explicitly algebraic. In an abstract equivariant graded persistence category, the ordinary cyclic spread and the spread of every single-operator image are zero, while depth-two ideal decoration is strictly richer.
That example has not been realized as a concrete filtered Hamiltonian Floer persistence module, so it does not yet provide a stronger Hofer-geometric obstruction. The geometric results remain within a closed, connected symplectically aspherical setting over a coefficient field, with additional assumptions for the cyclic extension.
Taken together, the paper presents cap decoration as a cohomology-sensitive refinement whose clearest geometric result is the genus-two separation: information absent from ordinary Floer barcodes and Schwarz selectors appears in the added cap profile.
Paper data and sources
Original title: On Cap-Decorated Floer Persistence modules
Authors: Wenmin Gong
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text