A structured audit of what HF=HM preserves

Definitions and mathematical statement

The item labeled Question 1.41 only records that Yi-Jen Lee discussed relating Ozsvath--Szabo and Seiberg--Witten theory. It contains no mathematical assertion or recoverable question. The aid identifies the later Kutluhan--Lee--Taubes $HF=HM$ series as relevant context, not as the answer to an invented precise workshop conjecture.

Its additional proposition is an abstract compatibility test. Over a fixed coefficient field $\mathbb k$, suppose two matched three-flavor exact sequences have $\mathbb k[U]$-module isomorphisms commuting with every exact-sequence map. Write the relevant portion as \[ A^\infty\xrightarrow{i_A}A^+\longrightarrow A^-,\qquad B^\infty\xrightarrow{i_B}B^+\longrightarrow B^-. \] Define the tower images and reduced quotients by \[ T_A=\operatorname{im}i_A,\quad A_{\mathrm{red}}=A^+/T_A, \qquad T_B=\operatorname{im}i_B,\quad B_{\mathrm{red}}=B^+/T_B. \] The report states that the commuting isomorphism carries $T_A$ onto $T_B$, induces an isomorphism of reduced quotients, and preserves the specified $U$-power filtrations, for example $U^kA^+$ and $U^kB^+$. In particular reduced-group vanishing is equivalent. When the usual rational-homology-sphere and coefficient hypotheses identifying such vanishing with the L-space condition apply, that condition is transported as well.

The source emphasizes limits of this formal data. A shifted tower has the same ungraded $U$-module but can have a different absolute correction level. A module isomorphism without distinguished-element compatibility need not identify marked contact elements. Objectwise isomorphisms of theories do not by themselves commute with cobordism maps or give a natural transformation. Absolute grading, marked classes and cobordism naturality therefore require additional explicit compatibility. These logical countermodels do not assert that the actual $HF=HM$ constructions fail those stronger properties. The proposition is contextual synthesis rather than a new contact-geometric question or a proof of the Floer comparison itself.

Short English statement

The source is only a discussion note. The aid adds an abstract test showing which invariants follow from an exact-sequence-compatible $U$-module comparison, and which require grading, marked-element or naturality data. It does not recover a missing workshop question or limit the actual comparison theorem beyond the stated formal hypotheses.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[436], item 1.41. Title: "A structured audit of what HF=HM preserves."
• AIM workshop list Holomorphic curves in contact geometry, supplied reference https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. Original attribution: Yi-Jen Lee's discussion of Ozsvath--Szabo and Seiberg--Witten theory. This draft uses the supplied record without accessing a PDF.
• Supplied research summary refers to the Kutluhan--Lee--Taubes $HF=HM$ series without full bibliographic entries. Attempt 1 supplies the compatibility proposition and shifted-tower, marked-module and objectwise-functor countermodels, whose detailed examples are not reproduced.
• Registry dated 14 August 2026: status "open," classification "OPEN-TRIAGE," statement status "exact"; attempt context only, research-run status not a problem, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.