FOOO filtered curved A-infinity identity theorem

F2 constructs the actual finite rational Morse operations on A and their compact oriented one-dimensional boundary. The raw disk-bubble and zero-edge seams are sewn before perturbing their matched thickening; only genuine Morse breaks over discrete generators are prescribed boundary products. Their signed finite boundary gives every curved identity on A, including positive-energy zero-input components, with the suspended cohomological insertion sign. This geometric input is not deduced from the conditional transfer lemma.

In the actual smooth passage compactification $U_p$ of the unstable cell, each boundary face is the product of compactified trajectories from p to q with $U_q$. Triangulate trajectory spaces in increasing index drop and $U_p$ in increasing index, prescribing ordered shuffle product triangulations on every face; these agree at corners because multi-factor monotone shuffles are associative. Under endpoint evaluation every positive-dimensional trajectory factor projects a shuffle simplex to a simplex with repeated consecutive vertices. It is therefore a literal singular degeneracy. Modulo degeneracies only index-drop-one faces remain, with the signed Morse coefficient. This gives an evaluated chain map from A to normalized singular chains. The explicit Moore normalization section, given in degree r by $P_r=(1-s_0{d}_1)\mathrm{.}\mathrm{.}\mathrm{.}(1-s_{(r-1)}{d}_r)$ with rightmost factor first, lifts it to a chain map u:A→S for the original unnormalized differential. Each generator has a nondegenerate interior simplex in its own disjoint unstable cell, proving injectivity. Its relative class on each skeletal pair is the oriented cellular generator, so the actual Morse/cellular/singular comparison proves u is a quasi-isomorphism. This is a geometric inclusion, not merely an arbitrary homology identification.

The quotient $Q=S\mathrm{/}u(A)$ is acyclic. Choose rational splittings and the inverse-differential contraction $h_Q$ with $d_Q{h}_Q+h_Q{d}_Q=1$. For a degreewise rational section σ of π:S→Q set $K=d_S\sigma -\sigma d_Q$ and ${\sigma }^{\mathrm{\prime }}=\sigma +K{h}_Q$. Since K takes values in u(A) and $d_{S}K+K{d}_Q=0$, the displayed contraction gives $d_S{\sigma }^{\mathrm{\prime }}={\sigma }^{\mathrm{\prime }}{d}_Q$ and πσ'=1. Define p by up=1−σ'π. Then pu=1, p is a chain map, and $S=u(A)\oplus N$ with $N={\sigma }^{\mathrm{\prime }}(Q)$ an acyclic chain subcomplex. Every map is rational and extends coefficientwise preserving energy; its value on any individual continuous simplex is a finite rational chain.

On the completed suspended S set $M_0=u$ $m_0^A$, $M_1=d_S+u(m_1^A-d_A)p$ and $M_k=u$ $m_k^A$ $p^{\otimes }k$ for k≥2, with the common suspension signs. These are exactly the geometric A operations on u(A), and only the differential acts on N. In particular $M_{(1,0)}=d_S$ on the entire original continuous-chain complex, and $M_0$ retains positive valuation. The strict maps $U_1=u$ and $P_1=p$, all other Taylor components including $U_0,P_0$ zero, satisfy every curved morphism equation by these formulas and pu=1; their linear terms are quasi-isomorphisms. Thus this specified full-chain extension is explicitly A-infinity equivalent to the geometric evaluated Morse model of the Definition.

For an input word in u(A), every nested product equals u times its geometric A product with the same signs. Its sum is zero by the actual A identity. For a word with an N input, an inner non-differential operation either contains it and vanishes, or leaves it for the outer non-differential operation, which then vanishes. Differentiating it keeps it in N. The remaining unary term is $d_S^2=0$. A curvature insertion never deletes an original N input; the empty-word identity is the geometric A identity lifted by u. Hence all curved identities hold on every full continuous-chain input.

At each arity and energy cutoff the actual A coefficients are finite rational counts. Each p input is a finite combination of its finite generator set; u,p preserve energy, and only finitely many coefficients of a Novikov-series input lie below the cutoff. Thus every $M_k$ is well-defined, continuous and filtration preserving. On an algebraic bar word of length n there are finitely many insertion places and output lengths at most n+1, so the preceding identities give D squared equal to zero on the declared algebraic direct sum. Neither strict comparison has a zero-input component, so these particular comparison maps also preserve that direct sum. No step sets m0 equal to zero. This establishes the original associated-model Statement for this specified geometric full-chain comparison.