Trefoil and 5_2 knot Floer genus and fibering cases

Given: Full AC, the right-handed trefoil and standard alternating knot $5_2$ with the original integral normalization and classical choice branch declared in the dependencies.

External input. Ozsváth–Szabó, Heegaard Floer homology and alternating knots, Geometry & Topology 7 (2003), Theorem 1.3, printed p.230 (PDF p.6 of arXiv:math/0209149v3): if $K$ is alternating and its normalized symmetric Alexander polynomial is $a_0+{\sum }_{s>0}{a}_s(t^s+t^{-s})$, then $\widehat{HFK}(S^3,K,s;\mathbb{Z})\cong {\mathbb{Z}}^{\mathrm{\mid }{a}_s\mathrm{\mid }},$ supported in Maslov degree $s+\sigma (K)\mathrm{/}2$. This is the INTEGRAL theorem, including freeness; no integral conclusion is inferred from the retained mod-two grid comparison. Its signature convention has left trefoil signature $+2$.

External finite table input. Knot Atlas, pages 3_1 and 5_2, section “Polynomial invariants / Alexander polynomial”, records respectively $t-1+t^{-1}$ and $2t-3+2t^{-1}$. These two finite entries and their standard knot diagrams identify the table knots. The same polynomials result from the standard genus-one disk/band Seifert matrices $V_3=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),V_5=\left(\begin{array}{cc}1& 1\\ 0& 2\end{array}\right)$ (up to simultaneous handedness/sign conventions): $\det (V_3-t{V}_3^T)=1-t+t^2$ and $\det (V_5-t{V}_5^T)=2-3t+2t^2$. Multiplying by $t^{-1}$ gives the stated symmetric polynomials. The TABLE/diagram identification is external input; the two determinant calculations are finite local checks, not an assertion that an abstract matrix by itself identifies a knot.

External input at the exact $S^3$, integral scope. Manolescu, An Introduction to Knot Floer Homology, printed/PDF p.5, Theorem 2.3, gives $g(K)=\max \{s:\widehat{HFK}(K,s;\mathbb{Z})\mathrm{\ne }0\}$ for every knot in $S^3$. Theorem 2.5 on that same page gives that a knot in $S^3$ is fibered if and only if its integral top genus group is $\mathbb{Z}$. These exact specializations of the established Ozsváth–Szabó/Ghiggini/Ni/Juhász results are imported. This does not assert completion of the held local general-$Y$ signed comparison or product-criterion proofs. The source explicitly states these as integral groups.

Actual reading limits: the complete Theorem 1.3 statement and its preceding conventions on printed pp.229–230 of the primary alternating paper; the two named Alexander-polynomial table entries (not all table invariants); the full Theorems 2.3 and 2.5 statements and surrounding geometric-applications text on Manolescu p.5. No reading or local reconstruction of the full alternating theorem proof, Ni proof, or entire survey is claimed. These are explicitly authorized external imports under the run-specific source boundary. The original full-choice branch is retained; no integral full-choice theorem is silently cast as a CC theorem.