A smooth slice knot has a knot Floer complex stably equivalent to the unknot

Given: A smooth concordance from $K$ to the unknot, the accepted normalized knot complex over $\mathrm{\Lambda }={\mathbb{F}}_2[U,U^{-1}]$, and ACω.

Smooth concordance is represented by the actual annulus in the Statement. Smooth knot concordance

The accepted normalized knot/grid and analytic comparisons identify the finite homogeneous bifiltered complex, total generator and both grading origins. Grid homology is independent of the grid and computes knot Floer homology, Regularity, compactness, and gluing for pointed Heegaard disks

External Floer input under CLAUDE.md §12: a smoothly slice knot has ${\mathrm{C}\mathrm{F}\mathrm{K}}^{\mathrm{\infty }}(K)\simeq {\mathrm{C}\mathrm{F}\mathrm{K}}^{\mathrm{\infty }}(U)\oplus D$ as a graded bifiltered Laurent chain complex, with $D$ finite-rank free and totally acyclic. The finite homogeneous filtered-basis splitting and the concordance-to-slice application are Hom, A survey on Heegaard Floer homology and concordance, Theorem1, Proposition3.11 and its proof, and proof of Theorem1, printed1–2,12–14. Coefficients are ${\mathbb{F}}_2$; acyclicity concerns total homology and not associated gradeds. The unknot factor has the usual Maslov-zero and $(0,0)$ origin. This is an external established input, not a completed local decorated-annulus/movie construction. The cited statements and finite splitting proof on printed1–2,12–14 were read; no complete reading of the survey or its upstream surgery references is claimed.

Choice scope of the external input: use it for the fixed knot diagram and finite smooth concordance/surface under consideration, not for a simultaneous choice of data for every knot. Integer domains and Spin-c labels form countable sets; all coefficient matrices and final graded basis changes are finite. Compact source charts give separable Sobolev and almost-complex parameter spaces with countable bases. Their fixed Fredholm problems admit a countable family of finite-cokernel perturbation charts, selected using ACω; the usual finite-dimensional Sard reduction and least-basis nested-ball Baire construction select one datum regular for the countable list. Compactness/gluing is used pointwise for these data, with finite covers of the compact low-index moduli and least-integer shrinking thresholds, not an arbitrary family of choices. Thus the imported fixed-data construction has a finite/countable specialization; full AC is not substituted for the declared ACω branch. The cited papers are not claimed to contain an explicit foundational choice audit.