Rational collared disk virtual correspondences and curved Morse operations

Fix a cutoff using $B(\beta ,k)=k+floor(\omega (\beta )\mathrm{/}h)$, where h is the actual component energy bound. Every contributing type has B≥1: zero-energy zero-input stable disks do not exist, and the no-disk unary zero-energy type is the supplied Morse differential. A boundary cut has $k_1+k_2=k+1$ and $E_1+E_2=E$, so $B_1+B_2\le B+1$ and each nonempty factor has $B_i\le B$. In particular positive curvature compensates for its companion's possible arity k+1. Fixed B≤N bounds energy and arity; the actual compactness, relative-class and stable-flag bounds give finitely many classes and combinatorial tree types. No discreteness of all symplectic periods is assumed.

Construct the tree THICKENINGS before choosing multisections. At a nodal disk face, match the two node evaluations on the component thickenings; their joint derivative is onto by the supported evaluation augmentation. This is a smooth fiber product with direct-sum obstruction space, not a product of perturbed zero sets. A finite gradient edge imposes $e{v}_{+}={\mathrm{\Phi }}_l(e{v}_{-})$; its finite-window and endpoint charts are the actual Morse matching charts. At $l=0$ this is the same nodal thickening. Include all its finite tuple choices and the actual inherited coordinate changes.

Sew a disk-smoothing half-collar to its zero-gradient half-collar before perturbation. On the latter set $l(t)=\exp (-\exp (1\mathrm{/}{t}^2))$, $t>0$, and $l(0)=0$. Its implicit normal correction is smooth in l, zero at $l=0$, and has bounded fixed derivatives on a compact smaller core. Every derivative after this substitution is bounded by exp(−exp(1/t²)) times fixed powers of exp(1/t²) and 1/t, hence vanishes faster than any power of t. The analytic disk side has the same infinite face jet by F1's actual common profile. Their section is the same obstruction projection, and their evaluations have the same face value plus flat corrections. Taylor's formula on the two halves proves smoothness in a signed coordinate τ. For r seams use the product signed coordinates; all mixed derivatives agree. Thus these faces are INTERNAL hypersurfaces, and their perturbations may be chosen jointly on the whole matched chart. They are not frozen products of previously counted standalone disks.

The genuine real boundary is an infinite Morse break. Its intermediate label is one of the finitely many critical generators. Its constrained charts are products over that DISCRETE label of lower pearl charts and, when present, the actual pure-Morse trajectory charts. The metric-end matching and hyperbolic-passage estimates supply their independent neck coordinates, all derivative bounds and corners; retain every broken history rather than collapsing several necks to one nonsmooth reciprocal of a sum. Interior sphere-node parameters are two real interior directions. These, the stable disk compactification and no-escape estimate exhaust the ends.

Order each finite budget lexicographically by actual energy E, arity k and then virtual dimension D of the fully Morse-constrained space. At an infinity split, both positive-energy factors have lower energy. If one factor has zero energy and is not the pure unary Morse type, its stable arity is at least two; the same-energy companion then has smaller arity. A zero-input factor has positive energy. The remaining same-energy/same-arity case has a supplied pure-Morse factor; its other factor has $D_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}=D-1-\dim (Morse)<D$ when nonempty. Therefore every required factor is already constructed. This ordering also covers negative dimensions; one must not reason only from $D_1+D_2=D-1$ when both could be negative. The finite set has a least dimension for each earlier type. Generic transverse negative-dimensional spaces are empty.

Make finite precompact reductions whose intersecting closures have comparable labels. A partition of unity on the compact zero space maps to its finite nerve. Pull back truncated open stars of vertices of its barycentric subdivision, which are nonempty label subsets. Intersecting stars belong to one barycentric simplex and hence give a chain of subsets. A common sufficiently small positive coordinate threshold still covers; finite further shrinking retains compact smaller supports and positive separation margins. The simultaneous tuple charts are the actual union-of-obstruction/auxiliary charts from F1.

Here is the relative tube extension used for these reductions. A prescribed retraction ${\pi }_0$ onto a neat subchart S gives a normal-quotient splitting with image $\ker (d{\pi }_0)$ along S. Extend those vectors locally as fields vertical for ${\pi }_0$; away from a closed smaller protected region use ordinary normal splitting fields. A scalar partition of unity preserves the quotient identity and verticality on every protected region, since all active prescribed retractions agree there. Include corner coordinates as preserved submersions, so the fields keep each face; use their product prescription on closed smaller collar strips. Average fields over the actual finite group. The time-one flow E(s,v) has derivative $(a,v)\mapsto a+j_s(v)$ at zero, an isomorphism. Finite compact shrinking makes it injective on a variable-radius tube. On the protected region ${\pi }_0(E(s,v))=s$, so the inverse tube's retraction equals ${\pi }_0$ EXACTLY; no averaging of manifold-valued retractions occurs.

Construct retractions in increasing CHAIN LENGTH, not by fixing arbitrary long tubes. For $I<K$, first prescribe ${\pi }_{IK}={\pi }_{IJ}{\pi }_{JK}$ on each compact region with intermediate $I<J<K$, where both shorter pairs are already fixed. If J and H are simultaneously active they are comparable; shorter triple identities give ${\pi }_{IJ}{\pi }_{JK}={\pi }_{IJ}{\pi }_{JH}{\pi }_{HK}={\pi }_{IH}{\pi }_{HK}$. Incomparable reduced closures have disjoint protected neighborhoods. Thus the composites form one prescribed retraction near their union, and step 7.1 extends it over the remaining overlap. Retain the product retractions of already constructed infinity faces throughout. Their corner products are the same maps over the fully broken tree, so agree with these composites. The actual $E_I={\oplus }_{i\in I}{E}_i$ give fixed direct-summand transports $B_{IJ}$ with $B_{JK}$ $B_{IJ}=B_{IK}$. Finite chain-length induction and finite shrinkings supply exact open-neighborhood triple identities and all protected face products.

A finite rational multisection is a pointwise finite probability distribution locally lifted by smooth sections; lifts need not differ by locally constant permutations. Extend a smaller distribution pointwise by $B_{IJ}\ast {\nu }_I({\pi }_{IJ}(y))$, zero in the extra obstruction summand. In normal coordinates $s_J(x,z)=(a(x,z),z)$, $a(x,0)=s_I(x)$, its zero equations reduce to $z=0$ and $s_I={\nu }_I$. The stronger equality $a=s_I\pi$ is unnecessary. The fixed control identities make these extensions agree on open overlaps and corners without pairing independently extended lifts. A distribution already defined on an open protected neighborhood can be multiplied by one smooth scalar cutoff, equal to one on a smaller neighborhood, and extended by the zero distribution off its support. Existing local lifts verify smoothness; there is no fictitious global branch labeling.

The protected infinity-face distributions are the products of the already constructed FULLY CONSTRAINED factors. Their derivative is block diagonal, because the common label is discrete, so it is onto. Empty negative-dimensional factors give zero-free products on smaller compact neighborhoods. On multiple infinity faces the prescriptions are the same tensor products in different parenthesizations, with the actual graded determinant permutation. Hence their open collar germs agree. This establishes the joint protected transversality needed for extension; transverse standalone disk sections and submersive unperturbed evaluations alone would not have established it. The internal seams in step 3.1 instead receive jointly chosen whole-tree distributions.

On the unprotected part take finite local cutoffs ${\chi }_a$ and bundle frames. For each chart group G use independent vectors $p_{a,g}$ in the section $T_a={\mathrm{\Sigma }}_g$ $g\cdot ({\chi }_a{p}_{a,g})$, then its whole orbit distribution ${h\cdot T_a}_h$. Every selected orbit branch has an active parameter onto E throughout the orbit-covered region. Convolve these finite distributions with the existing one using product rational weights. For remaining endpoint/Morse incidence constraints, the actual finite-window vector fields have arbitrary supported endpoint value; the joint evaluation augmentation and actual stable/unstable variables supply the corresponding constraint-normal directions. These parameters are independent of the section parameters. After prescribing a constraint-normal variation, section parameters correct its E variation. Thus the UNIVERSAL map (parameters,x)↦(s(x)−ν(x),H(x)) is transverse to zero×constraint on each stratum. This does not assert that the evaluation on a fixed zero branch is submersive.

The Sard–Smale assertion in [F5], applied to the projection of each universal constrained zero set to its finite parameter space, gives fixed parameters transverse for every branch, face and matching constraint. There are finitely many local presentations and strata near the compact controlled zeros; a countable patch exhaustion of their interiors adds only a countable union of meagre exceptional sets. Their complement meets every small parameter ball by the Baire assertion in [F5]. On protected neighborhoods the previously onto derivative persists by compactness and openness; all varying parameters vanish on smaller protected strips. Choose them below the positive separation margins, so zeros remain in the controlled interiors. Apply this finite relative procedure in chart order and then the order of step 5.1. Every fixed type stops changing on increasing budgets, giving one coherent system rather than a sequence of unrelated choices.

The constrained local zero branches are smooth manifolds with corners, not necessarily a globally smooth weighted union. Their controlled realization is compact Hausdorff: the actual extra-forcing-zero and recovered-map/tuple equalities give closed coordinate-change graphs on closed smaller supports; smallness keeps their limits inside the protected common chart. Any identifying zigzag reduces by comparable labels and exact tuple cocycles to a bounded chain and a finite group element. The relation is a finite union of those closed graphs, hence closed. Its quotient of the finite compact controlled presentations is Hausdorff. This is a controlled tuple-atlas argument, not a properness assumption on an arbitrary unrestricted atlas.

For an isolated zero sum the probability masses of all local branches attaining the section value, with their orientation signs, and divide by the actual point stabilizer. Equivalently sum every lift and divide by its chart group order. A forgotten-label cover has its actual free relative tuple-group number of lifts; that factor exactly compensates the ratio of group orders. Normal cancellation preserves branch mass and determinant sign. Hence the mass is independent of which comparable chart represents the point. Duplication and pointwise re-presentation preserve it. A compact transverse zero-dimensional system is locally finite and therefore finite. It gives a genuine finite rational zero-chain under the output evaluation, with each quotient point counted once.

For a one-dimensional system construct a compatible finite distribution of scalar height functions as bookkeeping, with total mass one. On predecessor tubes pull it back by the fixed retractions. Elsewhere use cutoff scalar polynomials through degree two with independently parameterized translates inside each branch and then the full group orbit, as in step 11.1. On every smooth zero curve these parameters span its first and second jets. The same parameter-projection argument using [F5] for its first derivative makes every critical point nondegenerate and keeps the derivative nonzero at boundary points. This test is independent of local lift choices: the closed agreement loci for the finitely many possible pointwise selections cover the domain; their interiors have dense union by the finite Baire property. Approaching a point through one such open agreement set shows that every fixed-order jet of a new lift equals one of the old jets. There are finitely many critical and boundary points by local isolation and compactness.

At a regular height c intersect the weighted zero curves with height $=c$ and use step 14.1 with the intersection sign of d(height). The rational count C(c) is locally constant away from the finitely many critical and boundary values. At an interior Morse critical point the local formula c±u² creates two points with opposite derivative signs and the SAME weight, so contributes no jump. A boundary half-arc contributes exactly its outward boundary mass. Chart covers preserve these masses by step 14.1. C is zero below and above its compact height range, so the sum of all jumps is zero. This proves the exact weighted one-dimensional boundary identity without calling the weighted union a manifold.

When an actual one-chain is needed, cut the bounded height range at its finitely many critical/boundary values and additional regular cuts. Over each compact regular interval, the finite local curve presentations are graphs over height. After clearing branch and group denominators, their unordered point distributions agree on overlaps. For each possible permutation the tuple-agreement locus is closed and these finitely many loci cover the interval. Their interiors have dense union. Choose each artificial cut in one such interior, where graph segments agree on an OPEN neighborhood after that permutation and duplication. Compactness permits finitely many such cuts. Add the actual local critical arcs and boundary half-arcs. The resulting finite rational path chains have cancelling artificial endpoints and boundary exactly the original oriented boundary zero-chain. Push them by the output evaluation. This uses no unique global branch covering and declares no arbitrary lower-rank singular simplex zero.

For fixed ordered Morse generators count the rigid systems by step14.1. A one-dimensional system has only genuine infinity products as boundary: its disk/zero-edge seams have opposite outward normals and identical branch masses and cancel in the signed family constructed in2.1–3.1; sphere strata have codimension two and are empty after transversality. Apply the orientation supplier's explicit incidence-line comparison [F1, proof 7.2–10.1] to THIS actual family. A finite edge has normal equation $e{v}_{+}-{\mathrm{\Phi }}_T(e{v}_{-})$, while4.1 supplies its actual infinite critical-label product and10.1 supplies its fully constrained product sections. The full constraint-normal comparison retains the unstable-normal determinant and every external normal block. Use exactly its fixed arity/virtual-dimension normalization $f(k,d)=k(k-1)\mathrm{/}2+kd+k-1$ and critical-line ray comparison $h(q)=q(q-1)\mathrm{/}2$ on this model; these formulas are chosen on whole charts, not separately on faces. Its explicit full-matrix calculation gives the raw exponent r+st+1+(2−s)Q, including the ordinary fiber-product/quotient permutations. It does not identify the diagonal with a stable/unstable product as a set. Its outward normal is increasing T; transport through the internal seam retains the supplier's one common outward minus. The explicit virtual marking/automorphism calculation there covers unary and nullary disk presentations, and its pure-flow calibration gives the actual Morse differential. With the supplier's fixed ${\beta }_k$ suspension convention, its comparison [F1, proof 10.1–12.1] therefore gives exactly (−1) raised to the sum of preceding suspended degrees $q_j-1$, including all curvature insertions. No independent choice of face sign is made here. Cover degrees and rational branch weights multiply on the discrete-generator products. Thus step16.1 says every coefficient of the curved nested-operation sum vanishes. These are actual finite rational geometric operations on A; neither the A-infinity identity nor a diagonal/Morse chain comparison was assumed to obtain the boundary sign.

Assign each tree its global free relative disk class $\beta$ by the actual rooted collapse and gradient paths of [F1]; its smoothing preserves that class, including its integral homology image with torsion. Put $E=\omega (\beta )$. The double convention [F6] makes $\mu$ homotopy invariant and additive under grafting: determinant clutching loops concatenate, and a sphere attachment contributes $2c_1$ of its tangent pullback. For completeness, trivialize the complex pullback over a smooth disk representative. The oriented boundary bundle admits a periodic real frame, whose complex determinant is a loop $f:S^1\to {\mathbb{C}}^{\ast }$. The determinant clutching of its double is $f\mathrm{/}\bar{f}$, so $\mu =2\mathrm{wind}(f)$ is even; changing the real oriented frame has positive determinant and changing the disk trivialization has zero boundary winding. The disk index $n+\mu$, its $k+1$ marks and three-dimensional automorphism quotient give dimension $n+\mu +k-2$. Each internal gradient edge adds one length variable and $n$ matching equations; summing over a tree gives the same total dimension. For inputs of cohomological degrees $q_i=n-\mathrm{index}(p_i)$ and output degree $q_0$, the constrained dimension is $\mu +k-2+q_0-{\sum }_i{q}_i$. Rigidity therefore gives $q_0={\sum }_i{q}_i+2-k-\mu$, hence bare suspended degree $1-\mu$. Weight the count by $T^E{e}^{\mu \mathrm{/}2}$ in the explicitly defined ring ${\mathrm{\Lambda }}_{\mathbb{Q},e}^0$: its degree is $\mu$, making every $m_k$ degree one. Additivity of area and $\mu$ gives the same weight on every boundary splitting, so the classwise identities of step 18.1 remain $D^2=0$ with unchanged signs, since $e$ is even. Below any energy cutoff only finitely many contributing classes occur by [F1], and each rational count is finite, so all sums and convolutions are well-defined and filtration preserving. The no-disk unary zero-energy type has $\mu =0$ and is exactly $d_A$. No stable zero-energy zero-input disk exists, so $m_{0,0}=0$, and curvature has energy at least $h$. Finally $e\mapsto 1$ is a continuous ring map: its sum over exponents of $e$ is finite below each energy cutoff. It preserves parity, every count and every identity, giving the stated energy-only version. These constructions supply the genuine geometric model for the following coefficientwise chain comparisons.

Cohomological grading of the external comparison. Put $n=\dim L$. Before coefficient extension use the rational complexes $A_{\mathbb{Q}}^q={⨁}_{\mathrm{ind}(p)=n-q}\mathbb{Q}p$ and $S_{\mathbb{Q}}^q=C_{n-q}(L;\mathbb{Q})$. Take $d_A$ to be the given signed Morse boundary and $d_S=\mathrm{\partial }$ to be the ORIGINAL singular boundary, with no extra sign. Each lowers homological degree by one, hence raises this cohomological degree q by one. Regrade the degeneracy subcomplex and normalized singular quotient in exactly the same way. In the following rational chain comparison A and S mean these regraded complexes. The top unstable-cell chain for p has homological degree $\mathrm{ind}(p)$, hence cohomological degree $n-\mathrm{ind}(p)$, exactly the degree of p in A. The homological Morse/cellular/singular comparison of [F3] is reindexed on both sides, so it gives $H^q(A)=H_{n-q}(L;\mathbb{Q})$ with the same matrices and orientation signs. After this rational comparison, extend coefficients and complete in energy as in the Statement; a monomial $T^{\lambda }{e}^j$ adds degree $2j$ on both sides. Construct the external HALF-INFINITE edge chain comparison on actual compactified unstable cells $U_p$. Their free-endpoint charts are obtained by cutting trajectories at the finitely many critical Morse bands and using the supplied hyperbolic passage/end matching charts; retain every broken history and its independent neck. Compactness of L and strict decrease between different critical values bound their number. The endpoint maps are smooth in these resolved corner charts, and their faces are the products of the actual compactified trajectory spaces M(p,q) with $U_q$. The determinant orientation is the supplied Morse orientation. These are smooth passage compactifications, not an assertion that an arbitrary topological disk parametrization is smooth.

Triangulate the trajectory spaces in increasing index drop and then $U_p$ in increasing index, prescribing each boundary face as the ordered SHUFFLE product of its already chosen factor triangulations. Multi-factor shuffle triangulations are associative: their simplices are the same monotone lattice paths after forgetting parentheses, with the same signs. Hence prescriptions agree at every corner, and the actual finite relative triangulation theorem fills the compact interiors. Let $c_p$ be the endpoint-evaluated top chain of $U_p$. On a face $M(p,q)\times U_q$ with positive-dimensional M(p,q), each shuffle simplex projects to its $U_q$ simplex with consecutive repeated vertices. Its pushforward is LITERALLY a singular degeneracy. In normalized singular chains these faces vanish; the zero-dimensional trajectory factors give precisely the signed Morse incidence. Thus $p\mapsto [c_p]$ is an actual chain map from A to normalized singular chains, not a claim that all thin raw singular chains vanish.

Let S be the full unnormalized continuous singular complex, D its degeneracy subcomplex and q:S→S/D. In degree r set $P_r=(1-s_0{d}_1)(1-s_1{d}_2)\cdots (1-s_{r-1}{d}_r)$, with rightmost factor first. The simplicial identities successively give image $P_r={\cap }_{i=1}^r$ ker $d_i$, $P_{r}D=0$, $\mathrm{\partial }{P}_r=P_{r-1}\mathrm{\partial }$ and $P_{r}x-x\in D$. Thus $J([x])=P_{r}x$ is a well-defined chain section of q. The degeneracy filtration by ${\mathrm{\Sigma }}_{i\le a}{s}_{i}S$ is contracted successively by (−1)^a $s_a$ on its successive quotients; starting with the largest a in a cycle and subtracting the corresponding boundary constructs a primitive in finitely many steps. Hence q and J are quasi-isomorphisms. Compose the normalized map of step 21.1 with J to obtain an actual geometric chain map u:A→S for the ORIGINAL singular differential. Here r is the homological simplex dimension and the cohomological degree is n−r; each normalization map preserves r and therefore has cohomological degree zero. Thus u has degree zero. The following degreewise quotient splittings are in this common cohomological grading: d has degree +1, its inverse-differential contraction has degree −1, and u,p have degree zero. Their coefficient extensions and their induced maps on the common suspensions keep these degrees, making every term of the later extension formulas well-typed.

The map u is injective: each unstable cell has a nondegenerate interior top simplex supported in that cell, whereas the same-index other cells are disjoint and their boundaries lie in lower-index cells. Its normalized relative class is the oriented cellular generator and cannot be a degeneracy. The actual Morse/cellular/singular comparison therefore identifies H(u) with its stated natural isomorphism, proving u a quasi-isomorphism. This uses the evaluated external unstable cells and the specified Morse orientations, not an arbitrary abstract identification of isomorphic homology groups.

Here is a rational chain splitting that retains THIS geometric u. The quotient $Q=S\mathrm{/}u(A)$ is acyclic. AC gives its degreewise boundary/complement splitting and a contraction $h_Q$ with $d_Q{h}_Q+h_Q{d}_Q=1$. For a degreewise section σ of π:S→Q put $K=d_S\sigma -\sigma d_Q$; it lands in u(A) and $d_{SK}+K{d}_Q=0$. Then ${\sigma }^{\mathrm{\prime }}=\sigma +K{h}_Q$ is a CHAIN section, since its differential defect is $K-K(d_Q{h}_Q+h_Q{d}_Q)=0$. Define p by $up=1-{\sigma }^{\mathrm{\prime }}\pi$. This gives $pu=1$ and $S=u(A)\oplus N$ as chain complexes, $N={\sigma }^{\mathrm{\prime }}(Q)$ acyclic. Each map is rational and takes an individual simplex to a finite chain. It extends coefficientwise preserving energy and the original $d_S$.

On completed suspended S define $M_0=u$ $m_0^A$, $M_1=d_S+u(m_1^A-d_A)p$ and $M_k=u$ $m_k^A$ $p^{\otimes k}$ for k≥2. On u(A) these are exactly its geometric operations; on N only $d_S$ acts. The strict maps $U_1=u$ and $P_1=p$, all other Taylor components including $U_0,P_0$ zero, satisfy all curved morphism equations directly from these formulas and $pu=1$. Their linear terms are quasi-isomorphisms. This constructs the full continuous-chain comparison model expressly allowed by the Definition: its geometric external-edge map is u, with its explicit normalization and rational retraction. These strict comparisons are not mislabeled as counts of k+1 independent finite external intervals, which would have the wrong parameter degree.

Every nested identity on u(A) is u times the geometric identity with the same signs. With an N input, any non-differential inner operation containing it vanishes; if it misses it, the outer operation still receives it and vanishes. A differential keeps it in N, leaving only $d_S$ squared. Curvature insertion never deletes a previous N input, and the empty-word identity is the lifted geometric curvature identity. Thus all full-chain curved identities hold. Each coefficient uses finitely many critical generators and the actual finite rational counts, and u,p preserve energy. Novikov-series inputs have only finitely many terms below each cutoff. Every $M_k$ is therefore continuous and filtration preserving. An algebraic bar word has finitely many insertion positions and output lengths at most its length plus one, so D squared vanishes on the stated algebraic direct sum, without completing in tensor length.

This is a chosen geometric Morse/evaluation CHAIN-MODEL correspondence, followed by an actual full-singular-chain comparison, as specified in the Definition. Arbitrary continuous simplices are never claimed transverse, and no independent raw simplexwise intersection enhancement is identified with it. No strict unit on all S is asserted: for an acyclic summand input the higher products vanish, so the displayed extension need not be strictly unital. Its induced cohomology product and any supplied cohomological unit are retained. The current Statement and Definition require the curved identities and this chosen comparison model, not strict units or a separate finite-interval realization. Such stronger requirements would need their own construction.

Use the specified finite action on the chosen disk atlas and its Cauchy–Riemann equation, as required by the orientation supplier; this includes its actual action on product collars. If that action also permutes the chosen Morse/edge model and preserves its lifted spin orientation, use full chart/forcing/parameter orbits. The rational weighted zero/one-chain construction then gives the prescribed equivariant counts and boundary identity. To retain equivariance of the external-cell comparison, average its chain map over the finite group. Each averaged generator still has its oriented fundamental relative class in its own unstable cell, hence the map remains injective and a quasi-isomorphism. Average the quotient chain section to obtain an equivariant p; averaging preserves both its chain property and $\pi {\sigma }^{\mathrm{\prime }}=1$. The strict extension formulas are then equivariant. No equivariant triangulation or averaging of a transverse single branch is invoked.

For a noninvariant fixed Morse model no additional strictification is concluded. Throughout, the true protected products were verified on the sewn atlas, actual rational zero/one chains supplied the geometric boundary calculation, and the external unstable-cell/normalization construction supplied the stated comparison. No Pardon functional or real de Rham integration was silently promoted to rational singular chains. The constructed system has all the claimed class, curvature, energy, orientation and specified symmetry properties.