Index-grouped descending paths give based framed handle movies

Fixed index cuts and the based spheres. Use the exact weak field family (g_s,X_s) after grouping, with its finite simple value crossings, finite separated same-index trajectory events, and exact birth blocks. On a closed parameter interval missing these events choose smooth regular cuts a_{i−1}(s),a_i(s) immediately below and above every index-i group. They exist because the index groups have a uniform strict gap. Identify the cut levels by the actual regular-band flow/trivialization, retaining their maps. At an isolated birth use the outer two cuts of its combined k,k+1 group; no regular cut through the birth critical point is asserted. For an index-i critical point p choose a sufficiently small local unstable sphere S_p≅S^{i−1} in its fixed chart. Every point of this sphere reaches the lower cut a_{i−1} in finite positive time at a parameter with no equal-index connection. Here is the full obstruction argument. If it did not, descending flow on the compact intervening band has a critical limit or a broken critical limit. The height-curve compactness supplied by the field theorem proves this without any slice-wise Morse–Smale hypothesis. An intervening critical point has index at least i, because all smaller indices lie below the lower cut. The first limiting orbit would therefore be index increasing or same-index. Both are absent at this parameter. There is no vertical-boundary limit: X is the fixed noncritical product field there, and its compact interior trajectory sections stay off that collar. A regular infinite-time limit is excluded by the strict descent bound on compact sets away from critical charts. These exhaust the possibilities, so the hitting time is finite. The union of the starting spheres over a compact event-free parameter interval is compact. If their hitting times were unbounded, the same uniform broken-trajectory extraction would give a forbidden index-increasing or same-index orbit at a limiting parameter. Thus the hitting times have a uniform finite bound. At the lower cut dg_s(X_s) is nonzero, so the parameter-dependent implicit function theorem makes the hitting time and hitting point smooth in the sphere variables and s. The flow differential carries the local normal framing. Shrink the full normal D^{d−i} radius uniformly: compactness of the source and the bounded-time flow imply that the full S^{i−1}×D^{d−i} hits the cut as an embedding. The same uniqueness argument gives disjoint core spheres for distinct index-i points: a common point would have one backward orbit with two different critical limits. There are finitely many compact disjoint core images, so their full normal radii can be chosen positive and pairwise disjoint, smoothly over the parameter interval. These are the actual based framed index-i attaching embeddings on the outgoing boundary of the lower-index stage. They are not globally unstable spheres in the original N across all smaller-index critical points. The lower cut is exactly the boundary after the already constructed smaller-index handles. This distinction is what permits positive-index-drop trajectories and their tangencies.

Product complements and the retained higher attaching data. Attach all index-i handles on that common lower cut, with the displayed full embeddings. The actual local Morse attachment charts identify their outgoing regions. On their complement use the regular descending flow to identify with the upper cut a_i. To justify this complement map, remove small invariant entry/exit blocks around the finitely many critical points. Every remaining trajectory between the cut faces is regular and, after removing smaller block collars, has a uniform strict descent bound. Its finite hitting map and its inverse are smooth. The side collars of the Morse blocks use their explicit product entry/exit maps; patching is by those same flow coordinates. This gives the ordinary handle-correspondence comparison for the whole index band, with corners rounded in the retained seam collars. Same-index connections are absent on the present interval, so no one block ends at another. The comparison varies smoothly by the uniform bounds of step 1.1. Let d_i(s) be this actual outgoing identification from the index-i attached stage to the upper cut. A later framed attaching embedding A on that upper cut is carried by d_i(s)^{-1}∘A to the new stage, and its terminal parametrization is composed with d_i(s) in the corresponding direction. The product complement map is therefore part of the data, not an equality of unparametrized diffeomorphism types. If incoming cut identifications are h_s and outgoing ones k_s, the written band map changes by the actual conjugation k_s^{-1}d_i(s)h_s. All later embeddings are carried by that conjugation. This explicitly controls every higher-index attachment even if its global trajectory meets a lower stable sheet or undergoes a positive-drop tangency. The weak elementary-band supplier already proves that its one-critical-band embedding is smooth; composing the present smooth outgoing comparisons preserves that property. No higher-index attaching sphere is required to avoid every lower-index passage region.

Isolation at the one same-index trajectory. Let s₀ be one of the finite equal-index orbit parameters, with its unique trajectory p_i→q_i. Choose its compact connecting segment between a source sphere near p and an entry face of a small q chart. The generic field supplier gives the exact transverse incidence in that entry face: its (i−1) source-sphere variables plus s span the i normal unstable directions. It also gives a normalized linear q model near s₀, away from every actual fold. The exact hyperbolic passage calculation then applies. The other index-i core sections can be kept away from this passage support. For r≠p,q a sequence of r core trajectories approaching the q stable entry sheet or the chosen connecting segment would, by the uniform broken-trajectory compactness, give at s₀ an orbit or broken chain from r to q or p before reaching the lower cut. Every intervening index is at least i. The no-increase property makes every component same-index, and a chain of two components would give two equal-index orbits at s₀. A one-component limit would give a second equal-index orbit distinct from p→q. Both contradict the supplied finite-event conditions. Thus these finitely many other core sections have a uniform positive clearance on a sufficiently small parameter interval. Shrink their full normal thickenings to keep that clearance. The portion of the p source sphere outside a small neighborhood of the one incidence likewise has uniform clearance: any contrary limiting incidence would be a second point of the same transverse connection, hence the same orbit meeting the chosen source sphere twice. The local source sphere is a one-exit section in its exact Morse chart, so this is impossible. Compactness permits a slightly larger removed patch and one uniform clearance for its closed complement. The q unstable disk itself is kept as the second attaching handle, rather than removed from the model. In q coordinates the stable entry patch is |v|=ε, with incoming u=u_0. Its exit map is $(u_0,v_0)⟼(ϵu_0\mathrm{/}\mathrm{\mid }{u}_0\mathrm{\mid },\mathrm{\mid }{u}_0\mathrm{\mid }{v}_0\mathrm{/}ϵ)\mathrm{.}$ The transversality permits the fiber-preserving straightening u_0=(x,s−s₀) on a smaller patch, by the already supplied local lemma. The two parameter sides cover the two complementary hemispheres of q's unstable sphere. Their nonzero v displacement chooses an actual parallel of that sphere, with the transverse direction e fixed by the stable entry point. After choosing two small regular parameter slices, reduce the normal radii so both this parallel patch and the q attaching thickening are disjoint. No positive-width attaching map at the singular parameter itself is required. From the q exit face down to the common lower cut there is no intervening critical point of smaller index, because its height is below that cut; an intervening same/higher-index limit is excluded just as in step 1.1. The compact q exit sphere therefore has uniformly finite smooth transport to the lower cut. It carries the actual parallel, its connecting band and its full normal framing. Other index-i cores remain disjoint by the preceding clearance, so the isolated local passage lemma supplies precisely the based framed handle slide at that cut. Its outgoing comparison is the product passage/collar map retained in that lemma. The higher attaching maps are then transported through that actual outgoing comparison by step 2.1. For d=4 the forward isolated-slide supplier is invoked only for i=1,2, inside its stated range 1≤i≤d−2. An index3 event is handled by reversing the entire local band: use (−g,−X), whose corresponding index is4−3=1 and which still has no index-increasing orbit. The incoming/outgoing marks are exchanged and then exchanged back. Thus its actual comparison is the reverse of the proved index1 slide, not an unsupported invocation of a forward index3 slide theorem. Index0 has no unstable sphere and index4 has no stable sphere, so no such event exists at those extremes. In a general d the same reverse interpretation handles i=d−1; the forward local lemma covers1≤i≤d−2.

Critical-value crossings and birth blocks. At a simple same-index critical-VALUE crossing there is no connecting orbit at that parameter: such a trajectory would strictly decrease the critical height. The event lists were also separated from all equal-index trajectory parameters. Hence step 1.1 gives based full thickenings smoothly through the value crossing and step 2.1 gives their product complements. The two attachments are disjoint on one common lower cut. Their quotient gluings commute, and the supplied elementary crossing comparison is exactly the collar-smoothed version of this commuting square. It retains the actual terminal maps. This is a value crossing, not a slide, and introduces no additional based move. At a birth of indices k,k+1 protect its exact cubic chart and use the two outer cuts. The unique cancelling trajectory, its framed intersection and the no-outside-return energy estimate are the actual elementary birth supplier. There is a genuine local cancelling handle pair. On the complement of a smaller birth block, there are no degenerate critical points; choose the other one-critical charts and regular cut transports as in that supplier. The lower groups and the old same-index branches have the uniform no-equal-incidence clearance from the newborn same-index branches, supplied by the weak field theorem near a birth. Their full based data are therefore regular up to the outer faces. For higher groups, including an allowed positive-drop trajectory into the fold, do not assert clearance from the birth block. Instead use the actual outgoing cancellation comparison from the two outer cuts and carry their entire elementary attaching embeddings through it as in step 2.1. This is precisely how the given elementary birth theorem identifies later handle data. It permits older higher attaching maps to pass through the block; it does not require them to remain fixed there. For s on the two-critical side one may choose an intermediate regular cut between the newborn heights, but it is not extended as a regular cut through s₀. The event is read in the combined outer band. The product cancellation chart then identifies the post-birth based presentation with the pre-birth presentation plus the cancelling k/(k+1) pair, with the outgoing diffeomorphism retained. This does not mean that older index-(k+1) attaching spheres automatically miss the born k-handle. In dimension four, for a 1/2 birth, put the older attaching curves transverse to its belt sphere. There are finitely many intersection points. At each, the product rectangle through the new one-handle joins that traversal to the opposite traversal of the cancelling two-handle; sliding over this cancelling handle removes that intersection and transports the complete framing. Choose parallel narrow bands and repeat. These are genuine additional two-handle slides used to put the birth in its isolated based form; their maps are composed with, rather than dropped from, the outgoing cancellation comparison. For a 2/3 birth use the reversed index1/2 calculation. For a 0/1 birth the same transport moves the finitely many other feet across the joining ball/tube, which is the rooted-tree comparison in the conversion proof; 3/4 is its reverse. Higher-index attaching data are carried after the combined band by the actual outgoing map as above. Thus positive-drop incidences at a fold introduce no omitted event type, while the finite normalizing slides are explicitly included in the mixed-handle movie. At a death use its inverse and carry all later data by that same inverse. Changes of the auxiliary cut levels are finite regular-flow comparisons with these stated maps. This is the exact cancellation/birth move in the mixed-handle movie, not a claim that the full later attaching data are pointwise unchanged.

Finite assembly and the endpoint presentations. The grouped family has finitely many births, value crossings and equal-index trajectory events by the field theorem and the added critical-height Sard construction. Cover each by one of step 3.1, step 4.1's disjoint parameter neighborhoods. Their complement is a finite union of compact regular intervals, and step 1.1, step 2.1 supply their based framed handle isotopies and product complement maps. Match the incoming identification of each piece to the actual outgoing identification of the preceding piece. The resulting finite mixed-handle movie has only the asserted births/deaths, same-index slides, commuting value interchanges and isotopies. All higher attaching data and the complete terminal parametrization have been carried by the actual maps at every piece. At an all-two-handle endpoint the descending field was prescribed from that given handle presentation. Its based unstable spheres are therefore its original attaching curves and full framings under the original handle-correspondence map. Index grouping has retained that field and only changed critical values. Moving the common lower cut follows the same field to the fixed incoming collar, so it inserts only the recorded regular-flow identification. Choose that identification to be the original incoming N. The endpoint link and its outgoing marking are consequently the given ones, rather than some merely diffeomorphic surgery output. This proves the based-movie bridge from the stated weak-field, elementary-comparison and grouping suppliers. It discharges the extra based-isolation hypothesis left explicit in the weak field proof. This establishes the stated based-movie interface with its actual maps; no further global conversion is used as a premise.